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Applicability and extensibility of linguistic geometry to modeling of real world intelligent systems

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Title:
Applicability and extensibility of linguistic geometry to modeling of real world intelligent systems
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Umanskiy, Oleg. ( author )
Place of Publication:
Denver, Colo.
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University of Colorado Denver
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English
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Doctorate ( Doctor of Philosophy)
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University of Colorado Denver
Degree Divisions:
Department of Computer Science and Information Systems, CU Denver
Degree Disciplines:
Computer Science and Information Systems
Committee Chair:
Alaghband, Gita
Committee Members:
Stilman, Boris
Altman, Tom
Mannino, Michael
Vu, Tam

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Artificial intelligence ( lcsh )
Game theory ( lcsh )
Linguistic geometry ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Linguistic Geometry (LG) is a powerful game theory and technology for generating winning strategies for real world systems. It has been successfully employed in numerous research, government and commercial projects. However, in order for this approach to be applied to each particular problem domain, one must first model this domain as LG Abstract Board Game (ABG). Furthermore, the tools and methods required to achieve such mapping from the oddities of the real world to the rigid mathematical definitions of the ABGs vary depending on the particular domain under consideration. This research addresses a wide spectrum of modeling issues required to successfully apply LG to such real world problems. The majority of these issues arise from the difficulties of mapping the vast breadth and depth of items, actions, and effects present in the real world into the formal notions of ABGs. In the course of this research, we developed techniques and methodologies that allowed the theory of LG to be extended to achieve the high level of applicability to the wide range of problem domains. These extensions are organized in a way that provides the highest benefit for current and future practical application of LG to various real world problems, and include examples of applications to several representative operational domains.
Thesis:
Thesis (Ph.D.)--University of Colorado Denver. Computer science and information systems
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Includes bibliographic references.
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Department of Computer Science and Engineering
Statement of Responsibility:
by Oleg Umanskiy.

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|University of Colorado Denver
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Auraria Library
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Full Text
APPLICABILITY AND EXTENSIBILITY OF LINGUISTIC GEOMETRY
TO MODELING OF REAL WORLD INTELLIGENT SYSTEMS
by
OLEG UMANSKIY
M.S., University of Colorado at Denver, 2001
B.S., University of Colorado at Denver, 2000
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Computer Science and Information Systems
2015


This thesis for the Doctor of Philosophy degree by
Oleg Umanskiy
has been approved for the
Computer Science and Information Systems Program
by
Gita Alaghband, Chair
Boris Stilman, Advisor
Tom Altman
Michael Mannino
Tam Vu
April 12, 2015


Umanskiy, Oleg (Ph.D., Computer Science and Information Systems)
Applicability and Extensibility of Linguistic Geometry to Modeling of Real World
Intelligent Systems
Thesis directed by Professor Boris Stilman
ABSTRACT
Linguistic Geometry (LG) is a powerful game theory and technology for generating
winning strategies for real world systems. It has been successfully employed in numerous
research, government and commercial projects. However, in order for this approach to
be applied to each particular problem domain, one must first model this domain as LG
Abstract Board Game (ABG). Furthermore, the tools and methods required to achieve
such mapping from the oddities of the real world to the rigid mathematical definitions of
the ABGs vary depending on the particular domain under consideration.
This research addresses a wide spectrum of modeling issues required to
successfully apply LG to such real world problems. The majority of these issues arise from
the difficulties of mapping the vast breadth and depth of items, actions, and effects
present in the real world into the formal notions of ABGs. In the course of this research,
we developed techniques and methodologies that allowed the theory of LG to be
extended to achieve the high level of applicability to the wide range of problem domains.
These extensions are organized in a way that provides the highest benefit for current and


future practical application of LG to various real world problems, and include examples of
applications to several representative operational domains.
The form and content of this abstract is approved. I recommend its publication.
Approved: Boris Stilman
IV


This work is dedicated to the memory of my Grandpa,
whose life has greatly inspired and influenced me.
v


ACKNOWLEDGEMENTS
It has been a very long journey and I would never have completed it without the
relentless encouragement of my advisor, Boris Stilman. You have encouraged my growth
as a scientist and a person for over 15 years, and I am constantly grateful for that.
Likewise, this would not have been possible without my family whose drive, by
their own example, towards scientific pursuits from the early age has led me down this
path. And thank you to everyone, especially my wife, Alia, for their tolerance of my
hermit-like existence during the last stretch of this trek.
VI


TABLE OF CONTENTS
CHAPTER
1 INTRODUCTION.................................................................1
1.1 Solving Real World Systems Using Game Approaches.......................1
1.2 Gaming Approaches......................................................2
1.3 Complexity of Discrete Games...........................................4
1.4 Human Approach to Complex Systems......................................6
1.5 Applicability of LG for Solving Real World Systems.....................8
2 LINGUISTIC GEOMETRY FOUNDATIONS AND HISTORY................................10
2.1 Abstract Board Games..................................................11
2.2 Hierarchy of Formal Languages.........................................20
2.2.1 Language of Trajectories..........................................21
2.2.2 Language of Zones.................................................32
2.3 LG Strategies.........................................................41
2.4 Historical Validation of LG...........................................42
3 APPLICABILITY AND EXTENSIBILITY OF LG.......................................47
3.1 Complexity of Modeling Real World Systems as ABGs.....................47
3.2 Concurrency...........................................................49
3.3 Spatial Discretization................................................50
vii


3.3.1 Regular Grids...........................................................50
3.3.1.1 2D Grids...........................................................51
3.3.1.2 3D Grids...........................................................53
3.3.1.3 Spherical Grids....................................................55
3.3.1.4 Curvature of the Earth.............................................58
3.3.2 Terrain Obstacles.......................................................59
3.3.2.1 Cell Types and Under-types.........................................61
3.3.2.2 Eggshell Model.....................................................64
3.3.2.3 Density Model......................................................67
3.3.2.4 Source Data Submodel...............................................71
3.3.3 Dynamic Obstacles.......................................................75
3.4 Mobility....................................................................77
3.4.1 Temporal Discretization.................................................77
3.4.2 Reachabilities..........................................................78
3.4.3 Trajectory Selection....................................................82
3.4.4 Direction Phase Spaces..................................................83
3.4.5 Speed Phase Space.......................................................86
3.4.6 Ballistic and Orbital Trajectories......................................87
3.5 Heterogeneous Systems.......................................................90
viii


3.5.1 LG Hypergames......................................................93
3.5.1.1 From ABGs to Hypergames........................................95
3.5.1.2 Common Inter-Linking Mappings.................................100
3.5.1.3 Benefits of Hypergames........................................101
3.5.2 Variable Step ABGs................................................102
3.5.2.1 Cell Sizes, Time Steps and Entity Movement Speed..............104
3.5.2.2 Variable Time Step............................................107
3.5.2.3 Practical Implications........................................109
3.6 Incomplete Information................................................110
3.6.1 WorldViews........................................................110
3.6.2 Deception Employment and Discovery................................112
3.6.3 Communication Groups..............................................113
3.6.4 Sensors...........................................................115
3.7 Weapon Systems........................................................118
3.7.1 Strikabilities....................................................118
3.7.2 Weapon Effects....................................................120
3.7.3 Paired/Prerequisite Trajectories..................................122
3.7.4 Synchronized Trajectories.........................................124
3.8 Mission Concepts......................................................125
IX


3.8.1 Goals and Missions
125
3.8.2 Prescribed Behaviors.................................................127
4 OPERATIONAL DOMAINS............................................................129
4.1 Operations in Urban Terrain..............................................130
4.1.1 Spatial Discretization...............................................131
4.1.1.1 Grids............................................................134
4.1.1.2 Obstacles........................................................137
4.1.1.3 Dynamic Obstacles................................................144
4.1.2 Mobility.............................................................145
4.1.2.1 Reachabilities...................................................145
4.1.2.2 Trajectory Selection.............................................152
4.1.3 Heterogeneous Systems................................................156
4.1.3.1 Variable Step ABGs...............................................156
4.1.3.2 LG Hypergames....................................................156
4.1.4 Incomplete Information...............................................158
4.1.5 Weapon Systems.......................................................161
4.1.5.1 Strikabilities...................................................161
4.1.5.2 Weapon Effects...................................................164
4.1.5.3 Paired and Synchronized Trajectories.............................166
x


4.1.6 Mission Concepts.....................................................168
4.1.6.1 Goals and Missions................................................168
4.1.6.2 Prescribed Behaviors..............................................170
4.2 Air and Naval Operations................................................171
4.2.1 Spatial Discretization................................................172
4.2.1.1 Grids.............................................................173
4.2.1.2 Obstacles.........................................................174
4.2.1.3 Dynamic Obstacles.................................................176
4.2.2 Mobility..............................................................176
4.2.2.1 Reachability......................................................176
4.2.2.2 Trajectory Selection..............................................179
4.2.3 Heterogeneous Systems.................................................180
4.2.4 Incomplete Information................................................184
4.2.5 Weapon Systems........................................................186
4.2.5.1 Strikabilities....................................................186
4.2.5.2 Weapon Effects....................................................187
4.2.5.3 Prerequisite Trajectories.........................................188
4.2.6 Mission Concepts......................................................189
4.3 Ballistic and Orbital Operations (BO)...................................190
XI


4.3.1 Spatial Discretization...........................................191
4.3.2 Mobility.........................................................192
4.3.3 Heterogeneous Systems............................................195
4.3.4 Incomplete Information...........................................196
4.3.5 Weapon Systems...................................................196
4.3.6 Mission Concepts.................................................198
4.4 Joint Forces Operations (JF)...........................................198
4.4.1 Heterogeneous Systems............................................198
4.4.2 Air-Ground Hypergame.............................................200
4.4.3 Litoral Air-Naval Hypergame......................................201
4.4.4 Joint Amphibious/Air Assault Operations..........................203
5 CONCLUSION................................................................207
REFERENCES...................................................................211
xii


LIST OF TABLES
Table
1 ABG definition.................................................................11
2 ABG definition for the game of chess...........................................17
3 ABG definition for combat simulations..........................................19
4 Grammar of shortest trajectories Gt(l).........................................23
5 Grammar of Zones Gz............................................................38
6 RAID Experimental Scoring Criteria.............................................44
7 RAID Experiment Results........................................................45
8 RAID Experiment Results........................................................46
xiii


LIST OF FIGURES
Figure
1JFACSEAD Mission.................................................................5
2 Comparison of searches for the same processing time...........................7
3 A Hierarchy of Formal Languages in LG.........................................10
4 TRANSITION^,x,y)..............................................................13
5 Bundle of trajectories........................................................22
6 Interpretation of the algorithm for nexti for the grammar Gt(l)...............26
7 Values of MAPss,p.............................................................27
8 Values of MAP36,p.............................................................27
9 SUM...........................................................................28
10 STi(88)......................................................................28
11 STi(78)......................................................................29
12ST2(88)........................................................................29
13 STi(68)......................................................................30
14 ST3(88)......................................................................30
15 STi(57)......................................................................30
16 ST4(88)......................................................................30
17 STi(47)......................................................................31
18 ST5(88)......................................................................31
19 LG zone for TC system with strikes...........................................33
20 Various types of LG zones....................................................36
XIV


21 Rectangular and Hexagonal Regular Grids
52
22 Rectangular and Hexagonal Line Rasterization......................................53
23 3D cell: hexagonal prism..........................................................55
24 A planet-level board with spherical hexagonal cells...............................58
25 Aircraft movement through using basic cell types..................................62
26 Eggshell hexagonal model of an urban environment..................................66
27 The 2D rectangular grid board and reachabilities of pieces........................79
28 Side view of the cruise missile reachability pattern..............................80
29 Defining reachability relationships based on turning radius.......................81
30 Bundles of shortest trajectories..................................................83
31 LG Zones with Trajectory Bundles for Aircraft Engagement..........................83
32 Directions with respect to a hex grid.............................................84
33 Trajectories of the aircraft changing direction...................................85
34 Ballistic trajectories on hexagonal shperical ABG.................................88
35 Orbital trajectories..............................................................90
36 Air (left) and Land (right) ABGs of an Air-Land hypergame.........................96
37 Air (left) and Navy (right) ABGs of an Air-Navy hypergame.........................98
38 Cells-per-turn and Time step vs Speed in km......................................107
39 Incomplete/false information.....................................................112
40 . A cruise missile illuminated by the aircraft radar.............................116
41 Strikability with obstacles......................................................119
42 Mission Execution Matrix for Operations in Urban Terrain.........................127
xv


43 Map data: Google, Bluesky.....................................................132
44 Rectangular 2D Grid for Grid Plan Urban Area..................................135
45 Hexagonal 2D Grid for Grid Plan Urban Area....................................136
46 Rectangular 2D Grid for Non-Grid Plan Urban Area..............................136
47 Hexagonal 2D Grid for Non-Grid Plan Urban Area................................136
48 LG Trajectories on Hexagonal Eggshell Urban Board.............................138
49 Urban area using 10m hexagonal grid...........................................140
50 Urban area using 20m hexagonal grid...........................................141
51 Urban Area Source Data Submodel...............................................142
52 Urban area using 30m hexagonal grid...........................................143
53 Dismounted outdoor reachability pattern.......................................147
54 Dismounted indoor reachability pattern........................................149
55 Vehicle road reachability pattern.............................................150
56 Vehicle off-road reachability pattern.........................................151
57 Vehicle threat map............................................................153
58 Dismounted threat map.........................................................154
59 Dismounted threat-avoidance trajectory........................................155
60 Overlapping Source Data Submodel LOS from 2 observers, 20m cells..............160
61 Cone SDZ for firing small arms direct-fire weapons............................162
62 Probability of Hit curve with rapid drop off..................................166
63 LG Zone with synchronized negation trajectories...............................168
64 Army tactical doctrinal taxonomy..............................................169
XVI


65 Bounding Overwatch
171
66 Coast terrain as hexagonal grid board...........................................175
67 Aircraft reachability at same altitude, climbing, and diving....................178
68 Basic ship reachability.........................................................178
69 Air threat-avoidance trajectories before and after SAM site destruction.........180
70 Littoral ABG with ships and UAVs LG Zone........................................182
71 Large scale air combat ABG with LG Zones........................................183
72 PD Curve for Ground Radar against Aerial Targets................................185
73 Air-to-ground and air-to-air strikabilities.....................................187
74 LG Zone for defense against cruise missiles with integrated fire control........189
75 Orbital trajectories on hexagonal planetary grid................................192
76 LG Trajectories for terrestrial pieces on a planetary grid......................193
77 Initial LG Zone for ballistic missiles..........................................194
78 LG Zone during ballistic missile engagement, with radar illumination............195
79 Air (top) and Ground (bottom) hypergame.........................................201
80 Littoral hypergame..............................................................203
81 Joint Amphibious/Air Assault Hypergame..........................................206
xvii


CHAPTER 1
INTRODUCTION
1.1 Solving Real World Systems Using Game Approaches
Real world adversarial systems, such as military operations, seemingly yield
themselves to easy and straight forward modeling as a game (similar to the game of
chess). Some of the aspects of such modeling are indeed straight forward an aircraft or
a tank can be represented as game piece, while the geographical space including
ground, oceans, aerospace, as well as underwater and outer space can be represented
as a large all-encompassing 3-dimenssional game board. However, this ease is quite
deceptive as we will explore within this work. The world, and, by extension, any operation
within it are exceedingly complex. Capturing every single minute aspect is therefore next
to impossible. However, there is an opposite problem as well. As we attempt to build
more and more accurate models of the real world, these problems increase in complexity
and can easily become computationally intractable.
The goal of modeling Defense Systems as a game is to model them in such a way
that yields itself to generation of winning strategies so as to benefit the end user the
consumer of the end results of the systems. A game strategy describes the behavior of all
actors and entities involved in terms of a sequence of moves. A move represents the
smallest activity of pieces discernible from the game point of view. Without being able to
use the game models to find winning strategies, such models can only be used for
1


visualization and display of the situation rather than to their full potential as an artificial
intelligence based decision support aids.
Thus, we are faced with two simultaneous challenges. The first is to discover
appropriate techniques to capture the real world systems as abstract models. Yet we
must also ensure that these models can be tractably analyzed by specific artificial
intelligence techniques to produce useful and meaningful results for the end user. The
search for the balance between these two diametrically opposed constraints is the focus
of this dissertation.
1.2 Gaming Approaches
The two primary impediments to generating winning strategies for real world
systems are exponential explosion due to the high dimensionality of the solution space
and modeling of active intelligent adversaries capable of asymmetric responses. The only
approach that allows introduction of the full-scale intelligent adversary is the gaming
approach [1-4]. There are various classifications of game-based approaches. Two such
dimensions of categorization are continuous vs. discrete, and strategic vs. extensive.
Continuous games are often described mathematically as pursuit-evasion
differential games [5], which is not suitable for dynamic, multi-agent models [6, 7]. There
is a small number of differential games, for which an exact analytical solution is calculable,
and another set of differential games, for which numerical solutions can be computed in
reasonable time under restrictive conditions. However, these games are poorly suited to
a general military problem as they are focused on one-to-one problems not the many-
to-many scenarios of the real world military operations. Additionally, they are unable to
2


model asymmetric non zero-sum games where the desires of the enemy are not the exact
opposite of the friendly desires. Furthermore, 3D modeling, limitation of the lifetime of
the agents, and presence of heterogeneous agents create additional barriers to ability to
use continuous games for the wide range of varied military problems.
Discrete strategic games are another category of games that are unsuitable for
practical real world military problems. Such games were introduced in [1, 2, 8] and later
developed by multiple researchers. This approach can only analyze the entire course of
action for each player at once each player must choose a plan of action once and for all
and is not informed about the plan of the enemy. The game may not be broken down into
individual sequences of moves it must be analyzed as an entire full game strategy.
This leaves the discrete extensive games as the candidate for mathematical
modeling of real world defense systems. One of the key beneficial aspects of such models
is the ability for each player to choose appropriate moves at any step in the game the
player can make a decision to alter the plan in response to the previous moves made by
the opposing player and themselves [8]. Another useful aspect is the requirement of
discretization of the problem domain. Such discretization can allow for real world entities
(such as airplanes and ground vehicles), and their actions and interactions (such as
movement and engagement) to be represented within these extensive discrete games. A
common representation of such games is as a game tree where each node represents
the state of the game after each successive move and the edges correspond to the
possible choices made by the players. Such an extensive game tree would include every
possible move of every strategy of every player. Because this game includes all possible
3


moves, it is theoretically trivial to look through the possible branches and variations and
find the optimal strategy. This is the strategy that guarantees best possible outcome for
each player regardless of what the other player is doing. Thus, classical theory of
extensive games is not concerned with tractability of the solution rather its concern is
with the existence of such solutions. That particular weakness makes extensive games not
useful for practical real world problems where solutions need to be found in a reasonable
amount of time.
This brings the discussion to the practical gaming approach which tries to search
discrete games trees [9]. While most suitable to practical uses, the primary obstacle of
this method is the sheer scale of such game trees, which is sometimes referred to as "the
curse of dimension". Even for small real world scenarios, these games trees are of such a
size that it is intractable to find a solution via conventional approaches such as brute force
or minimax search.
1.3 Complexity of Discrete Games
Since the size of the game tree to be searched using practical gaming approaches
to discrete games is the major factor which impacts the tractability of the problem, we
can use the size of such trees as the measure of the complexity of the game. Consider a
very small game with only 10 pieces, with each piece only capable of 10 legal moves at
anytime. If the game lasts for only 50 moves-the size of the game tree would be (lO10)50,
or lO500. Even for such unrealistically small game, the size of the tree is staggering. More
realistic military operations are even more prohibitively complex no computer could
4


possibly search through such trees in a practical amount of time or even in a lifetime.
Let us consider two of the real examples from previous research [3, 4, 10-12].
Total number of pieces Valid moves per piece (approximate) Depth of game tree (total moves) Size of tree
JFAC SEAD 30 18 70 ^q1861
RAID 70 24 480 1021255
TTTgpn
Figure 1JFAC SEAD Mission
For game trees of such size, even for the theoretically best cases, the most
powerful search algorithms such as alpha-beta pruning, would not produce significant
search reduction. In the best case, the number of moves to be searched employing alpha-
beta algorithm grows exponentially with the power of this exponent divided by two with
respect to the original game tree [13]. As one can see from the table above, such a
reduction is essentially meaningless in terms of problem tractability. To make matters
5


even worse, alpha-beta pruning is not applicable to totally concurrent games (where all
players may move all of their entities at the same time) it is only applicable to sequential
alternating games with players taking turns for moving an entity. Military war games are
necessarily totally concurrent and thus cannot benefit from such pruning.
Since fully exhaustive search is not possible, the common alternative is to search up
to a certain depth for instance 2 to 5 moves ahead and apply heuristic evaluation
functions to determine the value of the position at that point to determine best short
term plan. Of course, the accuracy of such approaches depends on the ability to correctly
evaluate such intermediate positions. These heuristics cannot really be generated to
provide correct assessment in all situations, and the algorithms that rely on shallow
search of this sort can be short sighted. This might result in picking a plan that seems
beneficial within the range of the next 2-5 moves, but does not lead to the ultimate victory
which requires 50-100 moves down the line. There are other alternatives that attempt to
search deeper along "more promising" branches and shallower in other directions.
However, such pruning is still based on heuristic analysis of the positions at the point of
the cutoff, and, therefore, is similarly susceptible to the limitations described above.
1.4 Human Approach to Complex Systems
In order to cope with such intractable search problems, Linguistic Geometry
approaches the problem from a different angle. The motivation behind such approach is
the difference between the human and the so-called computer based approaches to
solving such problems. The main advantage of the computers over humans is the ability
to perform extremely fast and precise combinatorial search which leads to the natural
6


application of exhaustive search algorithms that attempt to consider every possible
permutation of moves. However, despite humans' apparent handicap of computing
several orders of magnitude slower than a computer, human experts can still outperform
any computing system for many classes of real-world complex systems. The reason for
this is the difference of the human approach from the general computer search approach.
Rather than considering all possibilities, a human expert only considers one or two
possible moves for each position. With such a low branching factor, a human expert can
look very deep into the tree of variations much deeper than a much faster computer
can look using a brute force search.
The ability of a human expert to solve the problems in such an efficient manner is
likely based on their ability to decompose a complex problem into sub-problems. Such
problems can then be solved and their solutions reconstructed into the solution for the
original problem. This process is in fact quite complicated due to the complexities of the
real world systems. It is usually impossible to decompose the problem into truly
Figure 2 Comparison of searches for the same processing time
7


independent sub-problems. More likely, some inter-dependencies between such sub-
problems need to be taken into account when solving each individual one. Another
related problem is ensuring that the combined optimal solutions to the smaller problems
actually result in the optimal solution to the original larger problem. If care is not taken
to track all of the possible dependencies, it is quite possible to produce optimal solutions
for the sub problems which produce a subpar solution to the overall problem. The need
to account for such inter-connections has an adverse effect on the dimension-reducing
benefits of the original decomposition even to the point of completely eliminating the
benefits of the decomposition.
Typically, a human expert possesses the necessary knowledge and experience to
perform such an efficient decomposition for a particular set of problems within his field.
They may not be able to transpose such knowledge to a different problem domain. Due
to such specialization of particular "human heuristics" within each domain, we do not yet
possess a general methodology to translate human approach into computer algorithms.
Within each problem domain, with sufficient and careful study, a method could be found
that mimics the human expert approach of the problem decomposition; however,
attempting to transfer such knowledge to another domain is a different task altogether.
1.5 Applicability of LG for Solving Real World Systems
Chapter 2 of this dissertation describes the foundations of the Linguistic Geometry
as a method for tractable and practical generation of winning strategies, including a
summary of the complex problems addressed by this approach, its theoretical footing, as
well as experimental results confirming the validity and accuracy of this methodology. In
8


turn, Chapter 3 extends previous research as to the applicability and extensibility of
Linguistic Geometry as it pertains to solving complex real world problems. Specific
problems and the methodologies for addressing them are described in detail so as to
support the high degree of relevance of this research to the community of practice.
Chapter4 examines howthese methods can be applied to 4 specific operational domains.
While the focus of this work is on solving practical issues, theoretical component of this
research is also discussed to substantiate the rigorous nature of this presentation.
9


CHAPTER 2
LINGUISTIC GEOMETRY FOUNDATIONS AND HISTORY
The purpose of Linguistic Geometry is to develop a formal approach to a certain
class of multi-agent systems that involves breaking down a system into a hierarchy of
dynamic subsystems [14-19]. The methods of LG are formalized as a Hierarchy of Formal
Languages used to solve certain classes of problems. The languages in the hierarchy are
used to generate a hierarchy of structures: trajectories, zones, translations, searches, etc.
The class of problems that is addressed by these techniques is formally defined as abstract
board games. The LG method generates a "tree of translations", which is a string in the
Language of Translations. In order to do so, several strings of the Language of Zones or
Webs are generated. In turn, those LG Zones contain strings of the Language of
Trajectories. Together, this hierarchy creates decompositions of dynamic subsystems,
which is then used to solve the problem [18].
Figure 3 A Hierarchy of Formal Languages in LG
10


2.1 Abstract Board Games
Before the Hierarchy of Formal Languages at the heart of LG can be applied to a
problem, it needs to be represented as an abstract board game [14, 18]. This provides
formalization for a large class of problem domains. As a result, LG methods can be easily
and directly applied to any system that can be represented in this form. The formal
definition of ABGs is shown below. Following this definition, two examples are presented
to demonstrate the relationship between an ABG and a problem it represents.
Abstract board game is the following eight-tuple:
< X, P, Rp, SPACE, val, Sj, St, TR>
Table 1 ABG definition
x = {Xj} A finite set of points; locations of elements
P = {Pj} A finite set of elements; P is a union of two disjoint subsets Pi and P2 called the opposing sides
Rp(x, y) A set of binary relations of reachability in X (x and y are from X, p is from P);
Val A function on P with positive integer values describing the values of elements
11


Table 1 (cont.)
SPACE The state space. A state S e SPACE consists of the list of formulas, which employ a partial function of placement ON:P^Xand additional parameters. The value ON(p) = x means that element p occupies location x at state S. Thus, to describe function ON at state S, we write equalities ON(p) = x for all elements p, which are present at S. We use the same symbol ON for each such partial function, though the interpretation of ON may be different at different states. Every state S from SPACE is described by a list of formulas {ON(pj) = Xk} in the language of the first order predicate calculus, which matches with each relation a certain Well-Formed Formula (WFF).
Sj and St The sets of start and target states. Thus, each state from Si and St is described by a certain list of WFF {ON(pj) = Xk}. St is a union of three disjoint subsets St1, St2, and St3. St1, St2 are the subsets of target states for the opposing sides Pi and P2, respectively. St3 is the subset of target draw states.
12


Table 1 (cont.)
TR A set of transitions, TRANSITIONS, x, y), of the System from
one state to another (Fig. 2.2). These transitions are described
in terms of the lists of WFF (to be removed from and added to
the description of the state) and a list of WFF of applicability
of the transition. These three lists for state S e SPACE are as
follows
Applicability list: (ON(p) = x) n RP(x,y);
Remove list: ON(p) = x;
Add list: ON(p) = y,
where p e P. The transitions are defined and carried
out by means of a number of elements p from Pi, P2, or both.
This means that each of the lists may include a number of
items shown above.
13


Note, that this generic definition can be expanded for a particular class of ABGs.
Additional parameters of a state may be added. In addition to the data described above,
a state of an alternating ABG includes a binary function MT(S) e {1, 2}, a move turn to
distinguish between the states when player 1 (or 2) is allowed to move. A state of an ABG
with variable speed includes function of speed sp. Other information may be included
based on system requirements.
The transitions TR above may also be more complex. Transitions may be of several
types. A simple 'move' transition occurs when element p moves from x to y without
removing an opposing element. In this case, point y is not occupied by an opposing
element. A 'remove' transition occurs if element p moves from x to y and does remove
an opposing element q, i.e., OPPOSE(p, q) holds. For alternating serial systems, the
opposing element q has to be at location y before the move of p has commenced. In this
case, the Applicability list and the Remove list include additional formula ON(q) = y. For
concurrent systems, this is not necessary element q may arrive at y simultaneously with
p and be removed. These transitions are governed by the reachability relation Rp(x, y).
Further constraints can be imposed on the members of TR. In case of an
alternating ABG (see definition of SPACE above), if the move turn MT(S) = i, then only
elements p from Pj (from Pi or ?2, respectively) can participate in the transition.
Additional constraints may be introduced based on a particular class of system (e.g.,
systems in which no two game pieces can occupy the same location).
For real-world systems (especially Defense Systems), another type of transitions
is common a 'strike' transition. For this class of systems, another element is introduced
14


into the eight-tuple above binary relations of strikability Strkp(x,y), which is analogous
to relations of reachability. 'Strike' transition occurs when an opposing element q is
removed at a location different from y. This reflects a common situation, where a piece
can destroy an opposing piece located at a different position on the board by shooting. In
this case, Remove List may include additional formulas ON(qi) = zi, ON(q2)=Z2, etc, for all
locations Zi that are considered 'strikable' from location y (i.e., where Strkp(y,Zj) holds
true).
In an ABG, the goal of each side is to reach either one of its winning states (a state
in subsets St1 or St2, respectively), or, if impossible, a draw state from St3. The problem of
the optimal operation of the System is considered as a problem of search for a sequence
of transitions leading from the Start State of Si to a target state of St assuming that each
side makes only the best moves, i.e., such moves (transitions) that could lead the ABG to
the respective subset of target states. To solve an ABG means to find a strategy (an
algorithm to select moves) for one side, if it exists, that guarantees that the proper subset
of target states, St1, St2, or St3, will be reached assuming that the other side makes
antagonistic moves.
As mentioned above, a wide range of classes of ABGs is possible. These can be
categorized into 3 general classes: Alternating Serial, Alternating Concurrent, and Totally
Concurrent systems. In Alternating Serial (AS) systems, the opposing sides alternate turns
and only one element at a time can be moved. In Alternating Concurrent (AC) ABGs, the
opposing sides alternate turns; however, all, some, or none of the current player's
elements can be moved simultaneously. In a Totally Concurrent (TC) ABGs, players do not
15


alternate and all, some, or none of the pieces of both sides can be moved simultaneously.
It is important to note that, in general, the level of difficulty increases from AS to AC to TC
due to the need to consider all possible combinations of moves. For instance, consider a
game of 5 pieces against 5 opposing pieces, where each piece can make any one of 10
moves. Then, AS branching factor is 10, AC branching factor is 105, and TC branching factor
is 105+5=1010. Due to the nature of real-world Defense Systems, they usually need to be
modeled as Totally Concurrent ABGs.
To further illustrate, the relationship between a problem and an ABG, two
examples are presented below outlining the definitions of the eight-tuple for the game of
chess and a generic combat simulation.
16


Table 2 ABG definition for the game of chess
x = {Xj} 64 squares on the chess board
P = {Pj} White and black pieces
Rp(x, y) Reachability is defined by the rules of chess. I.e. Rp(x,y) is true if and only if a piece p is allowed to move from x to y according to the rules of chess. For example, if p is a King, RP(x,y) is true iff y is one of the immediate neighbors of x on the chess board.
Val Val(p) is the value of a piece. E.g., Val(Pawn)=l, Val(B)=3, Va/(K)=500, etc.
SPACE ON(p)=x, iff piece p stands on square x MT(S)=White or Black active player
Sj and St Si is the traditional chess start state or an arbitrary position that is to be analyzed. St1 and St2 are sets of all checkmate positions for the corresponding side. St3 is the set of all draw positions.
17


Table 2 (cont.)
TR
TRANSITION^, x, y) represents moving piece p from square x
to y. If opposing piece is present at y, capture is implied. Since
chess is an Alternating Serial system, only one piece of the
active player can be moved at a time (with exception of
castling).
In addition, chess is a Complex System with blocked beams
and destinations, which means that some moves may be
prohibited due to presence of another piece either at the
target location y, or on the beam from location x to location y.
Pawn promotion is also a transition which removes ON(p)=x,
and substitutes ON(p')=y, where x is the row before last, y
immediately ahead in the top row, and p' is promoted piece.
As the reader can see, there are certain discrepancies from a general ABG present
- several pieces are prevented from occupying the same location (system with blocked
beams and destinations), pawns can be promoted, etc. Such differences can be easily
incorporated into a special class of Complex Systems. Similarly, for other problem
domains, apparent discrepancies from a general abstract board game can be easily
accommodated.
18


Table 3 ABG definition for combat simulations
x = {Xj} 2D or 3D grid of the area of operations. Could be a simple 2D rectangular grid for land operations, or a complex 3D packing of aerospace, including orbit positions of satellites.
P = {Pj} Sets of resources of the opposing sides, where each element can represent individual or groups of airplanes, tanks, ships, infantry, satellites, etc.
Rp(x, y) Reachability is defined by the movement capabilities of different types of elements present. Ships can move on grid points corresponding to water only. Satellites can only move to a point further along its orbit. The maximum speed of a particular element defines how far it can move in one step.
Val Val(p) can be defined by the abilities of the element (more powerful element has a higher value), as well as by the value of the element to the operation (an airstrip that must be protected can have a higher value).
SPACE ON(p)=x, iff element p is at location x (at particular coordinates on the surface of Earth, volume of aerospace, or position in orbit depending on set X).
Sj and St Si is arbitrary position that is to be analyzed e.g., positions of the resources before the conflict. St is a set of target states based on mission objective e.g., certain targets destroyed (or defended).
TR TRANSITION^, x, y) represents element p moving from location x to y (on land, sea, air, etc). In addition to 'move' transitions, 'strike' transitions are needed to reflect long- distance shooting for objects Due to the nature of real world, these are Totally Concurrent systems any combination of pieces from both sides can move simultaneously during a single time step. Also, transitions may include a game piece 'producing' a new element, changing into a new element, splitting into several elements to reflect events such as firing missile, dropping paratroopers, re-loading expended ammunition, splitting into smaller combat units, etc.
19


Real-world Defense Systems are the focus of this thesis. Although they exhibit more
complex behavior than traditional board games, they are fully susceptible to LG
techniques.
2.2 Hierarchy of Formal Languages
As mentioned above, LG methods are formalized as a Hierarchy of Formal
Languages Language of Trajectories, Language of Zones, Language of Translations, and
Languages of Searches. This formal hierarchy is presented in [18] employing declarative
(non-constructive) definitions, followed by introduction of generating grammars for these
languages. Since the subject of this dissertation is extension and application of LG to real
world systems, this section is not intended as comprehensive LG theory presentation. The
purpose of this section is to present the overall principles of the LG methods and allow
the readerto follow further discussions in this dissertation. Languages of Trajectories and
Zones are introduced informally and constructively through their generating grammars.
The generating grammars used by LG theory are the so-called controlled grammars, which
are very flexible tools for producing strings of symbols with parameters [21]. The
definitions are followed with simplified examples to demonstrate application of these
grammars (on a chess-like ABG).
20


2.2.1 Language of Trajectories
The lowest level in the LG Hierarchy of Formal Languages is the Language of
Trajectories [14, 18]. The strings of this language represent a path or route of a game
piece from one location to the next. In general the strings are of the following form:
t0 = o(x)o(xi)...o(x/),
which represents a trajectory for some piece peP from location xeX to location x/eX of
length /. Informally, it is simply a string of symbols 'a' with parameters, where each
parameter is a location on the ABG. The main property of this string is that it represents
a valid path for piece p from x to x/. This implies that every point Xi is reachable in one step
from previous location xm, i.e., Rp(xi,Xi+i)=True for all i = 0,1,...,/-1. Usually, there is more
than one trajectory between any two locations. A set of trajectories for the element p
from location x to location y of length / is called a bundle of trajectories of length /, and
can be denoted by tp(x,y,/). An example, using a chess-like board, is shown below. The
game piece p in the example has a reachability relation analogous to a chess King, while
shaded locations are considered unreachable. Locations are represented by two digits
(xix2).
21


8
7
6
5
4
3
2
1
Figure 5 Bundle of trajectories
Trajectories for piece p from (88) to (36) of length 5 are as follows:
t0=o(88)o(78)o(68)o(57)o(47)o(36),
ti=o(88)o(78)o(68)o(58)o(47)o(36).
A bundle of trajectories for p from (88) to (36) of length 5 (note, that there are
no other trajectories of length 5):
tp(88,36,5)={to,ti} .
Next, the controlled grammar Gt*1* that generates shortest trajectories is
presented and demonstrated by the above example. This is the most basic type of
trajectories. A variety of other trajectory grammars is possible. For instance, admissible
trajectories of degree k represent trajectories consisting of k segments, each of which is a
shortest trajectory [22]. Detailed definitions, proofs of correctness, and discussions on
more advanced types of trajectory grammars is given in [18].
1 2 3 4 5 6 7 8
22


Table 4 Grammar of shortest trajectories Gt(l)
L Q Kernel, 7t^ nn ft Ff
1 Qi S{x,y,l) ~^A{x,y,l) two 0
2/ Q2 A{x,y,l) - a(x)A(next\ {x,l), y,f{l)) two 3
3 Qs A(x,y,l) - a{y) 0 0
VT={a}
VN={S,A}
VPR
Pred
Ql(x, y,/) = (MAPx#p(y) =/) (0 < /< n)
Q2(D = (/>D
QS = T
Var= {x, y, /}
F = Fcon u Fi/ar,
Fcon = {f, nexti,...,nextn} (n = |X|),
/(/) = 1-1, D(f) = Z+
Fi/or = {x0,y0,/0,p}
E = Z+u Xu P
Parnr. S^Var, A^Var, a>{x}
23


Table 4 (cont.)
L = {1,3} xx two, two = {21; 22, ..., 2n}
At the beginning of derivation:
x = xQ, y = y0, /= lo> xo from X, yQ from X, /Q from Z+, p from P.
nextj is defined as follows:
D{nextj) = X x Z+ x x Z+ x P
SUM = {v | v from X, MAPXO;P(v) + MAPy0;P(v) = /G},
ST|<(x) = {v | v from X, MAPX p(v) = k},
MOVE/(x) is an intersection of the following sets:
ST1(x), ST/0./+1(x0) and SUM.
if
MOVE/(x) = {mi, m2, ^0
then
nextj(x, I) = m/ for i nextj(x, I) = mr for r else
nextj(x, I) = x.
endif
There are several points that require clarification before an example is shown.
First, is the definition of the MAPx,p(y) function. MAPx,p(y) is equal to k, such that y is
24


reachable from x in k steps, but not reachable in k-1. For instance, MAPx,P(y)=l for all
points y such that Rp(x,y)=True; MAPx,P(y)=2 if there is a point z such that Rp(x,z)=True and
Rp(z,y)=True; and so forth.
A second necessary clarification is that the number of productions in this grammar
is not 3 as it seems at first glance. Rather, there are several productions 2i (the set two).
This number is limited by the size of set X; however, it can vary at different steps in the
generation process. There is one production 2\ for every different value of the function
next,.
The function next, is the 'heart' of this grammar. Function next, returns the ith
memberof the set MOVEi(x), which contains all possible locations forthe next step in the
trajectory. This set is determined as an intersection of three sets: STi(x), St/0-/+i(x0) and
SUM. The set SUM is a set of all points v such that MAPxo ^(v) + MAPy0 ^(v) = /Q, i.e., the
set of all points such that the distance from the beginning of the trajectory to this point
added to the distance from this point to the end of the trajectory is equal to the total
length of the trajectory. It can be easily shown that this is a set of all the points on all the
shortest paths of length lo from xo to yo. This set is shown as the long ellipse in Figure 6.
25


Figure 6 Interpretation of the algorithm for nexti for the grammar Gt(l)
To illustrate the meaning of the other two sets, consider a situation where the first
k-1 points of the trajectory have been constructed (xo, xi,...,Xk-i), and we are interested in
finding all possible points v for the kth step. Clearly v belongs to the set SUM. Since the
point v has to be on the kth location along the shortest path, the distance from xo to v has
to be equal to k. Therefore, v is in the set STk(xo)={v | MAPXo ^(v) = k} (shown as the
rectangle in the magnified view on the right side of the figure above). Furthermore, point
v is reached on the kth step from location xn; as a result, v has to be reachable from xn
in one step. That is, v is in the set STi(xki)={v | MAPXk x ^(v) = 1} (shown as the small circle
in the magnified view). In the example above, MOVE contains two elements. This implies
that at the kth step, the generation can branch into two separate trajectories by applying
either production 2i or 22.
To demonstrate how the controlled grammar Gt*1* actually generates shortest
trajectories, it has been applied to the situation in Figure 5 in a step-by-step fashion. The
start symbol is S(x,y,l) = S(88,36,5). We start with production 1, and follow to production
26


2i out of set two. Since Qi=True (MAPgg p(36) = 5), we took the branch corresponding to
Ft in the first production:
S(88,36,5) !=> A(88,36,5) 2i=> a(88)A(next/(88,5),36,f(5)).
At this point we need to compute the parameters using function f and nexti. The
first function is trivial: f(5)=5-l=4, which means that the remainder of the trajectory needs
to be of length 4. To compute nexti, we need to construct the set MOVE by computing 3
different sets mentioned above. Let us compute MAPXo ^(v) and MAPy0^(v), i.e.
MAPgg p(v) and MAP35 ^(v). These maps are necessary for the construction of the sets
SUM and STk.
Figure 7 Values of MAP8s,p Figure 8 Values of MAP3e,p
Now, we can compute the set SUM as a set of all points v on the board such that
MAPgg p(v)+MAP36 p(v)=5. In order to complete construction of the set MOVE, we need
2 more sets: ST^(x), ST/0-/+i(x0), i.e. ST^(88) and ST5_5+i(88) (in this case they are
identical). Note, that the same set SUM is used on every step of the generation. However,
the other two sets that form MOVE do change.
27


From the above figures, it is easy to see that the set MOVE={78}. As a result, there
is only one value of function nextj= 78 and there is only one production in the set two 2\.
Therefore: o(88)A(nexti(88,5),36,f(5)) = o(88)A(78,36,4). We can continue the derivation
by applying rule 2\ again:
o(88)A(78,36,4) 2'=> a(88)a(78)A(next//(78,4),36,f(4)).
At this step the set MOVE is found as intersection of sets SUM (same as above),
and STi(78) and ST5_4+i(88) (Figure 11 and Figure 12). As on the previous step, this set
contains only one value: MOVE={68}. As a result there is only one value of nexti and only
one rule 2i. Therefore, o(88)o(78)A(nexti(78,4),36,f(4)) can be rewritten as
o(88)o(78)A(68,36,3) and derivation continued as before:
o(88)o(78)A(68,36,3) 2i=> o(88)o(78)o(68)A(next/(68,3),36,f(3)).
Next, the sets ST^(68) and ST5_3+i(88) (Figures 2.11 and 2.12) need to be
computed. However, this time the intersection of the 3 sets form the set MOVE with more
than one element: MOVE={57,58}. Therefore, the function next can take on two different
28


values. This means that at this step the shortest trajectory can go two different ways. By
applying production 2i or 22 we can achieve the following derivations:
a(88)cr(78)A(68,36,3) 2i=> a(88)a(78)a(68)A(57,36,2), or
o(88)o(78)A(68,36,3) 22=> a(88)a(78)a(68)A(58,36,2).
In practice, the generation has to branch every time when the set MOVE contains
more than one element. The total number of shortest trajectories in the bundle is
multiplied every time such condition is encountered. In this case there will be at least two
shortest trajectories generated. For this demonstration, let us pick the first of these
trajectories and proceed with the generation.
8
7
6
5
4
3
2
1
8
7
]*
5
4
3
2
1
1 2 3 4 5 6 7 8
Figure 12 ST2(88)


:if > 1

29


8
7
6
5
4
3
2
1
The next iteration of applying the rule 2\ produces the following expansion:
o(88)o(78)o(68)A(57,36,2) 2'=> o(88)o(78)o(68)o(57)A(next/(57,2),36,f(2)).
8
7
6
5
4
3
2
1
The set MOVE={47} (Figures 2.13 and 2.14), therefore
o(88)o(78)o(68)o(57)A(next/(57,2),36,f(2))=o(88)o(78)o(68)o(57)A(47,36,l).
The next application of rule 2\ produces the following expansion, where nexti=36,
since MOVE={36} (Figures 2.15 and 2.16):
o(88)o(78)o(68)o(57)A(47,36,l) 2i=>
o(88)o(78)o(68)o(57)o(47)A(next/(47,l),36,f(l)) =
o(88)o(78)o(68)o(57)o(47)A(36,36,0).
30


8
7
6
5
4
3
2
1
Figure 18 ST5(88)
At this point, production 2\ can no longer be applied due to the fact that
Ch=(/>l)=(0>l)=False. Therefore, rule 3 from Ff of rule 2\ is applied. There are no more
non-terminal symbols present and generation is complete:
o(88)o(78)o(68)o(57)o(47)A(36,36,0) 3=> o(88)o(78)o(68)o(57)o(47)o(36).
The second trajectory that is generated by choosing o(88)o(78)o(68)A(58,36,2) on
the 4th step (instead of o(88)o(78)o(68)A(58,36,2)) is o(88)o(78)o(68)o(58)o(47)o(36).
Therefore the entire bundle of shortest trajectories for piece p from location 88 to 36 is
trajectories is tp(88,36,5)={to,ti} (Figure 2.3), where
t0=o(88)o(78)o(68)o(57)o(47)o(36),
ti=o(88)o(78)o(68)o(58)o(47)o(36).
Note, that this grammar is completely universal with respect to the problem
domain. It will produce the shortest trajectories for any system for which the set of
locations X and reachability relation RP(x,y) is defined. Other grammars can be applied in
similar fashion to generate longer trajectories, such as admissible trajectories of order k.
31


This demonstrates the general method of employing controlled grammars to generate
strings belonging to the languages in the LG Hierarchy of Formal Languages.
2.2.2 Language of Zones
The next level in the LG Hierarchy of Formal Languages is the Language of Zones
[14, 18]. The strings of this language represent a network or a set of interconnected
trajectories. One of these trajectories is known as the main trajectory, and the others are
negation trajectories. Intuitively, the main trajectory represents a path that the main
piece needs to take to accomplish a certain goal. The 1st negation trajectories represent
paths for the opposing pieces that can disrupt the main piece from arriving at its
destination. The purpose of kth negation trajectories is to prevent a trajectory of k-1
negation from accomplishing their interception. Formal definitions of zones and
supporting concepts are given in [18].
Consider the zone shown in Figure 2.17 for a Totally Concurrent ABG with strikes
(zero-time moves that destroy another object). The main goal within the zone is for the
Gray Bomber po to destroy the Black Tank qo. To accomplish this goal the Gray Bomber
needs to move along trajectory o(l)o(2)o(3)o(4). This brings it within the strike range of
the target which can be destroyed by the strike 4^5. The zone also contains two Black
elements that are capable of destroying piece po before it reaches its target qo namely,
qi and qi with 1st negation trajectories o(6)o(7)o(8) and o(9)o(10)o(ll) respectively.
However, during the construction of the zone, Language of Zones also generates 2nd
negation trajectory for the Gray Plane pi which allows it to intercept the Black Plane q2
and therefore, stop q2 from intercepting the Gray Bomber o(12)o(13)o(14). At the next
32


level, the generation is able to employ Black Plane q3, which was not able to intercept the
Gray Bomber due to time constraint. However, a 3rd negation trajectory can be
constructed for q3 to stop pi from intercepting q2, so that q2 can intercept po to protect
qo. Finally, a 4th negation trajectory is added seeing as Gray Plane P2 can assist the Gray
side by destroying q3 and canceling the chain of events above.
An LG zone has several constraints. The first constraint is that every trajectory
represents a valid path for the ABG (generated by the LG Language of Trajectories). The
second major constraint is that the zone timing is maintained. This timing constraint is
necessary to ensure that the interception is actually possible, i.e., the interceptor arrives
in time to destroy the target. In general, the length of any negation trajectory t must be
less than or equal to the number of moves that the acting piece on the trajectory negated
t.
r
Figure 19 LG zone for TC system with strikes
33


by t has to make for reaching the target location of t. A zone is strict if the length of t is
not only limited by the above number, but is strictly equal to it. Strict zones correspond
to the ABGs where a game element cannot 'wait' for its target to arrive at the location of
intercept, e.g., fighter plane may not be able to hover in the air waiting for the bomber to
come within range. In the zone above, the length of the 1st negation trajectory of qi is
equal to 2, which is equal to the number of steps po has to make to arrive at the intercept
location (3). Likewise, the length of the 4th negation trajectory of p2 is 1, which is equal to
the number of steps for q3 from (15) to (16). Different ABG may require modification of
the timing constraints. For instance, in chess the zone is constructed in a way that the
length of the trajectories of the pieces of the same color as the main piece cannot exceed
1 due to the Alternating Serial nature of the game.
As with the language of trajectories, there is usually not a single zone, but rather
a bundle of zones. Consider that the trajectory for the main piece to the target is usually
a bundle of trajectories with the same source and target. As a result, there is usually a
bundle of zones with each of those trajectories as the main trajectory. The same principle
holds for the negation trajectories.
When LG strategy is derived from the zones, either individual zones or the entire
bundles are analyzed. The important aspect of the zones is that they describe
relationships between the pieces in the ABG at the same time providing information on
how to exploit these relationships. For instance, if the above zone were generated within
a game, we would know immediately that the Black Side would win, because there is an
intercept trajectory with no counter-intercept trajectories possible. Moreover, we can
34


see exactly how the Black Forces must behave to accomplish this protection. For instance,
the plane qi is clearly the most important element. The Gray side can also see how
support fighters can attempt to increase Gray Bomber's chances by counter-interception.
Even though qi will be able to shoot at po, the rest of the zone may still be important if
the destruction is probabilistic (rather than unconditional as in chess). The bundle of
zones shows exactly how the interception and counter-interception are possible. The
analysis becomes more complex when zones with different goals are present and are
interconnected. In Defense Systems, usually more than one target may be given and
available resources must be distributed between different tasks. However, analysis of
zones may provide a strategy, which uses the same resource for several tasks (i.e., in
several zones) simultaneously. This is usually achieved by the tree evaluation procedure
which favors moves that allow involvement in several zones. In addition to attack zones
described above, other zones are possible, such as block/relocation, domination, retreat,
and unblock zones (see Figure 20). Block/relocation zones differ from attack zone in that
the main piece is not attempting to destroy a target, but rather just move to the given
endpoint. The opposite side is attempting to block this relocation. The domination zone
is essentially a relocation zone to the endpoint that allows the main piece to provide
domination of another game piece. Retreat and unblock zones have the goal of moving
the main element so as to save it or clear the path for another piece. Other similar
networks may be used for a decomposition of an ABG for a variety of problem domains.
35


The previous section presented formalization of trajectories as a string of symbols
in the Language of Trajectories. Similarly, a zone introduced above can be represented as
strings in the Language of Zones using the relation of trajectory connection. Two
36


trajectories are considered connected, if the endpoint of the first trajectory coincides with
an intermediate point of the second trajectory. In general, zones are represented in the
following form:
Z = t(po,to,To) t(pi,ti,Ti) ... t(pn,tn,Tn),
where t is a terminal symbol, pi are game pieces from the set P, ti are trajectories from
the Language of Trajectories forthe corresponding pieces, and Ti are non-negative integer
time bounds on the corresponding trajectories. The first symbol, t(po,to,To) corresponds
to the main trajectory of the zone, while t(pi,ti,Ti)...t(pn,tn,Tn) correspond to the negation
trajectories. The time bound Ti for ti represents the notion of time constraints presented
above, which insure that the negation trajectories can intercept on time. Intuitively, Ti is
the number of game moves that the piece pi has to intercept its target trajectory. The
particular way Ti is calculated depends on the class of the ABGs and the level of negation.
The Grammar of Zones computes these values automatically during derivation of a
particular zone. The zone from Figure 2.17 can be represented in the following way:
Z = t(pQ, o(l)o(2)o(3)o(4)o(5), 4) t(q1# o(6)o(7)o(8)o(3), 3)
t(q2, o(9)o(10)o(ll)o(3), 3) t(Pl, o(12)o(13)o(14)o(ll), 3)
t(q3, o(15)o(16)o(13), 2) t(p2, o(17)o(18)o(16), 2).
The concept of zones has been thoroughly presented above. The controlled
Grammar of Zones Gz that generates such zones for a given Alternating Serial ABG is
presented in Table 2.5. Since the method of application of such control grammars was
presented in the previous section, an example of applying Gz is not shown. Such examples
as well as declarative definitions of the Languages of Networks and Zones, related
37


concepts (e.g., trajectory connections and t time constraints) are given in [18].
Productions 1 and 2\ generate the main trajectory. Productions 3 and 4j are used to
generate the 1st negation trajectories, while production 5 provides a way to switch to
higher negation. Subsequently, combination of productions 3, 4j, and 5 is used to
generate trajectories of all higher levels of negation. The generation is terminated using
production 6 when no more trajectories of higher level of negation exist (predicate Qs is
False). The main intricacies of this grammar are generation of connection points and
maintaining time bounds on the negation trajectories to each of those points.
Table 5 Grammar of Zones Gz
L Q Kernel, (z e X) nn(z e X) Ft fF
1 Q1 S(u, v, w) ~^A(u, v, w) two 0
2/ q2 A(u, v, w) -> t(hj(u), l0+1) TIME(i) = DIST(i,hi(u))
3 0 4((0, 0, 0), g{hj{u),w), zero)
3 Qs A(u, v, w) A{f{u, v), v, w) NEXTTIME(i) =
four 5 initiu, NEXTTIMEU))
47 q4 A(u, v, w) -> t(hj(u), TIME(y))) NEXTTIME(i) = 3 3
A(u, v, g(hj(u), w)) ALPHA(i, hj(u),
77M£(y)-/+l)
5 Qs A{u, v, w) >4((0, 0, 0), w, zero) TIME(i) = 3 6
NEXTTIME(i)
6 Qe A(u, v, w) ->6 0 0
38


Table 5 (cont.)
Vj {£), - {5, A},
VPR
Pred ={Q],Q2,Q.3,Q-fyQ-5,Q-b}
Qi(u) = (ON(p0) = x) a(MAPx po(y) (3q ((ON(q) = y) a(OPPOSE(p0, q))))
Q2(u) = T
Q^u) = (x^ n) v (y ^n)
Q4(u) = (3p ((ON(p) = x) a (/ > 0) a (x *x0) a (x *y0)) a
((iOPPOSE(p0, p) a (MAPX p(y) = 1)) v
(OPPOSE(p0, p) a (MAPX p(y) < /)))
Q$(w) = (w ^zero)
Qe=T
Var = {x, y, I, z, 6, v, v2,..., vn, wx, w2,..., wn};
for the sake of brevity: u = (x, y, I), v = (v^, v2,..., vn),
w = (wlt w2,..., wn), zero = (0, 0,..., 0);
Con = {x0,y0,/0,p0};
Func = Fcon 'uFvar;
Fcon = {/x/y,//, gyg-2,... ,gn, hvh2,... ,hM,
V' h2,..., hM, DIST, init, ALPHA},
f= 0 = (Sxl' 0X2' 0xj'
M = | Lt^0(S) | is the number of trajectories in Lt^0(S);
Fvar = {x0,y0,/0,p0, TIME, NEXTTIME}
39


Table 5 (cont.)
E = Z+u XuPu L^(S) is the subject domain;
Parm: S > Var, A > {u, v, w), t > {py ^ 0};
L ={1,3,5,6} u two ufour, two ={21,22,--,2M},four = {41,42,...,4m}-
D(init) = X x X x Z+ x Z+
init(n. r)
2n, if u = (0,0,0),
r, if u ^ (0,0,0).
D(/) = (X x X x Z+ u {0, 0, 0}) x Z+n
J(x + l,y,/), (i£x n) a(/> 0)) v((y = n) a (/< 0))
fiu, v)-|(1> y+1> T]ME(y +1) x vy+1), if (x = n) v((/< 0) A(y n)).
D{DISTj = X x P x Lt/o(S).
Let t0 e Lt/o(S), t0 = o(z0)o(z1)...o(zm), t0 e
fpo(zO' zm' m);
If zm = Yo) a (p = p0) a (3 k (1 < k < m) a (x = z|<))) v
(zm Yo) v (p Po)) a (3 k (1 < k < m -1) a (x = z|<)))
then D/ST(x, p0, t0) = k+1
else D/ST(x, p0, t0) = 2n.
D(ALPHA) = X x P x 4/o(S) x Z.
ALPHAS, po, tQ, k) = 1 k
max (NEXTTIME(k\k),
NEXTTIME(E),
if(DIST(x, p0,t0) 2n)
a (NEXTTIME{x) 2n);
Z>ZHT(x, po, tQ) ^ 2n)
a (NEXTTIME{x) = 2n);
iE)IST(x, p0,t0) = 2n).
D(gr) = P x4/o(S) x Z+n, r e X.
fl, if Z)ZST(r,p0,t0) <2n,
MPoA0,w> if Z)ZSr(r,p0,t0) = 2n.
D (hj) = X x X x Z+; Let TRACKSpo = {p0} x (u L[Gt<2)(x, y, k, p0)];
1 < k < /
If TRACKSpo=0 then h(u)=e
else TRACKSPo = {(p0,t1),(p0,t2),...,(p0,tb)},(b < M) and
''Ah.
[(Po = tb), if i > b.
40


Table 5 (cont.)
D(hj) = X x X x Z+; Let TRACKSp = {p} x (u L[Gt<2)(x, y, k, p)];
1 < k < /
If TRACKSp = 0 then hj(u) = e
else TRACKSp ={(p1,t1),(p1,t2),...,(pm,tm)}, (m < M) and
f(p,,t, ), if i< m,
h.(u) = \
l(Pmtm), if i > m.
Trajectories tj should not be embedded (as sub-trajectories) in the trajectories of the same negation.
At the beginning of generation:
u = (x0' Vo' lo)w = zero>v = zero> xo 6 x> y0 6 nx,i0 e +, p0 6
TIME(i) = 2n, NEXTTIME(i) = 2n for all z from X.
2.3 LG Strategies
There are two methods that can be used to apply the Hierarchy of Formal
Languages to construct strategies. The first, more traditional approach is to construct
search trees employing the Languages of Translations and Searches using the generating
tools presented above. By following this technique, reduced search trees can be
generated of sizes significantly smaller than those of Alpha-Beta search. The second
approach is to construct a solution without any search at all. The construction of
strategies is achieved by decomposition of the State Space in the form of the State Space
Chart, which is based on the expansion of the terminal sets. Neither of these approaches
is presented here, as both of them are discussed in detail in [18], and a full understanding
of strategy construction is not essential for the reader to follow further discussions in the
subsequent sections. It is sufficient to mention that these methods make significant use
of Languages of Zones and Trajectories presented in previous sections.
41


2.4 Historical Validation of LG
Since the conception of Linguistic Geometry, this theory has been extended and
utilized in a multitude of real world applications. Considerable evidence, both theoretical
and experimental, has been gathered to demonstrate that the LG software tools provide
highly effective scalable solutions and a faithful model of an intelligent enemy [11, 23].
These applications have included complex military and industrial problems and have
garnered international recognition from such organizations as US Air Force Scientific
Advisory Board, US Army Science Board, and UK Defence Science and Technology
Laboratory [24]. These boards define national policy in the defense-related research and
its transition to the US Armed Forces. Since 1999, LG-based technology has been
successfully tested in more than 30 government and commercial defense-related projects
[25]. The most significant and thorough validation of the Linguistic Geometry was
accomplished in a series of war gaming experiments within the DARPA (Defense
Advanced Re-search Projects Agency) RAID (Real-time Adversarial Intelligence and
Decision-making) program [3, 25]. The LG-generated courses of action (COAs) significantly
exceeded that of COAs developed by the human commanders and staff. Results of those
experiments have led to multiple follow-up utilizations of the Linguistic Geometry across
various programs within the US Army research and development organizations for
intelligence, mission command and control, and training applications.
As previously mentioned, the RAID experiment events have provided the most
thorough opportunities to validate the accuracy and benefits of the LG-generated
solutions [3, 26]. These events consisted of a series of comparative trial runs. In each case
42


two teams of current and past military personnel Blue and Red participated in
wargames against each other employing OTB (OneSAF Testbed Baseline,
www.onesaf.org) simulation software to provide a representation of a real world
environment. In the baseline case, the Blue force commander was assisted by a team of
five advisers acting as Staff officers. During the LG portion of the experiment, the Blue
commander was assisted by the LG-based courses of action generation and analysis
software instead of the human aides. In both cases human or LG assisted, the Blue
commander controlled entities within the simulation by ordering OTB operators
(commonly referred to as pucksters) to execute his plan. On the Red side, the enemy
commander similarly employed pucksters to control Red forces within OTB in an attempt
to defeat the Blue forces. A model of 16 square kilometers of an actual city was utilized
to provide a complex urban operational environment. The Blue simulated force was
equivalent to a US company with about 30 to 35 infantry fire-teams, strykers and
helicopters. The Red force consisted of several kinds of insurgents with approximately 30
teams of various sizes. Four such experimental events took place in April 2005, July 2005,
February 2006, and July of 2006. LG-based software demonstrated intelligence by far
exceeding human-developed courses of actions as shown by the statistical analysis of the
sophisticated scoring of the outcomes of each experimental run [3].
43


Table 6 RAID Experimental Scoring Criteria
Attack Mission Weight Defense Mission Weight
Red Casualties 40% 10%
Collateral Damage 10% 10%
Blue Casualties 35% 35%
Advance To Objective 5% 0%
Time to Complete the Mission 10% 5%
Facility Protection 0% 40%
Out of the 18 paired simulation runs (2 hours each) conducted in Experiment 4,
the LG-assisted Blue commander outperformed the commander with a human staff 14
times (78%). In 5 out of these 14 paired runs, the human Blue team had lost to the Red
team, whereas the LG-assisted Blue commander had won. In many other paired runs out
of those 14, while both teams had won over the Red, the LG-assisted Blue team scored
significantly higher. On average, for all the 18 paired runs, the commander with LG
software achieved scores that exceeded the score of the commander with the staff by
about 10% one standard deviation. Out of the 4 paired runs where the staff
outperformed LG, the difference in score for 3 of them was under 3%, and only one run
had the difference of about 10%. Overall, the level of confidence in correctness of the LG-
generated COAs was 98% [26]. It is crucial to note that during this experiment, the Blue
commander was obligated to follow the LG-generated course of action, so that these
scores could be used to directly judge the quality of LG solutions as compared to a team
44


of subject matterexperts (SMEs). Table 7 and Table 8 summarize the experimental results
for two of the other experiments which further support the above conclusions [3].
Table 7 RAID Experiment Results
Experiments show that the RAID- Non-RAID
supported commander noticeably outperforms the staff-supported Pair ID RAID Score Score
1 71.91 69.76
commander.
Statistics: 2 78.34 75.07
- Number of Valid Run Pairs = 9 - Mean Difference = 2.78 3 84.66 86.81
- StDev of Difference = 5.21 4 74.22 78.11
RAID advantage is significant: 5 83.04 73.35
- 6 times out of 9 Outperformed
the Human team 6 80.25 81.41
- In 2 runs, the difference was 94.83 82.93
higher than StDev (86% and /
128% higher) 8 89.31 85.92
RAID disadvantage is negligible 81.09 79.30
- Out of 3 runs where the Human Si
Blue team outperformed RAID, the difference never exceeded Mean 81.96 79.18
StDev StDev 7.12 5.74
45


Table 8 RAID Experiment Results
Experiments show that RAID-
supported commander noticeably
outperforms the staff-supported
commander
Statistics:
- Number of Valid Run Pairs = 9
- Mean Difference 3.14
- StDev of Difference = 10.10
RAID advantage is significant:
- Outperformed the Human team
5 times out of 9
- In 2 runs, the difference was
higher than StDev (67% and 99%
higher)
RAID disadvantage is negligible
- Out of 4 runs where the Human
Blue team outperformed RAID,
the difference never exceeded
StDev
Pair ID
Pair ID RAID Score RAID Score
3 74.300 72.500
2 76.890 75.240
6 57.840 67.440
4 88.650 71.750
8 70.390 76.350
10 77.810 79.710
9 86.390 78.170
1 77.500 57.380
7 69.530 72.520
Mean 75.48 72.34
StDev 9.20 6.72
There was another interesting aspect of these experiments which served as a
certain variant of a TuringTest. During each of these runs, the Red team was isolated from
the Blue team and was not informed of or could in any way find out whether they were
playing against Blue staff generated or LG-generated courses of action. At the end of
every scenario, the Red commander was asked which kind of the opponent he was playing
against human team or LG-assisted commander. In 16 out of 36 cases (44%), the Red
Commander was wrong [26].
46


CHAPTER 3
APPLICABILITY AND EXTENSIBILITY OF LG
3.1 Complexity of Modeling Real World Systems as ABGs
Previous chapters have introduced the theoretical foundations of Linguistic
Geometry as well as some of the experimental evidence as to the high quality of the
solutions provided by this methodology. LG-based technology has been successfully
employed in numerous government and commercial defense-related projects [25];
however, each such application relies on more than just the core LG theory. In order for
this approach to be applied to each particular problem domain, one must first model the
real world characteristics of such environment as LG Abstract Board Games. Furthermore,
the tools and methods required to achieve such mapping from the complexity of the real
world to the rigid mathematical definitions of the ABGs vary depending on the particular
domain under consideration. For example, consider the difference in requirements
between naval vs ground, urban vs rural, or air vs outer space operations.
In this chapter, we explore and address a wide spectrum of modeling issues
required to successfully apply Linguistic Geometry to such real world problems. The
majority of these problems arise from the difficulty of mapping the vast breadth and
depth of items, actions, and effects present in the real world into the strict mathematics
of Abstract Board Games required as the basis for the LG application. This research will
build on and extend previous research as to the specific techniques and methodologies
that allow basic theory of Linguistic Geometry to be extended so as to achieve the high
47


levels of applicability to the wide range of problem domains. The key purposes for this
work is to present these extensions and methods in a rigidly organized and structured
fashion to provide the highest benefit for community of practice attempting to apply
Linguistic Geometry to both previously explored and novel problem domains.
The first requirement for solving the problem is to model it in the form that can
be solved using specific tools tools. The first step to applying LG is to adequately model
the real world system in discrete format of the Abstract Board Game, which can be
represented as 8-tuples < X, P, Rp, SPACE, val, Sj, St, TR>. Note, that this 8-tuple can be
further expanded based on the particular system requirements, such as adding strikability
relations for the long distance shooting and other extensions described in subsequent
sections. The first 3 elements of this 8-tuple -X, P, and Rp are the most critical elements.
Once the operational board, the set of pieces, and the capabilities of these pieces are
defined, the remaining game elements are easier to specify and LG can be applied to the
overall system.
As the purpose for modeling a real world system as an ABG is to employ LG to
generate winning strategies, we must keep in mind tractability as one of the primary
design constraints. Larger sizes of the sets X, P, and Rp cause increased requirements for
computer memory, processing power and time. A common theme through the rest of this
work is to find the balance between limiting complexities of the ABG for practical
performance considerations and achieving sufficient complexity to faithfully represent
the events of a real world system.
48


The initial development of LG was based on chess games, so unsurprisingly, chess,
and similar board games, can be easily modeled as ABGs. The main reason is that these
games have a well-defined and accepted set of rules, set of possible game elements on a
specific board, and standard goals. Let us now explore the challenges of such modeling
without the help of such existing frameworks.
3.2 Concurrency
Section 2.1 introduced the concept of concurrency categories as related to ABGs.
The games can be Alternating Serial (AS), Alternating Concurrent (AC), or Totally
Concurrent (TC) systems. Note, that even large AS games have significantly lower
branching factors than smaller TC games. For instance, an AS game with 2 players
controlling a total of 400 game pieces, each of which can make 5 distinct moves has the
branching factor of (400/2)*5=1000, whereas the same entities in a TC game would result
in the overall branching factor equal to 5400=3.9xl0279. As a matter of fact, the branching
factor of 1000, fora game with each piece capable of 5 distinct moves, would be exceeded
with just 5 game pieces in a TC ABG (branching factor of 55=3,125). AC games are less
complex than TC; however, they are much close in complexity to TC than to AS games.
For the same example above, the AC game branching factor would be 52=6.2xl0139.
The real world rarely presents a problem where only a single agent can move at
the same time typically all actors, such as cars, ships and people, can all move
simultaneously. Serial games are not well suited for modeling problems and concurrent
games must be employed. However, the added complexity of such games, as described
49


above, should serve as a reminder to minimize, as much as feasible, the size of the ABG
set of P and reduce complexity of Rp.
3.3 Spatial Discretization
Let us now consider the discretization problem. Linguistic Geometry as well as
other discrete game approaches requires both the space and time to be broken up into
discrete segments. Spatial and temporal discretization problems have to be addressed
together due to the effect that the tight interconnection between them has on the ability
to model movement and actions in the resultant ABG. For example, if the game board
employs cells of 1 km in size and a 1 min time step, then the smallest speed of movement
that can be represented is 1 km/min. However, if the time step is increased to 2 min, or
the cell size reduced to 0.5 km, then this speed is reduced to 0.5 km/min.
3.3.1 Regular Grids
The Abstract Board in the ABG is just a finite set of points. However, it is typically
more convenient to model it as a type of a regular grid due to the need of mapping the
LG strategies backto the real world. This modeling requires breaking up the entire surface
or volume into sub-regions or cells. One such grid, akin to a chess board, is a regular
rectangular grid with square cells. Another type of grid may employ hexagonal cells. Let
us recall that the purpose of such discretization of space is to model movement and
actions of entities as discrete transitions (jumps) from one cell to another during a single
time step. As such, it can be observed that there are distortions of space inherent to both
of such typical grids.
50


3.3.1.1 2D Grids
In the case of a rectangular grid, a distance to the adjacent cell in a diagonal
direction is about 41.4% longer than a distance to the adjacent cell in the orthogonal
(rank-and-file) directions. Using a reachability of moving one cell at a time in all 8
directions for a real world entity would be very misleading as such an entity would be able
to move 41.4% faster in some directions leading to incorrect movement paths. Essentially,
an entity could "cheat" by moving diagonally as much as possible in order to take
advantage of such a speed boost. This relative distortion is reduced as the radius of such
intended approximated circle is increased. For example, the longest range of the edge of
the dark grey area in Figure 21 is only 13% longer than the shortest range to the edge.
While only a partial improvement, we could employ a technique such that all the
movements utilize at least 4, 5, or more cells per single time step to minimize effects of
distortions. However, such improvement comes at the cost of significant increase in
overall size of the board which is usually correlated with performance or computing
memory requirements. For instance, reducing the cell size by 4 in order to model
movement employing 4 cells instead of 1 would effectively increase the total number of
cells for the same area by a factor of 16.
Using a hexagonal grid can help reduce some of such problems due to the primary
property of such grids that each cell is surrounded by six equidistant neighbors. Therefore,
movement of 1 cell per time step has no distortion of distance. However, such lack of
distortion is misleading-as it is limited to single cell neighbors. If you consider, the radius
of 2 cells, the longest range is about 15% longer than the shortest. The distortion can be
51


reduced similarly to rectangular grids by employing larger ranges to better approximate
circles.
Figure 21 Rectangular and Hexagonal Regular Grids
The problems with grid based discretization do not stop at just linear distance
distortions angular distortion and obstacle representation both pose a problem to
discrete based modeling of real world environments and accurate movement
representation. Consider representation of movement along a given vector. If such a
vector is aligned with some primary directions 8 in rectangular grid or 6 in hexagonal
grid the lines can be represented accurately. Yet the majority of lines would not follow
one of such directions, and as can be seen in Figure 22, such a line has to be 'rasterized'
which creates jagged, zigzag movement ratherthan a true straight line. It should be noted,
that just as with distance distortions, the line rasterization distortion can be addressed by
reducing the sizes of cells and modeling movement as 4 or more cells at a time, rather
than 1 per game turn.
52


Figure 22 Rectangular and Hexagonal Line Rasterization
3.3.1.2 3D Grids
While 2D Abstract Boards may be sufficient for some problems, a much higher
level of precision can be achieved with 3D discretization. For some Defense Systems this
extra degree of realism is essential. One class of such systems is air operations. As
mentioned above, some problems with air units can be approximated with a 2D game;
however, that is not always the case. Consider a scenario, where a group of airplanes is
flying through a mountainous area with surface-to-air defenses. A 3D representation will
allow us to handle complex behavior such as flying through the canyons to stay out of line
of sight of surface radars. Furthermore, the reachability relations for cells at higher
altitudes may allow faster flight than those at the lower altitudes above ground level.
Some weapons can only be fired from aircraft at targets that are within a certain cone of
attack, so that the plane has to adjust its position before strike. If we attempt to model
these types of games with a pseudo-2D ABG, we would have to introduce the second
pseudo-layer in which any position above a given cell will have identical properties. While
we can model one of the aspects mentioned above (such as canyons by noting ground
level at every 2D cell), we cannot model all of the potential different behaviors over that
53


cell. Using a 3D model, we can distinguish and plan our strategy to take advantage of such
effects as flying through this location at 100 ft above ground level to avoid radar, moving
to attack altitude, or accelerating to burst speed at 20,000 ft and keep track of line-of-
sight visibilities at all times.
Several approaches can be used to model 3D space. One way is to use a dense
packing of space by certain shapes (usually approximating spheres). The goal of such
packing is to achieve an effect similar to hexagonal grids on a plane every cell is
equidistant from all its neighbors. Such discretizations are usually complex and
counterintuitive for most people. More importantly, they are unnecessary and often not
appropriate for real-world Defense Systems. The reason is that in real life 2 horizontal
directions are very similar, while altitude is quite different. The aircraft may be able to
move at 500 miles per hour in any direction at any given altitude; however, it is usually
only able to change altitude at the rate of about 20 miles per hour. For the same reason,
it is also common to use a different scale in the horizontal and vertical dimensions. Since
motion in vertical direction is so different from horizontal directions, it makes more sense
to model 3D Abstract Boards as a stack of 2D Boards. If the 2D Board is discretized as a
rectangular grid, a 3D cell could be a cube or a rectangular prism. Hexagonal 2D
discretization produces hexagonal 3D prisms (Figure 23) [2, 3,11, 27, 28]. Each of the cells
represents a 'chunk' of 3D space and stores the properties of the section of space
contained in that region, such as air, water, ground, etc. The reachability relations are
then designed to conform to the properties of game elements-an airplane can only move
through cells 'filled with air', while a ground unit can only exist in an 'air' cell immediately
54


above a 'ground' cell. Furthermore, similarly to 2D games, the concept of Phase Space can
also be extended for the 3D games, thus making them 4D (or higher).
Figure 23 3D cell: hexagonal prism
3.3.1.3 Spherical Grids
So far, all the discretizations have only dealt with either 2- or 3-dimensional areas
above a planar surface. We have assumed that the 2D grid is covering a uniform portion
of a flat surface. However, the Earth is not flat. Plane approximations assumed above
work well for relatively small area. On the other hand, some Defense Systems may require
an operational district that is large enough to notice the curvature. Others may require
considering the entire surface of Earth as the operational district. In this case, we have to
discretize the entire surface of a sphere (for large, yet not full-Earth models, we can use
subsets of full spherical mappings) [2, 28, 29]. There are two general classes of
approaches to this problem. The first is to map the surface of a sphere onto a plane and
then discretize this planar representation. The second approach is to discretize the
surface of the sphere directly. The main difficulty of both methods is maintaining the
uniformity of cells. As mentioned in the previous section, an important property is for the
distances between neighboring cells to be uniform. We would also like each cell to have
approximately equal area and shape. A comprehensive survey of Discrete Global Grids
can be found in [30].
55


Consider the first approach. The first step in this approach is to map the surface
of the sphere onto a planar structure with as little distortions of distances, areas, and
shapes as possible. This problem has been studied extensively by researchers in the area
of map projections. Unfortunately, there is no projection which maintains distances,
areas, and shapes. There are projections that may be able to maintain some of the
properties (e.g., equal area projections); however, there is no projection that is
equidistant in all directions. As a result, we may have cells, which are not spaced uniformly
on the sphere. One of the better projections is the Snyder equal area projection onto a
polyhedral globe [31]. The second step of this method is to discretize the planar structure
into 2D cells. The cells can be of any shape; however, as for 2D Boards, hexagons have
some of the most desired properties. The difficulty of this step is usually due to the shape
of the planar structure used. For instance, consider projections onto a polyhedral globe.
These structures consist of several planar faces that can be triangles, squares, pentagons,
hexagons, etc. Each of these faces then needs to be broken down into cells. However, if
we attempt to discretize a pentagon as a tiling of hexagons, we will encounter problems
along the edges and at the corners. The easiest shape to discretize is either a square
(which usually has large projection distortions), ora triangle. Furthermore, there is usually
a tradeoff between the quality of the first and second steps. Snyder equal area projection
onto an icosahedron (20 triangular faces) has much higher distance distortions than a
projection onto a truncated icosahedron (20 hexagonal and 12 pentagonal faces). On the
other hand, the icosahedron is easy to discretize consistently into hexagons with no edge
56


problems, while pentagons and hexagons do not allow for such a consistent
discretization. One of the better approaches of this type is presented in [32].
The second approach consists of discretizing the surface of the sphere directly.
One popular method is to start by approximating the sphere as polyhedron (usually a
platonic solid). Then each face is broken down into several smaller parts to construct a
more complex polyhedron. By such successive subdivision, a sphere is approximated by a
polyhedron with a very large number of faces. The key element of this process is the
subdivision of the face to produce a number of smaller faces. At each subdivision, some
new vertices are produced which need to be projected back to the sphere creating the
new polyhedral structure. This process is somewhat similar to the map projections
mentioned above. The difficulty lies in creating the new vertices so that the distances
between neighboring vertices remain fairly close to each other. The final cells are created
from this collection of small faces. Either a cell can incorporate the area covered by 1 or
more faces, or a cell may be centered on a vertex and contain points that are closest to
it. This approach with an extra optimization step was presented in [33, 34]. The grid may
not exhibit all of the desired properties (such as equal distances and areas); however, it
can be used as a starting state for a dynamic optimization technique based on the
requirements for the grid. The cells are adjusted slightly to maximize an evaluation
function.
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y Global [aiicaigo_kill]
[ji Simulation View Mark Window Help Test
:is &\my r i$ t
Ready 25.0N:
;ja5lail||0Blobal |aitcaigo_kill]
Figure 24 A planet-level board with spherical hexagonal cells.
33.1.4 Curvature of the Earth
While previous sections presented either grids in the rectangular space or the
spherical "whole Earth" space, the reality of the modeling real world scenarios is that
even the rectangular grids represent some portion of the curved Earth surface. The
rectangular nature of grids presented in Sections 3.3.1.1 and 3.3.1.2 is typically employed
as an approximation of space above the surface of the planet. Consider that for an
observer standing on the ground with the eye level at 1.7 meters, or 5 feet 7 inches, the
horizon is only 4.7 kilometers, or 2.9 miles, away. That means that even a small "flat" area
10 x 10 km2 cannot be accurately represented as a truly 2-dimensional game board. This
problem similarly must be considered for higher elevations such as the flight of an
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aircraft at various altitudes. The great circle distance (as the bird flies) between 2 locations
that are 10 meters above sea level is significantly different than if those locations were
10 kilometers above sea level. Such curvature must be accounted for in the ABG through
either a modification of the board structure, such as including the dropping horizon and
increasing distances at various altitudes as an integral feature of the 3D board, or by
appropriately accounting for it during the reachability and line of sight calculations.
Similarly, the concept of the regular 2D grid is harder to maintain on a spherical
surface. Simply using latitude and longitude as the X and Y coordinates for the grid will
yield cell sizes that are significantly different as one moves away from the equator. At
equator, one degree of longitude corresponds to about 111 kilometers, while one degree
of longitude at 60 degrees latitude (e.g., southern Alaska) is only 56 kilometers across. A
more common approach to construct a grid with regularly spaced and equidistant cells is
to employ some sort of a map projection. The most common approach used today is
Universal Transverse Mercator, or UTM; however, there are many other projections
based on different optimization criteria or applicability parameters. For instance, UTM is
only applicable between 80 South and 84 North latitudes. Locations near south or north
poles require a different projection system, such as Universal Polar Stereographic.
Additionally, each projection provides various levels of area equality, distance equality,
or angular equality across a certain size of an Earth surface covered.
3.3.2 Terrain Obstacles
In the real world, the spaces represented by the game boards are not typically as
wide open as those on the chess board. There would usually be obstacles or other
59


features both natural and manmade that affect movement and the ability to perform
other actions. Natural obstacles may include impassible mountains or rivers and lakes,
trees and other vegetation. There could also be areas which affect movement speed for
instance, swamps, dirt or rocks. Manmade features such as buildings, fences, and roads
may similarly affect movement and possibility of actions. Such terrain characteristics
typically need to be represented on the game board by assigning properties to individual
cells for example, a cell could be marked as water and thus be impassible, or marked as
road and cause faster movement [3]. Unfortunately, the problem of rasterization
reappears when one attempts to represent the edges of such features on the game board.
This results in jagged edges of all such terrain elements. The problem is intensified when
multiple features overlap a single cell, or some features are so small that only a small
portion of the cell is covered.
We will investigate multiple methodologies of various levels of complexity that
can be used to address the terrain obstacles problem. Some of the key factors to consider
when evaluating suitability of one of these techniques to a particular situation are size of
obstacles that can be captured, accuracy of representation of an obstacle, and size of cells
needed to achieve desired accuracy. The latter consideration is particularly important as
it is directly correlated to the total number of cells needed to represent a given real world
geographic area. For practical uses, the storage, memory, and performance requirements
are typically tightly dependent on the total number of cells, also referred to as "game
board size".
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3.3.2.1 Cell Types and Under-types
The simplest approach to obstacle representation is to assign one or more "types"
to each cell. In the simplest case, each 3-dimensional grid location (such as a hexagonal
prism, presented above) could be assigned one of just two types open or closed. In this
manner, the cell could either represent a location through which movement is allowed,
or a cell which is an obstacle and cannot be moved into or through. Such representation
may be quite sufficient for certain types of operations.
For instance, when modeling air operations, solid ground can be represented as
closed cells, while the air as open cells. Movement patterns can then be easily set up
that track movement of aircraft that avoids crashing into the mountains, and LG can then
be applied to find the most efficient routes and analyze the strategies for any scenarios
in this model [12]. Line of sight can also be easily computed in such space.
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Figure 25 Aircraft movement through using basic cell types
An easy expansion to this model is to include various types of closed cells to
represent different types of terrain elements such as roads, rivers, swamps, and others
[3]. This would allow for representation of surface based entities with reachability
patterns that are different depending on the surface type. A boat may be allowed to move
across rivers and lakes, but not any other terrain types; while a vehicle may be able to
move rapidly across roads, much slower while off-roads, and not able to traverse any
water features at all. It should be noted that instead of using the type of the closed cell
immediately underground to represent various types of terrain, a concept of "under-
type" can be applied to the open cells immediately above ground. The open cells would
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then have a type attribute for the volume represented by the cell, as well as one (or
perhaps more) under-type attributes to represent the type of the terrain under the cell.
A further expansion would include introduction of various types of open cells -
air, forest, building, and others. Just as with under-types, various entities would then have
different movement patterns based on the type of the cell, e.g., movement could be
slower through the forest and the vehicles can be completely restricted from entering
"building" open cells.
This model allows for a very complex environment to be represented; however,
the limiting factor is the size of individual grid cells relative to the size of the key terrain
features to be represented. Effectively, no terrain element smaller than the cell size can
be adequately modeled. For instance, consider 2 roads running parallel to a river or a
canal between them. If the width of each of these roads and rivers is 10 meters, and the
size of the cell is 50 meters, all 3 of these features will be represented on a single cell and
it becomes impossible to set up reachability relations that faithfully represent movements
of various entities in such terrain. The only solution is to reduce the cell size until it is small
enough to represent necessary elements.
Two key problems of this approach must now be considered alignment and
rasterization (see Section 3.3.1.1). Even if the size of the cell is small enough for any one
feature, the edges or location of these features would not necessarily coincide with the
edges or area of the cell. For instance, the edge of a large forest that spans multiple cells
may fall in the middle of the cell. Similarly, even a small building or road may be located
on the edge between two cells. The rasterization problem occurs when trying to represent
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the edges of the features (or the entire linear features, such as roads) on a regular grid.
These problems all essentially produce the same effect a single cell that is split in half,
whereas one part is covered by one type of a feature while the other part is not. Presented
another way a particular feature partially covers one or more cells. In such cases, there
is no clear method as to whether such cell is to be marked with the particular feature
type. It may be just as incorrect to assign the type as not to assign it leaving no effective
resolution.
It is important to note that the alignment and rasterization problems can never be
completely resolved by a further reduction of the size of the regulargrid cells-there may
always be features straddling the cell boundaries regardless of how small individual cells
are. The effect of these problems can be significantly reduced, but could never be
completely eliminated. Typically, the most troublesome features in this respect are of
manmade- roads and buildings. It is usually not practically feasible to reduce the size of
the cells sufficiently to address rasterization and alignment of such small and
geometrically precise features. Thus, most of the methods presented in the following
sections are focused on the manmade elements.
3.3.2.2 Eggshell Model
In the previous section we introduced the concept of under-type to represent the
type of the terrain under the cell in addition to any type information for the space
occupied by the cell itself. In essence this is identical to assigning a type to the bottom
face of the 3-dimensional prism representing the board cell-such as the hexagonal prism
in Figure 23. This can be further extended by assigning a type to each face of such cell -
64


the bottom, the top, and any sides. For a hexagonal 3D cell, this would associate 9 types
with each cell one for the volume of the cell, and 8 for all of the walls, floor and ceiling.
We will refer to this method as the "eggshell" model due to the emphasis on the external
perimeter of the 3D cell.
The primary benefit of this method is to improve modeling of manmade structures
such as buildings, including their internals. Building hulls pose a higher degree of
difficulties due to the small thickness of the walls. As we have already discussed, no
obstacle smaller than a cell can be correctly represented. Thus, in order to represent a
building hull we would need the cell size to be comparable with the thickness of the wall
so that there would be a "solid" cell representing the wall, and 2 "open" cells on either
side representing locations just inside the building and just outside. This approach
would require the size of the cell to be on the order of several centimeters exclusively due
to the need to represent the walls as entire cells. In practice this is excessively wasteful,
as such resolution is not needed for any other part of the terrain tens of meters for a
single cell is typically sufficient to adequately represent all other features such as forests,
roads, and others.
The eggshell model allows one to deemphasize the walls when selecting
appropriate cell sizes the walls of buildings would actually be represented as walls of
the board cells [3, 35]. Note, that the types associated with the various faces of the prism
can have multiple values such as "solid wall", "window", "door", as well as "floor" and
"ceiling" to represent various floors in the building. This allows for additional fidelity in
modeling by associating different characteristics with each such type for instance,
65


"window" face type can be seen through but not traveled through, while the "door" face
type may imply the converse. The use of the cell faces is not limited to building walls alone
- other possible uses include other types of obstacles such as fences and barriers.
Figure 26 Eggshell hexagonal model of an urban environment
These benefits have to be considered in light of the alignment and rasterization
introduced in previous sections. It is indeed easily possible to employ 20 meter cells to
model the space inside and outside of buildings, while relying on cell walls to represent
much thinner obstacles presented by walls, of various types. However, if a wall passes
directly through the middle of a 20 meter cell, it can only be represented as a cell wall
66


that is located 10 meters to either side due to lack of alignment. Similarly, a straight fence
can only be represented as a jagged sequence of cell edges. Whether these errors in
representations are acceptable depends on the particular application of the models. In
some cases, a wall misplaced by a certain distance to the nearest cell edge may be quite
acceptable, while in others this discrepancy may require correction most frequently and
easily by (adaptive) reduction in the overall cell size.
3.3.2.3 Density Model
Every situation calls for a different level and type of discretization. Previous
sections were focused on modeling obstacles by using open/closed cells and various types
of walls to represent various real world objects as closely as possible. In this section, we
present an alternate methodology that instead attempts to capture the aggregate effect
of obstacles rather than minute details of each individual one. This can be done by
assigning additional characteristic of "density" to each cell in addition to the types
presented above [36]. Completely blocked cells would be assigned density of 100%, while
completely open ones would be considered 0% dense, with anything else in between
those values.
The density parameter can be computed in the simplest manner by calculating the
ratio of the space or volume (2D or 3D) occupied by the obstacles to the overall area or
volume of the cell. However, this simple method can be expanded to provide additional
fidelity by considering types and densities of the obstacles themselves. One such
example would be to consider a park or a sparse forest such an obstacle has its own
density associated with it. Therefore, when calculating the cell density, the percentage of
67


the cell space occupied by the forest needs to be scaled by the density of the forest itself.
It is interesting to note that even buildings could be considered to possess different
densities. Consider the difference between an airy building with paper walls, floor-to-
ceiling windows, and an open floor plan compared to a steel and concrete apartment
building with dense arrangement of internal walls. These buildings could be considered
to have different density when the effect of these obstacles on line of sight and mobility
is analyzed.
This methodology can further be extended by adding additional characteristics
that describe aggregated effect of the obstacles in the cell. For example, various
categories of densities can be used to represent density of obstacles that provide cover
and interfere with weapons engagement, densities of objects that interfere with
movement of various kinds of entities, density of line of sight obstructions, and others.
The effect of the density-based cells is, of course, different from the binary nature
of the open or closed model. Rather than blocking the movement, visibility, or other
actions, the cells of various non-zero densities merely impede these actions to some
degree. This impedance may be instantaneous or cumulative. For instance, a vehicle may
not be able to travel at all through locations of density greater than 75%; however, it
could travel through 4 cells of density 20% in a single game turn (depending on the cell
size, game turn length and other parameters). Similarly, this model captures how the
ability to see is negatively affected by the number and relative density of objects in all the
cells along the source-to-target vector.
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The main advantage of this model is that the size of the cells no longer needs to
be close to the smallest object being represented; however, some consideration must still
be given to the cell size. There must be enough board locations to represent realistic
positioning and movement of entities during the game; and each such board cell should
be small enough to capture local variances in the density. The usefulness of the model
significantly decreases if all cell densities are approximately equal. For instance, if a typical
downtown area is modeled using 500 meter cells, all cells will be very similar in value -
the effect of roads and other narrow open spaces will be completely diluted by the overall
mass of the buildings in each such 500 meter block.
The most apparent disadvantage of this model is its relative lack of precision.
Efficiency of the larger acceptable cell sizes comes at the price of replacing individual
obstacles with aggregated effect of all objects within a particular cell. However, when
considered in light of the alignment and rasterization problems of the previous models,
this penalty is actually not as significant, or completely annulled. Essentially, the fuzziness
introduced by the density representation is not dissimilar to the imprecision of
representing individual obstacles by the nearest cell or cell boundary. Consider the error
of line of sight calculation based on the cell density of 50%, due to the building occupying
half of the cell volume, with the error due to "snapping" the building boundary from its
position in the middle of the cell to the nearest cell boundaries. It is not possible to
quantify the inaccuracy of each method in a general case, therefore each particular
modeling problem needs to be considered separately. The best overall guidance is that
the eggshell model is especially beneficial when cell size is much smaller than overall
69


objects, e.g., representing one or just a few buildings using cells small enough to capture
hallways and other internal features. On the other hand, the density model is best used
when modeling an area with a very large numberof small obstacles, where each individual
obstacle is not as significant as their cumulative effect.
It is important to note a significant weakness of this model, which may be less
immediately apparent. We have focused the discussion above on two categories of
obstacles small objects, such as those that are completely contained within a cell and
take up only a portion of the cell's space, and large objects, which completely cover
multiple cells and partially cover other cells on the perimeter of the object. There is
another type of an object that combines both of those characteristics a long and skinny
one, such as a fence or another type of barrier. One such obstacle can easily span across
very many cells while only occupying a tiny portion of each one. The density of each cell
through which it passes would only be slightly above 0% yet for all intents and purposes,
the fence must completely block the movement. An easy solution would be to mark the
entire cell as impassible (100% dense); however, this would completely defeat the
purpose of introducing density-based model in the first place. Furthermore, by reverting
to essentially basic open vs. closed cell methodology, we reintroduce the problems of
alignment and rasterization, as well as the overarching requirement for small cells.
Marking an entire large density-based cell as 100% dense just because 2% of it is occupied
by the fence would introduce too much of a modeling error, e.g., by erroneously
forbidding any movement along the fence. This problem can be partially addressed by cell
70


size reduction similarto previous methods or by introducing special case for handling such
obstacles outside of the overall board cell structure.
33.2.4 Source Data Submodel
This section presents a radically different approach to mapping real-world
environments into the ABG for using LG [37, 38]. Let us recall that spatial discretization is
not an end in itself. The primary purpose of any such effort is to enable application of LG
by producing the following two components from the ABG definition (Table 1):
X = {xi},a finite set of points; locations of elements;
Rp(x, y), a set of binary relations of reachability (x and y are from X, p is from P).
LG does not require any kind of orderly arrangement of locations just an abstract
set of points. Any organization of these points is then achieved by the means of
reachability relations which define for an entity and a start location whether or not a given
target location is reachable in one game move. As discussed in Section 3.6.4, the concept
of reachability can be expanded to other relations, such as:
Visibility an entity's ability to "see" (sense) from start to target location and
detect an enemy piece there;
Strikability an entity's ability to shoot from start to target location and destroy
the enemy piece there.
Note, that each reachability relation defines movement from one point on the
board to another in one game move over one time interval T. Given that most real world
entities have a particular maximum speed S, it implies that all reachable locations are
within physical distance SxT of the start location. While this can be accomplished with a
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non-uniform arrangement of points on the board, this requirement of equidistance of
neighboring cells lends itself easily to fulfilling this requirement. It is also interesting to
consider that a non-regular distribution of cells that satisfies the equidistance
requirement would necessarily exhibit a somewhat uniform distribution similar to the
regular grid. Indeed, Section 3.3.1.3 describes a method of producing a collection a board
points on the surface of the sphere which begins with an non-equidistant distribution and
employs optimization techniques to shuffle such a collection until a required level of
uniformity is achieved. Thus the regularity of the grid, while not mandated by the LG ABG
concept, is a common technique to provide the set of locations that can be operated on
by the LG algorithms.
The previous sections focused on mapping real life obstacles into the structure of
the board cells. However, there is actually no requirement for the obstacles to be
represented within the ABGs only that the reachabilities are defined between any pair
of cells as either true or false. Representation of the real world objects on the board
structure serves one function only it provides a model that can then be utilized to
efficiently evaluate various reachability relations. The board structures described above
then help satisfy two key requirements of the ABGs they provide a set of locations that
can be occupied by entities and a method for computing reachabilities.
Let us now consider an alternative approach that decouples those two goals. We
can still utilize a board structure similar to those described in the previous sections so as
to produce a finite set of board positions. However, in order to generate reachability
relations, we will use a separate Source Data Submodel. In orderto apply any of the board
72


discretization techniques from the previous sections, a continuous or pseudo-continuous
type of source data must exist. For instance, this data could include the following
elements:
Ground elevation data as either regular grid posts or mesh, such as a TIN
(Triangulated Irregular Network);
Roads as either polygons or linear features (with possible additional attributes);
Rivers as either polygons or linear features (with possible additional attributes);
Vegetation as polygonal areas with attribution such as density and height;
Buildings as either 2D polygons with height, or 3D meshes.
This data can be applied directly to establish relations between various positions
on the board. For instance, given 2 board locations, relations based on the 3D line-of-sight
can be calculated between them by performing ray intersection tests against the various
polygons of the geometrical features above. Similarly, to evaluate whether a particular
vehicle can reach one location from another within a single game move, one can perform
very short range path finding using the full fidelity of the source geometry limiting such
search to the distance that can be traveled by the vehicle within the allotted game step
time.
This methodology ensures that the various reachability relations reflect the full
fidelity of the source data provided rather than suffer from any of the previously
discussed errors incurred by calculating these relations using the discretized board
models derived fromthe source data.This is notto implythatthe discrete model will have
the full accuracy of the continuous model just that for any pair of locations on the LG
73


board, the reachabilities are as accurate as the source data. Each board cell represents an
area or volume of the real world space; however, for the purpose of source data
utilization, a representative single location is chosen. Commonly the cell center is used,
although more complex techniques can employ a set of positions, rather than a single
location.
The discretization, i.e., introduction of the finite set of locations on the board, can
still present difficulties due to the persistent rasterization and alignment problems due to
the selection of the set of locations. For instance, consider a straight road, surrounded by
walls, running through a regular grid of points. Once this road is rasterized, some points
along the road will be located directly on the road, while other just next to it, and others,
possibly, a half of the cell width away. When reachability or line of sight is calculated
between the points along this road, each such relation will cause a jump on and off the
actual road possibly causing breaks in the connectivity.
As usual, this can be alleviated by the reduction in the grid size so as to increase
the size of the board (a finite set) and provide more actual locations for the Source Data
Submodel to produce the greatest benefit. Overall, this system drastically improves the
fidelity offered by the discrete model and allows for the LG strategy algorithms to be
based on the reachability relations calculated at the highest fidelity available from the
source data. While additional modeling errors can be encountered due to discrepancies
between source data and the real world, such issues are not related to the discretization
needed for LG algorithms; they are common to any Geographical Information System, and
are beyond the scope of this work.
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As a final note on modeling terrain obstacles, let us mention that all of the
methods described above can be leveraged and brought into various combination
approaches to achieve the compromise between:
Performance;
Accuracy;
Ease of modeling.
Cell types and under-types can be used to capture basic terrain properties,
densities to capture overall effect of buildings and forests, while employing the Source
Data Submodel to adjudicate any of the more troublesome cases such as fences
mentioned in the previous section.
3.3.3 Dynamic Obstacles
In addition to the obstacles that are based on some inherent or static
characteristics of the terrain, such as mountains and oceans, one must also consider
various versions of dynamic obstacles [4, 39]. Of course, the simplest example of such a
need is "terrain deformation", whereas certain terrain features can be destroyed (a
building or a part of a forest) or created (a bridge across the river). However, the more
interesting uses for dynamic obstacles include additional movement constraints that are
not necessarily related to the physical properties of the terrain itself. Specific examples
would include restriction of aircraft flight to specific altitudes or corridors or avoidance of
certain dangerous regions, such as those saturated with SAM (surface-to-air missile)
systems defenses, or severe weather areas.
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These restrictions are dynamic due to their ability to change during the scenario
execution: dangerous weather may move from one location to another, SAM sites can be
destroyed, thus changing the shape of the dangerous region, and various corridors can
become available for air travel. These restrictions can apply to some entities within the
ABG but not others. Additionally, different parts of the ABG may be affected: movement
(i.e., reachability), visibility, and sensor or weapon employment. Such obstacles must be
well integrated into various concepts of ABGs, such as reachabilities, to allow for the LG
to account for their effect by directly affecting the generation of trajectories and zones in
ways similar to static obstacles described in previous sections.
It is likewise important to note that these restrictions can be specifically defined
by the user of the system, e.g., as the no flight zones, or automatically generated by the
LG during strategy generation. For instance, a particular group of entities (such as an army
platoon) may not be restricted in its choice of a route to the goal destination. However,
once that first main trajectory is chosen by LG, the rest of the group must all operate
within a certain width corridor around the main path to maintain unit cohesion, which
can be achieved by considering anything outside of the corridor as an "obstacle". Similarly
to this automatic generation of such obstacles, they can also be automatically removed if
LG detects that the survivability in this corridor is too low with such restrictions. This
approach allows for various common real world use cases to be represented within an
ABG and reasoned upon by the LG algorithms.
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3.4 Mobility
3.4.1 Temporal Discretization
Another important part of ABG modeling is temporal discretization. As previously
mentioned, real-world Defense Systems generally have to be modeled employing Totally
Concurrent ABGs; however, any movement and actions must still occur in discrete
intervals commonly referred to as a time steps. The size of the time step may vary and
usually depends on the particular application.
Temporal and Spatial discretizations are tightly interconnected. In the discussion
of the spatial discretization above we have introduced the idea of a necessary
compromise: the cell must be small enough to be able to represent necessary level of
detail of the real world terrain, yet large enough so that the computational burden is
tractable. However, this spatial resolution parameter is also directly related to the time
step used. Consider representing a vehicle moving at 1 mile per minute. If we use a time
increment of 1 minute, we may want to use 1-mile cells. On the other hand, this spatial
scale may not be sufficient due to the nature of the terrain. If we decrease the size of the
grid to Va mile per cell, we have a choice of decreasing the time scale to 30 or 15 seconds
(so that the reachability is 2 or 1 cell/turn) or maintaining the spatial scale (using 4
cells/turn reachability). Conversely, if there are other events that may happen more
frequently in the game, we may need the time step of 30 seconds per turn, which may
require a change in the spatial scale. As the reader can see there is a tight interconnection
between temporal and spatial discretization. However, the time step may be affected by
factors other than just the relationship between cell size and agent speeds. There may be
77


additional non-movement events affecting the time scale, such as weapon effects and
sensor updates.
Additional consideration will be given to the various effects of temporal
discretization, as it relates to the diversity of the modeled entities and their actions, in
Chapter 4. This discrete time interval must be an integral part of any discussion of
movement, or other state changes within the ABG.
3.4.2 Reachabilities
The movement pattern within LG is defined employing reachability relations on
the game board [2, 3, 27]. As a result, the ability to represent certain patterns is subject
to our choice of spatial (and temporal) discretization. Conversely, the choice of the board
must reflect the movement patterns that have to be represented within the game. It is
not feasible in practice to precisely represent the mobility of all entities, thus, a
compromise solution must be chosen. Error is unavoidable and, therefore, the priority
must be given to selecting an appropriate spatial and temporal resolution that supports
generation of the game strategies corresponding to the real world strategies. The focus
of such analysis should be on determining which errors have tactical implications (e.g.,
how many game moves it would take for an entity to travel to a particular key location
and what route it would take), rather than on physically precise modeling of the
movement.
78



Y f

jj
6
Y
4


K
4 5 6 7 8 9 10 11 41
Figure 27 The 2D rectangular grid board and reachabilities of pieces
Consider a case where we need to model jet planes moving at 20 miles per minute
and cargo planes moving at 6 miles per minute using a time step of 1 minute. If we use a
cell size of 1 mile, the size of the board may be larger than optimal for performance
considerations. On the other hand if the cell size is 6 miles, the cargo plane can be easily
modeled, but the speed of the jet can only be represented as 3 or 4 cells i.e., 18 or 24
miles per minute. Another choice might be 5 miles/cell, so that the jet has a reachability
of 4 cells per move and the cargo plane of 1 cell per move, corresponding to 5 miles per
minute. For a particular problem, an expert in the problem domain must be consulted to
determine the minimum precision necessary to represent the real world. For example, 18
or 24 miles may not be an appropriate approximation for the jet, but 5 miles per minute
may be acceptable instead of 6 for the cargo plane.
Furthermore, consider a scenario with jet airplanes moving at 1,200 mph and
cargo ships moving at 20 mph. The size of the cell cannot be larger than 20 mph x
TimeStep, so that the movement of a ship could be represented. However, this requires
reachability relations for a jet of 60 cells per game move as a minimum. This corresponds
79


to more than 10,000 target locations reachable from any start location in one game move,
for a full circular reachability on a 2D board. As such computational burden is likely to be
too large for practical use; these 2 types of pieces cannot be represented on the same
board. Section 3.5 presents LG hypergame and variable step ABG approaches to address
such situations.
In addition, the movement of real objects is affected by other physical factors -
such as gravity and inertia. For 3D boards we need to define reachability relations to
include vertical motion as well as horizontal based on their parameters. A plane may be
able to descent faster than to ascent. This may be reflected by considering cells one level
above and two levels below current position reachable in one time step [40].
200 ft l Hex cell, side view
Approx. 2 n.m.
Figure 28 Side view of the cruise missile reachability pattern
Furthermore, inertia may prevent the plane from turning 180 in one game move,
and the reachability must be defined based on physical properties of the agents such as
minimum turning radius [27, 41]. Similar properties may include higher speed at higher
altitudes for airplanes and different reachabilities over different terrain types (such as
amphibious vehicles). Another property of movement for real objects is fuel limitations.
However, this property may not always need to be modeled. For instance, a truck with
period of operation that is less than 5 hours can be assumed to have unlimited fuel; while,
a missile in cannot be in flight for that entire time period. Moreover, some game
80


elements, such as an airplane in flight, may not remain on the same location for 2
consecutive game moves. This may impose certain restrictions on the generation of
trajectories and zone timing principles. For instance, in an LG zone a piece can arrive at
the attack (intercept) location at any time before the target piece reaches it. However, if
a piece cannot hover or stay in one place, the trajectories must match up exactly both
pieces must arrive at exactly the same time. This implies that more complex LG
trajectories and zones must be generated.
Protector Kuwait
5 Game Setup
Board Attrbutes
ReechabiBy Dele
Elevation Data
Mnmum Elevation [5
Manmun Elevation j 1200
(Feet
(Feet
3
BljeMC2A
Bkje Aecraft Carrier
Blue Cargo She
Bbe Fugate
HjsFu
Bkje Land Base
Bue land Cfctect
BLieLRIM
Bkje Checkpoint
Red CM
Red Striker
Red Lard Base
Rad Checkpoint
19 Piece Models and Icons
Piece Categories
19 Piece Reachabilities
3 Fightei
0.000000 -to- 0.0001
B weepen Information
Weapon Models and Icons
Weapon Categories
weapon StrkjMOes
a Sensor Information
Sensor I'todeis and Icons
Sensor Categories
Sensor Vsfcltles
P Above Sea Level i- Above Giound
P Keep destination mthin elevation lango
Automation Parameters
UteAutomatmPaianieteis
Horcontal F0V
Vertical FOV Jo
^J9J2

si <1
r
Figure 29 Defining reachability relationships based on turning radius
81


3.4.3 Trajectory Selection
One of the common problems of modeling movement comes from the
inaccuracies of spatial discretization. Consider rectangular and hexagonal 2D boards with
reachabilities of one cell in every direction. Any path for a piece in this framework will be
jagged similar to a straight line being drawn on a rasterized display. A path on a discrete
board is considered to pass through the centers of cells; as a result even the shortest
trajectory is not a straight line on a 2D Euclidean plane but a collection of several
segmented trajectories each with a number of straight segments. Clearly, on a 2D plane,
the trajectories on the outside of the bundle appear longer, while some of the internal
trajectories are extremely jagged (a turn on every step). For some applications this may
not be an important issue. This is especially true if the model is used for higher level
planning while another system handles low level control (e.g., through the LG hypergame,
Section 3.5.1). However, sometimes these zigzag trajectories have to be addressed. One
way such problems can be addressed is through Trajectory and Move Evaluation Function,
which is a standard component of LG [27]. Quality of individual moves and trajectories is
constantly evaluated based on parameters such as simultaneous participation in multiple
zones and avoiding interception. We can easily incorporate other criteria into this
evaluation, in particular, the 2D Euclidean "smoothness" and deviation from the true 2D
shortest direction. This allows us to discard trajectories that would not be considered
reasonable by human experts, while still evaluating entire bundles in case the unusual
trajectories prove useful. The exact parameters of this evaluation depend on the problem
domain.
82


Figure 30 Bundles of shortest trajectories
File Test 2D View LGView 3D View View Window Help
Move 11
StrategyLGO: survivability estimate (Blue zone)=0.7276
survivability estimate (Red zone)=0.6207
'BlueStriker' distributed to 'RedHQ2'
'BlueFighterl' distributed to 1 RedFighter2'
'BlueFighter2' distributed to 'RedFighter3'
'BlueFighterl' distributed to 'RedStriker'
3
IB Start | M H sJ S3 11 ^Windows Com... ||SflSEAD -
Figure 31 LG Zones with Trajectory Bundles for Aircraft Engagement
3.4.4 Direction Phase Spaces
There is one more issue we need to consider with regard to modeling movement
of the realistic entities on the LG boards, both 2D and 3D. An army unit may be able to
83


Full Text

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APPLICABILITY AND EXTENSIBILITY OF LINGUISTIC GEOMETRY TO MODELING OF REAL WORLD INTELLIGENT SYSTEMS by OLEG UMANSKIY M.S., University of Colorado at Denver, 2001 B.S., University of Colorado at Denver, 200 0 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Computer Science and Information Systems 2015

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This thesis for the Doctor of Philosophy degree by Oleg Umanskiy has been approved for the Computer Science and Information Systems Program by Gita Alaghband, Chair Boris Stilman, Advisor Tom Altman Michael Mannino Tam Vu April 1 2 2015 ii

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Umanskiy, Oleg (Ph.D., Computer Science and Information Systems) Applicabilit y and Extensibility o f Linguistic Geometry to Modeling o f Real World Intelligent Systems Thesis directed by Professor Boris Stilman ABSTRACT Linguist ic Geometry (LG) is a powerful game theory and technology for generating winning strategies for real world systems. It has been successfully employed in numerous research, government and commercial projects. However, in order for this approach to be applied to each particular problem domain, one must first model this domain as LG Abstract Board Game (ABG). Furthermore, the tools and methods required to achieve such mapping from the oddities of the real world to the rigid mathematical definitions of the ABGs v ary depending on the particular domain under consideration. This research addresses a wide spectrum of modeling issues required to successfully apply LG to such real world problems. The majority of these issues arise from the difficulties of mapping the va st breadth and depth of items, actions, and effects present in the real world into the formal notions of ABGs. In the course of this research we developed techniques and methodologies that allowed the theory of LG to be extended to achieve the high level of applicability to the wide range of problem domains. These extensions are organized in a way that provides the highest benefit for current and iii

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future practical application of LG to various real world problems, and include examples of applications to seve ral representative operational domains. The form and content of this abstract is approved. I recommend its publication. Approved: Boris Stilman iv

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This work is dedicated to the memory of my Grandpa, whose life has gre atly inspired and influenced me v

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ACKNOWLEDGEMENTS It has been a very long journey and I would never have complete d it without the relentless encouragement of my advisor, Boris Stilman. You have encouraged my growth as a scientist and a person for over 15 years, and I am constantly grateful for that. Likewise, this would not have been possible without my family whose drive by their own example, towards scientific pursuits from the early age has led me down this path. And thank you to everyone, especially my wife, Alia, for their tolerance of my hermit like existence during the last stretch of this trek. vi

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TABLE OF CONTENT S CHAPTER 1 INTRODUCTION ................................................................................................................ 1 1.1 Solving Real World Systems Using Game Approaches ......................................... 1 1.2 Gaming Approaches ............................................................................................. 2 1.3 Complexity of Discrete Games ............................................................................. 4 1.4 Human Approach to Complex Systems ................................................................ 6 1.5 Applicability of LG for Solving Real World Systems ............................................. 8 2 LINGUISTIC GEOMETRY FOUNDATIONS AND HISTORY ................................................. 10 2.1 Abstract Board Games ....................................................................................... 11 2. 2 Hierarchy of Formal Languages .......................................................................... 20 2.2.1 Language of Trajectories ............................................................................. 21 2.2.2 Language of Zones ...................................................................................... 32 2.3 LG Strategies ....................................................................................................... 41 2.4 Historical Validation of LG .................................................................................. 42 3 APPLICABILITY AND EXTENSIBILITY OF LG ..................................................................... 47 3.1 Complexity of Modeling Real World Systems as ABGs ...................................... 47 3.2 Concurrency ....................................................................................................... 49 3.3 Spatial Discretization .......................................................................................... 50 vii

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3.3.1 Regular Grids ............................................................................................... 50 3.3.1.1 2D Grids ............................................................................................... 51 3.3.1.2 3D G rids ............................................................................................... 53 3.3.1.3 Spherical Grids ..................................................................................... 55 3.3.1.4 Curvature of the Earth ......................................................................... 58 3.3.2 Terrain Obstacles ........................................................................................ 59 3.3.2.1 Cell Types and Under types ................................................................. 61 3. 3.2.2 Eggshell Model ..................................................................................... 64 3.3.2.3 Density Model ...................................................................................... 67 3.3.2.4 Source Data Submodel ........................................................................ 71 3.3.3 Dynamic Obstacles ...................................................................................... 75 3.4 Mobility .............................................................................................................. 77 3.4.1 Temporal Discretization .............................................................................. 77 3.4.2 Reachabilities .............................................................................................. 78 3.4.3 Tr ajectory Selection .................................................................................... 82 3.4.4 Direction Phase Spaces ............................................................................... 83 3.4.5 Speed Phase Space ...................................................................................... 86 3.4.6 Ballistic and Orbital Trajectories ................................................................. 87 3.5 Heterogeneous Systems ..................................................................................... 90 viii

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3.5.1 LG Hypergames ........................................................................................... 93 3.5.1.1 From ABGs to Hypergames .................................................................. 95 3.5.1.2 Common Inter Linking Mappings ...................................................... 100 3.5.1.3 Benefits of Hypergames .................................................................... 101 3.5.2 Variable Step ABGs ................................................................................... 102 3.5.2.1 Cell Sizes, Time Steps and Entity Movement Speed .......................... 104 3.5.2.2 Variable Time Step ............................................................................. 107 3.5.2.3 Practical Implications ......................................................................... 109 3.6 Incomplete Information ................................................................................... 110 3.6.1 World Views .............................................................................................. 110 3.6.2 Deception Employment and Discovery ..................................................... 112 3.6.3 Communication Groups ............................................................................ 113 3.6.4 Sensors ...................................................................................................... 115 3.7 Weapon Systems .............................................................................................. 118 3.7.1 Strikabilities ............................................................................................... 118 3.7.2 Weapon Effects ......................................................................................... 120 3.7.3 Paired/Prerequisite Trajectories ............................................................... 122 3.7.4 Synchronized Trajectories ......................................................................... 124 3.8 Mission Concepts ............................................................................................. 125 ix

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3.8.1 Goals and Missions ................................................................................... 125 3.8.2 Prescribed Behaviors ................................................................................ 127 4 OPERATIONAL DOMAINS ............................................................................................. 129 4.1 Operations in Urban Terrain ............................................................................ 130 4.1.1 Spatial Discretization ................................................................................ 131 4.1.1.1 Grids ................................................................................................... 134 4.1.1.2 Obstacles ........................................................................................... 137 4.1.1.3 Dynamic Obstacles ............................................................................ 144 4. 1.2 Mobility ..................................................................................................... 145 4.1.2.1 Reachabilities ..................................................................................... 145 4.1.2.2 Trajectory Selection ........................................................................... 152 4.1.3 Heterogeneous Systems ........................................................................... 156 4.1.3.1 Variable Step ABGs ............................................................................ 1 56 4.1.3.2 LG Hypergames .................................................................................. 156 4.1.4 Incomplete Information ............................................................................ 158 4.1.5 Weapon Systems ....................................................................................... 161 4.1.5. 1 Strikabilities ....................................................................................... 161 4.1.5.2 Weapon Effects .................................................................................. 164 4.1.5.3 Paired and Synchronized Trajectories ............................................... 166 x

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4.1.6 Mission Concepts ...................................................................................... 168 4.1. 6.1 Goals and Missions ............................................................................ 168 4.1.6.2 Prescribed Behaviors ......................................................................... 170 4.2 Air and Naval Operations ................................................................................. 171 4.2.1 Spatial Discretization ................................................................................ 172 4.2.1.1 Grids ................................................................................................... 173 4.2.1.2 Obstacles ........................................................................................... 174 4.2.1.3 Dynamic Obstacles ............................................................................ 176 4.2.2 Mobility ..................................................................................................... 176 4.2.2.1 Reachability ....................................................................................... 176 4.2.2.2 Trajectory Selection ........................................................................... 179 4.2.3 Heterogeneous Systems ........................................................................... 180 4.2.4 Incomplete Information ............................................................................ 184 4.2.5 Weapon Systems ....................................................................................... 186 4.2.5.1 Strikabilities ....................................................................................... 186 4.2.5.2 Weapon Effects .................................................................................. 187 4.2.5.3 Prerequisite Trajectories ................................................................... 188 4.2.6 Mission Concepts ...................................................................................... 189 4.3 Ballistic and Orbital Operations (BO) ............................................................... 190 xi

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4.3.1 Spatial Discretization ................................................................................ 191 4.3.2 Mobility ..................................................................................................... 192 4.3.3 Heterogeneous Systems ........................................................................... 195 4.3.4 Incomplete Information ............................................................................ 196 4.3.5 Weapon Systems ....................................................................................... 196 4.3.6 Mission Concepts ...................................................................................... 198 4.4 Joint Forces Operations (JF) ............................................................................. 198 4.4.1 Heterogeneous Systems ........................................................................... 198 4.4.2 AirGround Hypergame ............................................................................. 200 4.4.3 Litoral AirNaval Hypergame ..................................................................... 201 4.4.4 Joint Amphibious/Air Assault Operations ................................................. 203 5 CONCLUSION ................................................................................................................ 207 REFERENCES .................................................................................................................... 211 xii

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LIST OF TABLES Table 1 ABG definition ................................................................................................................ 11 2 ABG definition for the game of chess ............................................................................ 17 3 ABG definition for combat simulations ......................................................................... 19 4 Grammar of shortest trajectories Gt(1) ......................................................................... 23 5 Grammar of Zones GZ ..................................................................................................... 38 6 RAID Experimental Scoring Criteria ............................................................................... 44 7 RAID Experiment Results ............................................................................................... 45 8 RAID Experiment Results ............................................................................................... 46 xiii

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LIST OF FIGURES Figure 1 JFAC SEAD Mission ........................................................................................................... 5 2 Comparison of searches for the same processing time ................................................... 7 3 A Hierarchy of Formal Languages in LG ......................................................................... 10 4 TRANSITION(p,x,y) ......................................................................................................... 13 5 Bundle of trajectories .................................................................................................... 22 6 Interpretation of the algorithm for nexti for the grammar Gt(1) .................................. 26 7 Values of MAP88,p .......................................................................................................... 27 8 Values of MAP36,p .......................................................................................................... 27 9 SUM ................................................................................................................................ 28 10 ST1(88) .......................................................................................................................... 28 11 ST1(78) .......................................................................................................................... 29 12 ST2(88) .......................................................................................................................... 29 13 ST1(68) .......................................................................................................................... 30 14 ST3(88) .......................................................................................................................... 30 15 ST1(57) .......................................................................................................................... 30 16 ST4(88) .......................................................................................................................... 30 17 ST 1 (47) ......................................................................................................................... 31 18 ST 5 (88) ......................................................................................................................... 31 19 LG zone for TC system with strikes .............................................................................. 33 20 Various types of LG zones ............................................................................................ 36 xiv

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21 Rectangular and Hexagonal Regular Grids .................................................................. 52 22 Rectangular and Hexagonal Line Rasterization ........................................................... 53 23 3D cell: hexagonal prism .............................................................................................. 55 24 A planet level board with spherical hexagonal cells. .................................................. 58 25 Aircraft movement through using basic cell types ...................................................... 62 26 Eggshell hexagonal model of an urban environment .................................................. 66 27 The 2D rectangular grid board and reachabilities of pieces ........................................ 79 28 Side view of the cruise missile reachability pattern .................................................... 80 29 Defining reachability relationshi ps based on turning radius ....................................... 81 30 Bundles of shortest trajectories ................................................................................... 83 31 LG Zones with Trajectory Bundles for Aircraft Engagement ....................................... 83 32 Directions with respect to a hex grid ........................................................................... 84 33 Trajectories of the aircraft changing direction ............................................................ 85 34 Ballistic trajectories on hexagonal shperical ABG ........................................................ 88 35 Orbital trajectories ....................................................................................................... 90 36 Air (left) and Land (right) ABGs of an Air Land hypergame ......................................... 96 37 Air (left) and Navy (right) ABGs of an Air Navy hypergame ........................................ 98 38 Cells per turn and Time step vs Speed in km ............................................................ 107 39 Incomplete/false information .................................................................................... 112 40 A cruise missile illuminated by the aircraft radar. ................................................... 116 41 Strikability with obstacles .......................................................................................... 119 42 Mission Execution Matrix for Operations in Urban Terrain ...................................... 127 xv

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43 Map data: Google, Bluesky ........................................................................................ 132 44 Rectangular 2D Grid for Grid Plan Urban Area .......................................................... 135 45 Hexagonal 2D Grid for Grid Plan Urban Area ............................................................ 136 46 Rectangular 2D Grid for Non Grid Plan Urban Area .................................................. 136 47 Hexagonal 2D Grid for NonGrid Plan U rban Area .................................................... 136 48 LG Trajectories on Hexagonal Eggshell Urban Board ................................................ 138 49 Urban area using 10m hexagonal grid ....................................................................... 140 50 Urban area usin g 20m hexagonal grid ....................................................................... 141 51 Urban Area Source Data Submodel ........................................................................... 142 52 Urban area using 30m hexagonal grid ....................................................................... 143 53 Dismounted outdoor reachability pattern ................................................................ 147 54 Dismounted indoor reachability pattern ................................................................... 149 55 Vehicle road reachability pattern .............................................................................. 1 50 56 Vehicle off road reachability pattern ........................................................................ 151 57 Vehicle threat map ..................................................................................................... 153 58 Dismounted threat map ............................................................................................. 154 59 Dismounted threat avoidance trajectory .................................................................. 155 60 Overlapping Source Data Submodel LOS from 2 observers, 20m cells ..................... 160 61 Cone SDZ for firing small arms direct fire weapons .................................................. 162 62 Probability of Hit curve with rapid drop off ............................................................... 166 63 LG Zone with synchronized negation trajectories ..................................................... 168 64 Army tactical doctrinal taxonomy .............................................................................. 169 x vi

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65 Bounding Overwatch ................................................................................................. 171 66 Coast terrain as hexagonal grid board ....................................................................... 175 67 Aircraft reachability at same altitude, climbing, and diving ...................................... 178 68 Basic ship reachability ................................................................................................ 178 69 Air threat avoidance trajectories before and after SAM site destruction ................ 180 70 Littoral ABG with ships and UAVs LG Zone ................................................................ 182 71 Large scale air combat ABG with LG Zones ................................................................ 183 72 PD Curve for Ground Radar against Aerial Targets .................................................... 185 73 Air to ground and airto air strikabilities ................................................................... 187 74 LG Zone for defense against cruise missiles with integrated fire control ................. 189 75 Orbital trajectories on hexagonal planetary grid ...................................................... 192 76 LG Trajectories for terrestrial pieces on a planetary grid .......................................... 193 77 Initial LG Zone for ballistic missile s ............................................................................ 194 78 LG Zone during ballistic missile engagement, with radar illumination ..................... 195 79 Air (top) and Ground (bottom) hypergame ............................................................... 201 80 Littoral hypergame ..................................................................................................... 203 81 Joi nt Amphibious/Air Assault Hypergame ................................................................. 206 xvii

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CHAPTER 1 INTRODUCTION 1.1 Solving Real World Systems Using Game Approaches Real world adversarial systems, such as military operations, seemingly yield themselves to easy and straight forward modeling as a game (similar to the game of chess). Some of the aspects of such modeling are indeed straight forward an aircraft or a tank can be represented as game piece, while the geographical space including ground, oceans, aerospace, as well as underwater and outer space can be represented as a large allencompassing 3 dimenssional game board. However, this ease is quite deceptive as we will explore within this work. The world, and, by extension, any operation within it are exceedingly complex. Capturing every single minute aspect is therefore next to impossible. However, there is an opposite problem as well. As we attempt to build more and more accurate models of the real world these problem s increase in complexity and can easily become computationally intractable. The goal of modeling Defense System s as a game is to model them in such a way that yields itself to generation of winning strategies so as to benefit the end user the consumer of the end results of the systems. A game strategy describes the behavior of all actors and entities involved in terms of a sequence of moves A move represents the smallest activity of pieces discernible from the game point of view. Without being able to use the game models to find winning strategies, such models can only be used for 1

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visualization and display of the situation rather than to their full potential as an artificial intelligence based decision support aids. Thus we are faced with two simultaneous challenges. The first is to discover appropriate techniques to capture the real w orld systems as abstract m odels. Yet we must also ensure that these models can be tractably analyzed by specific artificial intelligence techniques to produce useful and meaningful results for the end user. The search for the balance between these two diametrically opposed constraints is the focus of this dissertation. 1.2 Gaming Approaches The two primary impediments to generating winning strategies for real world systems are exponential explosion due to the hi gh dimensionality of the solution space and modeling of active intelligent adversaries capable of asymmetric responses. The only approach that allows introduction of the full scale intelligent adversary is the gaming approach [1 4]. There are various classifications of game based approaches. Two such dimensions of categorization are continuous vs. discrete, and strategic vs. extensive. Continuous games are often described mathematically as pursuit evas ion differential games [5], which is not suitable for dynamic, multi agent models [6, 7]. There is a small number of differential games, for which an exact analytical solution is calculable, and another set of differential games, for which numerical solutions can be computed in reasonable time under restrictive conditions. However, these games are poorly suited to a general military problem as they are focused on one to one problems not the many to many scenarios of the real world military operations. Additionally, they are unable to 2

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m odel asymmetric non zero sum games where the desires of the enemy are not the exact opposite of the friendly desires. Furthermore, 3D modeling, limitation of the lifetime of the agents, and presence of heteroge neous agents create additional barriers to ability to use continuous games for the wide range of varied military problems. Discrete strategic games are another category of games that are unsuitable for practical real world military problems. Such games were introduced in [1, 2, 8] and later developed by multiple researchers. This approach can only analyze the entire c ourse of action for each player at once each player must choose a plan of action once and for all and is not informed about the plan of the enemy. The game may not be broken down into individual sequences of moves it must be analyzed as an entire full game strategy. This leaves the discrete extensive games as the candidate for mathematical modeling of real world defense systems. One of the key beneficial aspects of such models is the ability for each player to choose appropriate moves at any step in the game the player can make a decision to alter the plan in response to the previous moves made by the opposing player and themselves [8]. Another useful aspect is the requirement of discretization of the problem domain. Such discretization can allow for real world entities (such as airplanes and ground vehicles), and their actions and interactions (such as movement and engagement) to be represented within these extensive discrete gam es. A common representation of such games is as a game tree where each node represents the state of the game after each successive move and the edges correspond to the possible choices made by the players Such an extensive game tree would include every possible move of every strategy of every player. Because this game includes all possible 3

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moves, it is theoretically trivial to look through the possible branches and variations and find the optimal strategy. This is the strategy that guarantees best possible outcome for each player regardless of what the other player is doing. Thus, classical theory of extensive games is not concerned with tractability of the solution rather its concern is with the existence of such solutions. That particular weakness mak es extensive games not useful for practical real world problems where solutions need to be found in a reasonable amount of time. This brings the discussion to the practical gaming approach which tries to search discrete games trees [9]. While most suitable to practical uses, the primary obstacle of this method is the sheer scale of such game trees, which is sometimes referred to as the curse of dimension. Even for small real world scenarios, these games trees are of such a s ize that it is intractable to find a solution via conventional approaches such as brute force or min i max search. 1.3 Complexity of Discrete Games Since the size of the game tree to be searched using practical gaming approaches to discrete games is the major factor which impacts the tractabil ity of the problem, we can use the size of such trees as the measure of the complexity of the game. Consider a very small game with only 10 pieces, with each piece only capable of 10 legal moves at any time. If the game lasts for only 50 moves the size of the game tree would be (1010)50, or 10500. Even for such unrealistically small game, the size of the tree is staggering. More realistic military operations are even more prohibitively complex no computer could 4

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possibly search through such trees in a practical amount of time or even in a lifetime. Let us consider two of the real examples from previous research [3, 4, 10 12] Total number of pieces Valid moves per piece (approximate) Depth of game tree (total moves) Size of tree JFAC SEAD 30 18 70 10 1861 RAID 70 24 480 10 21255 Figure 1 JFAC SEAD Mission For game trees of such size even for the theoretically best cases, the most powerful search algorithms such as alphabeta pruning, would not produce significant search reduction. In the best case, the number of moves to be searched employing alpha beta algorithm g rows exponentially with the power of this exponent divided by two with respect to the original game tree [13]. As one can see from the table above, such a reduction is essentially meaningless in terms of problem tractability. To make matters 5

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even worse, alpha beta pruning is not applicab le to totally concurrent games (where all players may move all of their entities at the same time) it is only applicable to sequential alternating games with players taking turns for moving an entity. Military war games are necessarily totally concurrent and thus cannot benefit from such pruning. Since fully exhaustive search is not possible, the common alternative is to search up to a certain depth for instance 2 to 5 moves ahead and apply heuristic evaluation functions to determine the value of the position at that point to determine best short term plan. Of course, the accuracy of such approaches depends on the ability to correctly evaluate such intermediate positions. These heuristics cannot really be generated to provide correct assessment in all situations and the algorithms that rely on shallow search of this sort can be short sighted. This might result in picking a plan that seems beneficial within the range of the next 2 5 moves but does not lead to the ultimate victory which requires 50 100 moves down the line. There are other alternatives that attempt to search deeper along more promising branches and shallower in other directions. However, such pruning is still based on heuristic analysis of the positions at the poin t of the cutoff, and, therefore, is similarly susceptible to the limitations described above 1.4 Human Approach to Complex Systems In order to cope with suc h intractable search problems, Linguistic G eometry approaches the problem from a different angle. The m otivation behind such approach is the difference between the human and the so called computer based approaches to solving such problems The main advantage of the computers over humans is the ability to perform extremely fast and precise combinatorial search which leads to the natural 6

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application of exhaustive search algorithms that attempt to consider every possible permutation of moves. However, despite humans apparent handicap of computing several orders of magnitude slower than a computer, human expe rts can still outperform any computing system for many classes of real world complex systems. The reason for this is the difference of the human approach from the general computer search approach. Rather than considering all possibilities, a human expert o nly considers one or two possible moves for each position. With such a low branching factor, a human expert can look very deep into the tree of variations much deeper than a much faster computer can look using a brute force search. Brute Force Search Alpha Beta Search Human Expert Search Figure 2 Comparison of searches for the same processing time The ability of a human expert to solve the problems in such an efficient manner is likely based on their ability to decompose a complex problem into sub problems. Such problems can then be solved and their solutions reconstructed into the solution for the original problem. This process is in fact quite complicated due to the complexities of the real world systems. It is usually impossible to decompose the problem into truly 7

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independent subproblems. More likely, some inter dependencies between such subprob lems need to be taken into account when solving each individual one. Another related problem is ensuring that the combined optimal solutions to the smaller problems actually result in the optimal solution to the original larger problem. If care is not tak en to track all of the possible dependencies, it is quite possible to produce optimal solutions for the sub problems which produce a subpar solution to the overall problem. The need to account for such inter connections has an adverse effect on the dimensi on reducing benefits of the original decomposition even to the point of completely eliminating the benefits of the decomposition. Typically, a human expert possesses the necessary knowledge and experience to perform such an efficient decomposition for a particular set of problems within his field. They may not be able to transpose such knowledge to a different problem domain. Due to such specialization of particular human heuristics within each domain, we do not yet possess a general methodology to translate human approach into computer algorithms. Within each problem domain, with sufficient and careful study, a method could be found that mimics the human expert approach of the problem decomposition; however, attempting to transfer such knowledge to another domain is a different task altogether. 1.5 Applicability of LG for Solving Real World Systems Chapter 2 of this dissertation describes the foundations of the Linguistic Geometry as a method for tractable and practical generation of winning strategies, including a summary of the complex problems addressed by this approach, its theoretical footing, as well as experimental results confirming the validity and accuracy of this methodology. In 8

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turn, Chapter 3 extends previous research as to the applicability and extensibility of Linguistic Geometry as it pertains to solving complex real world problems. Specific problems and the methodologies for addressing them are described in detail so as to support the high degree of relevance of this research to the community of practice. Chapter 4 examines how these methods can be applied to 4 specific operational domains. While the focus of this work is on solving practical issues, theoretical component of this research is also discussed to substantiate the rigorous nature of th is presentation. 9

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CHAPTER 2 LINGUISTIC GEOMETRY FOUNDATIONS AND HISTORY The purpose of Linguistic Geometry is to develop a formal approach to a certain class of multi agent systems that involves breaking down a system into a hierarchy of dynamic subsystems [14 19]. The methods of LG are formalized as a Hierarchy of Formal Languages used to solve certain classes of problems. The languages in the hierarchy are used to generate a hierarchy of structures : trajectories, zones, translations, searches, etc. The class of problems that is addressed by these techniques is formally defined as abstract board games. The LG method generates a tree of translations which is a string in the Language of Translations In order to do so several strings of the Language of Zones or Webs are generated. In turn, those LG Zones contain strings of the Languag e of Trajectories. Together, this hierarchy creates decomposition s of dynamic subsystems, whi ch is then used to solve the problem [18]. Figure 3 A Hierarchy of Formal Languages in LG LanguageLanguageof WebsLanguageof Translations 32 4 5 6 1 LG Tree Node (State) LG Search Tree of Trajectories 10

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2.1 Abstract Board Games Before the Hier archy of Formal Languages at the heart of LG can be applied to a problem, it needs to be represented as an abstract board game [14, 18] This provides formalization for a large class of problem domains. As a result, LG methods can be easily and directly applied to any system that can be represented in this form. The formal definition of ABGs is shown below. Followin g this definition, two examples are presented to demonstrate the relationship between an ABG and a problem it represents. Abstract board game is the following eight tuple: < X, P, R p SPACE, val S i S t TR> Table 1 ABG definition X = {x i } A finite set of point s ; locations of elements P = {p i } A finite set of element s ; P is a union of two disjoint subsets P 1 and P2 called the opposing side s R p (x, y) A set of binary relations o f reachability in X (x and y are from X, p is from P); Val A function on P with positive integer values describing the values of elements 11

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Table 1 (cont.) SPACE The state space. A state S placement ON:P X and additional parameters. The value ON(p) = x means that element p occupies location x at state S. Thus, to describe function ON at state S, we write equalities ON(p) = x for all elements p, which are present at S. We use the same symbol ON for each such partial function, though the interpretation o f ON may be different at different states. Every state S from SPACE is described by a list of formulas {ON(pj) = xk} in the language of the first order predicate calculus which matches with each relation a certain Well Formed F ormula (WFF) S i and S t The sets of start and target states Thus, each state from S i and St is described by a certain list of WFF {ON(pj) = xk}. St is a union of three disjoint subsets St 1, St 2, and St 3. St 1, St 2 are the subsets of target states for the opposing sides P1 and P2, respectively. St 3 is the subset of target draw states. 12

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Table 1 (cont.) TR A set of transitions, TRANSITION (p, x, y), of the System from one state to another (Fig. 2.2). These transitions are described in terms of the lists of WFF (to be removed from and added to the description of the state ) and a list of WFF of applicability of the transition. These three lists for state S SPACE are as follows Applicability list : (ON(p) = x) Rp(x,y); Remove list : ON(p) = x; Add list : ON(p) = y, where p P. The transitions are defined and carried out by means of a number of elements p from P1, P2, or both. This means that each of the lists may include a number of items shown above. TRANSITION S1S2x y x yX X p q p Figure 4 TRANSITION(p,x,y) 13

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Note, that this generic definition can be expanded for a particular class of ABGs Additional parameters of a state may be added. In addition to the data described above, a state of an alternating ABG includes a binary function MT(S) {1, 2}, a move turn to distinguish between the states when player 1 ( or 2 ) is allowed to move. A state of an ABG with variable speed includes function of speed sp Other information may be includ ed based on system requirements. The transitions TR above may also be more complex. Transitions may be of several types. A simple move transition occurs when element p moves from x to y without removing an opposing e lement. In this case, point y is not occupied by an opposing element. A remove transition occurs if element p moves from x to y and does remove an opposing element q, i.e., OPPOSE(p, q) holds. For alternating serial systems, the opposing element q has to be at location y before the move of p has commenced. In this case, the Applicability list and the Remove list include additional formula ON(q) = y. For concurrent systems, this is not necessary element q may arrive at y simultaneously with p and be removed. These transitions are governed by the reachability relation R p (x, y). Further constraints can be imposed on the members of TR. In case of an alternating ABG (see definition of SPACE above), if the move turn MT(S) = i, then only elements p from P i (from P 1 or P 2 respectively) can participate in the transition. Additional constraints may be introduced based on a particular class of system (e.g. systems in which no two game pieces can occupy the same location). For realworld systems (especially Defense Systems), another type of transitions is common a strike transition. For this class of systems, another e lement is introduced 14

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into the eight tuple above binary relations of strikability Strkp(x,y), which is analogous to relations of reachability. Strike transition occurs when an opposing element q is removed at a location different from y. This reflects a common situation, where a piece can destroy an opposing piece located at a different position on the board by shooting. In this case, Remove List may include additional formulas ON(q1) = z1, ON(q2)=z2, etc for all locations zi that are considered strikable from location y (i.e. where Strkp(y,zi) holds true). In an ABG the goal of each side is to reach either one of its winning states (a state in subsets St 1 or St 2, respectively), or, if impossible, a draw state from St 3. The problem of the optimal operation of the System is considered as a problem of search for a sequence of transitions leading from the Start State of Si to a target state of St assuming that each side makes only the best moves, i.e., such moves (transitions) that could lead the ABG to the respective subs et of target states. To solve an ABG means to find a strategy (an algorithm to select moves) for one side, if it exists, that guarantees that the proper subset o f target states, St 1, St 2, or St 3, will be reached assuming that the other side makes antagonistic moves. As mentioned above, a wide range of classes of ABGs is possible. These can be categorized into 3 general classes: Alternating Serial, Alternating Conc urrent, and Totally Concurrent systems. In Alternating Serial (AS) systems, the opposing sides alternate turns and only one element at a time can be moved. In Alternating Concurrent (AC) ABGs the opposing sides alternate turns; however, all, some, or none of the current players elements can be moved simultaneously. In a Totally Concurrent (TC) ABGs players do not 15

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alternate and all, some, or none of the pieces of both sides can be moved simultaneously. It is important to note that in general, the level o f difficulty increases from AS to AC to TC due to the need to consider all possible combinations of moves. For instance, consider a game of 5 pieces against 5 opposing pieces, where each piece can make any one of 10 moves. Then, AS branching factor is 10, AC branching factor is 105, and TC branching factor is 105+5=1010. Due to the nature of real world Defense Systems, they usually need to be modeled as Totally Concurrent ABGs To further illustrate, the relationship between a problem and an ABG, two examples are presented below outlining the definitions of the eighttuple for the game of chess and a generic combat simulation. 16

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Table 2 ABG definition for the game of chess X = {x i } 64 squares on the chess board P = {p i } White and black pieces R p (x, y) Reachability is defined by the rules of chess. I.e. R p (x,y ) is true if and only if a piece p is allowed to move from x to y according to the rules of chess. For example, if p is a K ing, Rp(x,y) is true iff y is one of the immediate neighbors of x on the chess board. Val Val (p) is the value of a piece. E.g. Val ( Pawn)=1, Val (B)=3, Val (K)=500, etc. SPACE ON(p)=x, iff piece p stands on square x MT(S)=White or Black active player S i and S t S i is the traditional chess start state or an arbitrary position that is to be analyzed. St 1 and St 2 are sets of all checkmate positions for the corresponding side. St 3 is the set of all draw positions. 17

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Table 2 (cont.) TR TRANSITION (p, x, y) represents moving piece p from square x to y. If opposing piece is present at y, capture is implied. Since chess is an Alternating Serial system, only one piece of the active player can be moved at a time (with exception of castling). In addition, chess is a Complex System with blocked beams and destinations, which means that some move s may be prohibited due to presence of another piece either at the target location y, or on the beam from location x to location y. Pawn promotion is also a transition which removes ON(p)=x, and substitutes ON(p)=y, where x is the row before last, y immed iately ahead in the top row, and p is promoted piece. As the reader can see there are certain discrepancies from a general ABG present several pieces are prevented from occupying the same location (system with blocked beams and destinations), pawns c an be promoted, etc. Such differences can be easily incorporated into a special class of Complex Systems. Similarly, for other problem domains, apparent discrepancies from a general abstract board game can be easily accommodated. 18

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Table 3 ABG definition for combat simulations X = {x i } 2D or 3D grid of the area of operations. Could be a simple 2D rectangular grid for land operations, or a complex 3D packing of aerospace, including orbit positions of satellites. P = {p i } Sets of resource s of the opposing sides, where each element can represent individual or groups of airplanes, tanks, ships, infantry, satellites, etc. R p (x, y) Reachability is defined by the movement capabilities of different types of elements present. Ships can move on grid points corresponding to water only. Satellites can only move to a point further along its orbit. The maximum speed of a particular element defines how far it can move in one step. Val Val (p) c an be defined by the abilities of the element (more powerful element has a higher value), as well as by the value of the element to the operation (an airstrip that must be protected can have a higher value). SPACE ON(p)=x, iff element p is at location x ( at particular coordinates on the surface of Earth, volume of aerospace, or position in orbit depending on set X). S i and S t S i is arbitrary position that is to be analyzed e.g., positions of the resources before the conflict. St is a set of target states based on mission objective e.g., certain targets destroyed (or defended). TR TRANSITION (p, x, y) represents element p moving from location x to y (on land, sea, air, etc). In addition to move transitions, strike transitions are needed to reflect long distance shooting for objects Due to the nature of real world, these are Totally Concurrent systems any combination of pieces from both sides can move simultaneously during a single time step. Also, transitions may include a game piece producing a new element, changing into a new element, splitting into several elements to reflect events such as firing missile, dropping paratroopers, re loading expended ammunition, splitting into smaller com bat units, etc. 19

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Real world Defense Systems are the focus of this thesis. Although they exhibit more complex behavior than traditional board games, they are fully susceptible to LG techniques. 2.2 Hierarchy of Formal Languages As mentioned above, LG methods are formalized as a Hierarchy of Formal Languages Language of Trajectories, Language of Zones, Language of Translations, and Languages of Searches. This formal hierarchy is presented in [18] employing declarative (non constructive) definitions, followed by introduction of generating grammars for these languages. Since the subject of this dissertation is extension and application of LG to real world systems this sec tion is not intended as comprehensive LG theory presentation The purpose of this section is to present the overall principles of the LG methods and allow the reader to follow further discussions in this dissertation. Languages of Trajectories and Zones are introduced informally and constructively through their generating grammars. The generating grammars used by LG theory are the so called controlled grammars, which are very flexible tools for producing strings of sy mbols with parameters [21]. The definitions are followed with simplified examp les to demonstrate application of these grammars (on a chess like ABG). 20

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2.2.1 Language of Trajectories The lowest level in the LG Hierarchy of Formal Languages is the Language of Trajectories [14, 18]. The strings of this language represent a path or route of a game piece from one location to the next. In general the strings are of the following form: to = a (x)a (x1) a (xl) which represents a trajectory for some piece p P from location x X to location xl X of length l Informally, it is simply a string of symbols a with parameters, where each parameter is a location on the ABG. The main property of this string is that it represents a valid path for piece p from x to xl. This implies that every point xi is reachable in one step from previous location xi 1, i.e. Rp(xi,xi+1)=True for all i = 0,1,, l 1. Usually, there is more than one trajectory between any two locations. A set of trajectories for the element p from location x to location y of length l is called a bundle of trajectories of length l and can be denoted by tp(x,y, l ). An example, using a chess like board, is shown below. The game piece p in the example has a reachability relation analogous to a chess King, while shaded locations are considered unreachable. Locations are represented by two digits (x1x2). 21

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Figure 5 Bundle of trajectories Trajectories for piece p from (88) to (36) of length 5 are as follows: t0= a (88) a (78) a (68) a (57) a (47) a (36), t1= a (88) a (78) a (68) a (58) a (47) a (36). A bundle of trajectories for p from (88) to (36) of length 5 (note, that there are no other trajectories of length 5): tp(88,36,5)={t0,t1} Next, the controlled grammar Gt (1) that generates shortest trajectories is presented and demonstrated by the above example. This is the most basic type of trajectories. A variety of other trajectory grammars is possible. For instance, admissible trajectories of degree k represent trajectories consisting of k segments, each of which is a shortest trajectory [22]. Detailed definitions, proofs of correctness, and discussions on mo re advanced types of trajectory grammars is given in [18]. 22

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Table 4 Grammar of shortest trajectories Gt(1) L Q Kernel, k F F 1 Q 1 S (x,y, l ) A (x,y, l ) two 2 i Q 2 A (x,y, l ) a (x) A ( next i (x, l ), y, f ( l )) two 3 3 Q 3 A (x,y, l ) a (y) V T ={ a } V N ={ S A } V PR Pred ={ Q 1 ,Q 2 Q 3 }, Q 1 (x, y, l ) = (MAP x,p (y) = l ) (0 < l < n) Q 2 ( l ) = (l 1) Q 3 = T Var = {x, y, l } F = Fcon Fvar Fcon = { f next 1 ,..., next n } (n = |X|), f ( l ) = l 1, D( f ) = Z + Fvar = {x o ,y o ,l o ,p} E = Z + X P Parm : S Var A Var a {x} 23

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Table 4 (cont.) L = {1,3} two two = {2 1 2 2 ... 2 n } At the beginning of derivation: x = x o y = y o l = l o x o from X, y o from X, l o from Z + p from P. next i is defined as follows: D ( next i ) = X Z + X 2 Z + P SUM = {v | v from X, MAP x o ,p (v) + MAP y o ,p (v) = l o }, ST k (x) = {v | v from X, MAP x,p (v) = k}, MOVE l (x) is an intersection of the following sets: ST 1 (x), ST l o l +1 (x o ) and SUM if MOVE l (x) = { m 1 m 2 ..., m r } then next i (x, l ) = m i for i r and next i (x, l ) = m r for r < i n, else next i (x, l ) = x. endif There are several points that require clarification before an example is shown. First is the definition of the MAPx,p(y) function. MAPx,p(y) is equal to k, such that y is 24

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reachable from x in k steps, but not reachable in k 1. For instance, MAPx,p(y)=1 for all points y such that Rp(x,y)=True; MAPx,p(y)=2 if there is a point z such that Rp(x,z)=True and Rp(z,y)=True; and so forth. A second necessary clarification is that the number of productions in this grammar is not 3 as it seems at first glance. Rather, there are several productions 2i (the set two ). This number is limited by the size of set X; however, it can vary at different steps in the generation process. There is one production 2i for every diff erent value of the function nexti. The function nexti is the heart of this grammar. Function nexti returns the ith member of the set MOVEl(x), which contains all possible locations for the next step in the trajectory. This set is determined as an intersection of three sets: ST 1 (x), St lol+ 1 (x o ) and SUM. The set SUM is a set of all points v such that MAP x o ,p (v) + MAP y o ,p (v) = l o i.e. the set of all points such that the distance from the beginning of the trajectory to this point added to the distance from this point to the end of the trajectory is equal to the total length of the trajectory. It can be easily shown that this is a set of all the points on all the shortest paths of length l0 from x0 to y0. This set is shown as the long ellipse in Figure 6 25

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MAP (v) + MAP (v) =MAP (v) = k x x x MAP (v) = 1 next next y xx ,p 0y ,p 0 0x ,p 0 1 2x ,p k-1 k-1 1 00 0l Figure 6 Interpretation of the algorithm for nexti for the grammar Gt(1) To illustrate the meaning of the other two sets, consider a situation where the first k 1 points of the trajectory have been constructed (x0, x1,,xk 1), and we are interested in finding all possible points v for the kth step. Clearly v belongs to the set SUM. Since the point v has to be on the kth location along the shortest path, the distance from x0 to v has to be equal to k. Therefore, v is in the set STk(x0)={v | MAP x0,p (v) = k} (shown as the rectangle in the magnified view on the right side of the figure above) Furthermore, point v is reached on the kth step from location xk 1; as a result, v has to be reachable from xk 1 in one step. That is, v is in the set ST1(xk 1)={v | MAP xk 1,p (v) = 1} (shown as the small circle in the magnified view). In the example above, MOVE contains two elements. This implies that at the kth step, the generation can branch into two separate trajectories by applying e ither production 21 or 22. To demonstrate how the controlled grammar Gt (1) actually generates shortest trajectories, it has been applied to the situation in Figure 5 in a step by step fashion. The start symbol is S(x,y,l) = S(88,36,5). We start with production 1, and follow to production 26

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21 out of set two Since Q1=True ( MAP 88,p (36) = 5), we took the branch corresponding to FT in the first production: S(88,36,5) 1 => A(88,36,5) 2i=> a(88)A(nexti(88,5),36,f(5)). At this point we need to compute the parameters using function f and next1. The first function is trivial: f(5)=5 1=4, which means that the remainder of the trajectory needs to be of length 4. To compute nexti, we need to construct the set MOVE by computing 3 different sets mentioned above. Let us compute MAP x0,p (v) and MAP y0,p (v), i.e. MAP 88,p (v) and MAP 36,p (v). These maps are necessary for the construction of the set s SUM and STk. Figure 7 Values of MAP88,p Figure 8 Values of MAP36,p Now, we can compute the set SUM as a set of all points v on the board such that MAP 88,p (v)+MAP 36,p (v)=5. In order to complete construction of the set MOVE, we need 2 more sets: ST 1 (x), ST l o l +1 (x o ), i.e. ST 1 (88) and ST 5 5+1 (88) (in this case they are identical). Note, that the same set SUM is used on every step of the generation. However, the other two sets that form MOVE do change. 27

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Figure 9 SUM Figure 10 ST1(88) From the above figures, it is easy to see that the set MOVE={ 78}. As a result, there is only one value of function nexti=78 and there is only one production in the set two 21. Therefore: a (88)A(next1(88,5),36,f(5)) = a (88)A(78,36,4). We can continue the derivation by applying rule 2i again: a (88)A(78,36,4) 2i=> a (88) a (78)A(nextii(78,4),36,f(4)). At this step the set MOVE is found as intersection of sets SUM (same as above), and ST 1 (78) and ST 5 4+1 (88) (Figure 11 and Figure 12 ). As on the previous step, this set contains only one value: MOVE={68}. As a result there is only one value of next1 and only one rule 21. Therefore a (88) a (78)A(next1(78,4),36,f(4)) can be rewritten as a (88) a (78)A(68,36,3) and derivation continued as before: a (88) a (78)A(68,36,3) 2i=> a (88) a (78) a (68)A(nexti(68,3),36,f(3)). Next, the sets ST 1 (68) and ST 5 3+1 (88) (Figures 2.11 and 2.12) need to be computed. However, this time the intersection of the 3 sets form the set MOVE with more than one element: MOVE={57,58}. Therefore, the function next can take on two differe nt 28

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values. This means that at this step the shortest trajectory can go two different ways. By applying production 21 or 22 we can achieve the following derivations: a (88) a (78)A(68,36,3) 21=> a (88) a (78) a (68)A(57,36,2), or a (88) a (78)A(68,36,3) 22=> a (88) a (78) a (68)A(58,36,2). In practice, the generation has to branch every time when the set MOVE contains more than one element. The total number of shortest trajectories in the bundle is multiplied every time such condition is encountered. In this case there w ill be at least two shortest trajectories generated. For this demonstration, let us pick the first of these trajectories and proceed with the generation. Figure 11 ST1(78) Figure 12 ST2(88) 29

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Figure 13 ST1(68) Figure 14 ST3(88) The next iteration of applying the rule 2i produces the following expansion: a (88) a (78) a (68)A(57,36,2) 2i=> a (88) a (78) a (68) a (57)A( nexti(57,2),36,f(2)). Figure 15 ST1(57) Figure 16 ST4(88) The set MOVE={47} (Figures 2.13 and 2.14), therefore a (88) a (78) a (68) a (57)A(nexti(57,2),36,f(2))= a (88) a (78) a (68) a (57)A(47,36,1). The next application of rule 2i produces the following expansion, where next1=36, since MOVE={36} (Figures 2.15 and 2.16): a (88) a (78) a (68) a (57)A(47,36,1) 2i=> a (88) a (78) a (68) a (57) a (47)A(nexti(47,1),36,f(1)) = a (88) a (78) a (68) a (57) a (47)A(36,36,0). 30

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Figure 17 ST 1 (47) Figure 18 ST 5 (88) At this point, production 2i can no longer be applied due to the fact that Q2=( l 1)=(0 1)=False. Therefore, rule 3 from FF of rule 2i is applied. There are no more non terminal symbols present and generation is complete: a (88) a (78) a (68) a (57) a (47)A(36,36,0) 3 => a (88) a (78) a (68) a (57) a (47) a (36). The second trajectory that is generated by choosing a (88) a (78) a (68)A(58,36,2) on the 4th step (instead of a (88) a (78) a (68)A(58,36,2)) is a (88) a (78) a (68) a (58) a (47) a (36). Therefore the entire bundle of shortest trajectories for piece p from location 88 to 36 is trajectories is tp(88,36,5)={t0,t1} (Figure 2.3), where t0= a (88) a (78) a (68) a (57) a (47) a (36), t1= a (88) a (78) a (68) a (58) a (47) a (36). Note, that this grammar is completely universal with respect to the problem domain. It will produce the shortest trajectories for any system for which the set of locations X and reachability relation Rp(x,y) is defined. Other grammars can be applied in sim ilar fashion to generate longer trajectories, such as admissible trajectories of order k 31

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This demonstrates the general method of employing controlled grammars to generate strings belonging to the languages in the LG Hierarchy of Formal Languages. 2.2.2 Languag e of Zones The next level in the LG Hierarchy of Formal Languages is the Language of Zones [14, 18]. The strings of this language represent a network or a set of interconnected trajectories. One of these trajectories is known as the main trajectory and the others are negation trajectories Intuitively, the main trajectory represents a path that the main piece needs to take to accomplish a certain goal. The 1st negation trajectories represent paths for the opposing pieces that can disrupt the main piece from arriving at its destination. The purpose of kth negation trajectories is to prevent a trajectory of k 1 negati on from accomplishing their interception. Formal definitions of zones and supporting concepts are given in [18]. Consider the zone shown in Figure 2.17 for a Totally Concurrent ABG with strikes (zero time moves that destroy another object). The main goal within the zone is for the Gray Bomber p0 to destroy the Black Tank q0. To accomplish this goal the Gray Bomber needs to move along trajectory a (1)a (2)a (3)a (4). This brings it within the strike range of the target which can be destroyed by the strike 4 5. The zone also contains two Black elements that are capable of destroying piece p0 before it reaches its target q0 namely, q1 and q1 with 1st negation trajectories a (6)a (7) a (8) and a (9)a (10) a (11) respectively. However, during the construction of the zone, Language of Zones also generates 2nd negation trajectory for the Gray Plane p1 which allows it to intercept the Black Plane q2 and therefore, stop q2 from intercepti ng the Gray Bomber a (12) a (13) a (14). At the next 32

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level, the generation is able to employ Black Plane q3, which was not able to intercept the Gray Bomber due to time constraint. However, a 3rd negation trajectory can be constructed for q3 to stop p1 from i ntercepting q2, so that q2 can intercept p0 to protect q0. Finally, a 4th negation trajectory is added seeing as Gray Plane p2 can assist the Gray side by destroying q3 and canceling the chain of events above. Figure 19 LG zone for TC system with strikes An LG zone has several constraints. The first constraint is that every trajectory represents a valid path for the ABG (generated by the LG Language of Trajectories). The second major cons traint is that the zone timing is maintained. This timing constraint is necessary to e nsure that the interception is actually possible i.e., the interceptor arrives in time to destroy the target. In general, the length of any negation trajectory t must be less than or equal to the number of moves that the acting piece on the trajectory negated 33

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by t has to make for reaching the target location of t A zone is strict if the length of t is not only limited by the above number but is strictly equal to it. Strict zones correspond to the ABGs where a game element cannot wait for its target to arrive at the location of intercept, e.g. fighter plane may not be able to hover in the air waiting for the bomber to come within range. In the zone above, the length of the 1st negation trajectory of q1 is equal to 2, which is equal to the number of steps p0 has to make to arrive at the intercept location (3). Likewise, the length of the 4th negation trajectory of p2 is 1, which is equal to the number of steps for q3 from (15) to (16). Different ABG may require modification of the timing constraints. For instance, in chess the zone is constructed in a way that the length of the trajectories of the pieces of the same color as the main piece cannot exceed 1 due to the Alternating Serial nature of the game. As with the language of trajectories, there is usually not a single zone, but rather a bundle of zones. Consider that the trajectory for the main piece to the target is usually a bundle of trajectories with the same source and target. As a result, there is usually a bundle of zones with each of those trajectories as the main trajectory. The same principle holds for the negation trajectories. When LG strategy is derived from the zones, either individual zones or the entire bundles are analyzed. The important aspect of the zones is that they describe relationships between the pieces in the ABG at the same time providing information on how to exploit these relationships. For instance, if the abo ve zone were generated within a game we would know immediately that the Black Side would win because there is an intercept trajectory with no counter intercept trajectories possible. Moreover, we can 34

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see exactly how the Black Forces must behave to accomp lish this protection. For instance, the plane q1 is clearly the most important element. The Gray side can also see how support fighters can attempt to increase Gray Bombers chances by counterinterception. Even though q1 will be able to shoot at p0, the r est of the zone may still be important if the destruction is probabilistic (rather than unconditional as in chess). The bundle of zones shows exactly how the interception and counter interception are possible. The analysis becomes more complex when zones w ith different goals are present and are interconnected. In Defense Systems usually more than one target may be given and available resources must be distributed between different tasks. However, analysis of zones may provide a strategy, which uses the sam e resource for several tasks (i.e. in several zones) simultaneously. This is usually achieved by the tree evaluation procedure which favors moves that allow involvement in several zones. In addition to attack zones described above, other zones are possible, such as block / relocation domination, retreat and unblock zones ( see Figure 20 ). Block/relocation zones differ from attack zone in that the main piece is not attempting to destroy a target, but rather just move to the given endpoint. The opposite side is attempting to block this relocation. The domination zone is essentially a relocation zone to the endpoint that allows the main piece to provide domination of another game piece. Retreat and unblock zones have the goal of moving the main element so as to save it or clear the path for another piece. Other similar networks may be used for a decomposition of an ABG for a variety of problem domains. 35

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1 3 4 2 q 2 1 3 4 p 2 2 q Attack Block or Relocation 1 3 4 q p 2 2 q 2 Domination q 0 p 2 q 1 p 1 p 0 q 1 p 0 q 3 Unblock Retreat q p 0 q 3 q 3 q 1 p 0 2 1 p 1 Figure 20 Various types of LG zones The previous section presented form alization of trajectories as a string of symbols in the Language of Trajectories. Similarly, a zone introduced above can be represented as strings in the Language of Zones using the relation of trajectory connection. Two 36

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trajectories are considered connect ed, if the endpoint of the first trajectory coincides with an intermediate point of the second trajectory. In general zones are represented in the following form: Z = t (p0,t0,0) t (p1,t1,1) t (pn,tn,n) where t is a terminal symbol, pi are game pieces from the set P, ti are trajectories from the Language of Trajectories for the corresponding pieces, and i are non negative integer time bounds on the corresponding trajectories. The first symbol, t (p0,t0,0) corresponds to the main trajec tory of the zone, while t (p1,t1,1) t (pn,tn,n) correspond to the negation trajectories. The time bound i for ti represents the notion of time constraints presented above, which insure that the negation trajectories can intercept on time. Intuitively, i is the number of game moves that the piece pi has to intercept its target trajectory. The particular way i is calculated depends on the class of the ABGs and the level of negation. The Grammar of Zones computes these values automatically during derivation of a particular zone. The zone from Figure 2.17 can be re presented in the following way: Z = t (p o a (1)a (2)a (3)a (4)a (5), 4) t (q 1 a (6)a (7)a (8)a (3), 3) t (q 2 a (9)a (10) a (11) a (3), 3) t (p 1 a (12) a (13) a (14) a (11) 3) t (q 3 a (15) a (16) a (13) 2) t (p 2 a (17) a (18) a (16), 2) The concept of zones has been thoroughly presented above. The controlled Grammar of Zones GZ that generates such zones for a given Alternating Serial ABG is presented in Table 2.5. Since the method of application of such control grammars was presented in the previous section, an example of applying GZ is not shown. Such examples as well as declarative definitions of the Languages of Networks and Zones, related 37

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concepts (e.g. trajectory connections and time constraints) are given in [18]. Productions 1 and 2i generate the main trajectory. Productions 3 and 4j are used to generate the 1st negation trajectories, while production 5 provides a way to switc h to higher negation. Subsequently, combination of productions 3, 4j, and 5 is used to generate trajectories of all higher levels of negation. The generation is terminated using production 6 when no more trajectories of higher level of negation exist (pred icate Q5 is False). The main intricacies of this grammar are generation of connection points and maintaining time bounds on the negation trajectories to each of those points. Table 5 Grammar of Zones GZ L Q Kernel, k ( z X) (z X) F T F F 1 Q 1 S ( u, v, w ) A ( u, v, w ) two 2 i Q 2 A ( u, v, w ) t ( h i o (u), l o +1) TIME (z) = DIST (z, h i o ( u )) 3 A ((0, 0, 0), g ( h i o ( u ), w ), zero ) 3 Q 3 A ( u, v, w ) A ( f ( u, v ), v, w ) NEXTTIME (z) = four 5 init ( u NEXTTIME (z)) 4 j Q 4 A ( u, v, w ) t ( h j ( u ), TIME (y))) NEXTTIME (z) = 3 3 A ( u, v g ( h j ( u ), w )) ALPHA (z, h j (u), TIME (y) l +1) 5 Q 5 A ( u, v, w ) A ((0, 0, 0), w, zero ) TIME (z) = 3 6 NEXTTIME (z) 6 Q 6 A ( u, v, w ) 38

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Table 5 (cont.) V T = { t }, V N = { S A }, V PR Pred ={ Q 1 ,Q 2 ,Q 3 ,Q 4 ,Q 5 Q 6 } Q 1 ( u ) = (ON(p o ) = x) (MAP x,p o (y) l l o ) ( q ((ON(q) = y) (OPPOSE(p o q)))) Q 2 ( u ) = T Q 3 ( u ) = (x n) (y n) Q 4 ( u ) = ( p ((ON(p) = x) ( l > 0) (x x o ) (x y o )) (( OPPOSE(p o p) (MAP x,p (y) = 1)) (OPPOSE(p o p) (MAP x,p (y) l ))) Q 5 ( w ) = ( w zero ) Q 6 = T Var = {x, y, l, , v 1 v 2 ..., v n w 1 w 2 ..., w n }; for the sake of brevity: u = (x, y, l), v = ( v 1 v 2 ..., v n ), w = ( w 1 w 2 ..., w n ), zero = (0, 0,..., 0); Con = {x o ,y o l o ,p o }; Func = Fcon Fvar ; Fcon = { f x f y f l g 1 g 2 ... g n h 1 h 2 ... h M h 1 o h 2 o ... h M o DIST, init, ALPHA }, f = ( f x f y f l ), g = ( g x 1 g x 2 ... g x n ), M = |L t lo (S)| is the number of trajectories in L t lo (S); Fvar = {x o ,y o l o ,p o TIME NEXTTIME } 39

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Table 5 (cont.) E = Z + X P L t l o (S) is the subject domain; Parm : S Var A { u, v, w }, t {p,, }; L ={1,3,5,6} two four two ={2 1 ,2 2 ,...,2 M }, four = {4 1 ,4 2 ,...,4 M }. D( init ) = X Z + Z + (0,0,0). if r, (0,0,0), if 2n, r) ( init u u u D( f ) = (X X Z + {0, 0, 0}) Z + n f(u, v)=(x1,y,l), if ((xn)(l>0))((y=n)(l0))(1, y1, TIME(y1)vy1), if (xn)((l0)(yn)). DIST ) = X P L t l o (S). Let t o L t lo (S), t o = a (z o ) a (z 1 )...a (z m ), t o t p o (z o z m m); If ((z m = y o ) ( p = p o ) ( k (1 k m) (x = z k ))) (((z m y o ) ( p p o )) ( k (1 k m 1) (x = z k ))) then DIST (x, p o t o ) = k+1 else DIST (x, p o t o ) = 2n. D( ALPHA ) = X P L t l o (S) Z + ALPHA(x,po, t o k ) max ( NEXTTIME ( x ) k ) if ( DIST ( x, p o t o ) 2 n) ( NEXTTIME ( x ) 2 n ) ; k, if DIST ( x, p o t o ) 2 n) ( NEXTTIME ( x ) 2 n ) ; NEXTTIME ( x) if DIST ( x, p o t o ) 2n ) g r ) = P L t l o (S) Z + n r X. gr ( p o t o w ) = 1 if DIST ( r p o t o ) 2 n, w r if DIST ( r p o t o ) 2 n. h i o ) = X X Z + ; Let TRACKS p o = {p o } ( L[ G t ( 2 ) (x, y, k, p o )]; 1 k l If TRACKS p o = b. i ), t (p b, i ), t (p h M) (b )}, t (p ),..., t (p ), t {(p TRACKS ) ( b o i o i b o 2 o 1 o oopif if else then ) ( u andu hoi 40

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Table 5 (cont.) D( h i ) = X X Z + ; Let TRACKS p = {p} ( L[ G t ( 2 ) (x, y, k, p)]; 1 k l If TRACKS p = then h i ( u ) = else TRACKSp { ( p1, t1) ( p1, t2) . ( pm, tm) } ( m M ) and hi( u) ( pi, ti) if i m ( pm, tm) if i m At the beginning of generation : u = (x o y o l o ), w = zero v = zero x o X, y o X, l o Z + p o P, TIME (z) = 2n, NEXTTIME(z) = 2n for all z from X. 2.3 LG Strategies There are two methods that can be used to apply the Hierarchy of Formal Languages to construct strategies. The first, more traditional approach is to construct search trees employing the Languages of Translations and Searches using the generating tools presented above. By following this technique, reduced search trees can be generated of sizes significantly smaller than those of Alpha Beta search. The second approach is to construct a solution without any search at all. The construction of strategies is achieved by decomposition of the State Space in the form of the State Space Chart, which is based on the expansion of the terminal sets. Neither of these approaches is presented here, as both of t hem are discussed in detail in [18], and a full understanding of strategy construction is not essential for the reader to follow further discussions in the subsequent secti ons It is sufficient to mention that these methods make significant use of Languages of Zones and Trajectories presented in previous sections. 41

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2.4 Historical Validation of LG Since the conception of Linguistic Geometry, this theory has been extended and utilized in a multitude of real world application s Considerable ev idence, both theoretical and experimental, has been gathered to demonstrate that the LG software tools provide highly effective scalable solutions and a faithful model of an intelligent ene my [11, 23] These applications have included complex military and industrial problems and have gar nered international recognition from such organizations as US Air Force Scientific Advisory Board US Army Science Board and UK Defence Science and Technology Laboratory [24]. These boards define national policy in the defense related research and its tra nsition to the US Armed Forces. Since 1999, LG based technology has been s uccessfully tested in more than 30 government and commercial defense related projects [25]. The most significant and thorough validation of the Linguistic Geometry was accomplished in a series of war gaming experiments within the DARPA (Defense Advanced Re search Projects Agency) RAID (Real time Adversarial Intelligence and Decision making) program [3, 25]. The LG generated courses of action (COA s ) significantly exceeded that of COA s developed by the human commanders and staff. Results of those experiments h ave led to multiple follow up utilizations of the Linguistic Geometry across various programs within the US Army research and development organizations for intelligence, mission command and control, and training applications. As previously mentioned, the R AID experiment events have provided the most thorough opportunities to validate the accuracy and benefits of the LG generated solutions [3, 26] These events consisted of a series of comparative trial runs. In each case 42

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two teams of current and past military personnel Blue and Red participated in wargames against e ach other employing OTB (OneSAF Testbed Baseline, www.onesaf.org ) simulation software to provide a representation of a real world environment. In the baseline case, the Blue force commander was assisted by a team of five advisers acting as Staff officers. During the LG portion of the experiment, the Blue commander was assisted by the LG based courses of action generation and analysis software instead of the human aid e s. In both cases human or LG assisted, the Blue c ommander controlled entities within the simulation by ordering OTB operators (commonly referred to as pucksters) to execute his plan. On the Red side, the enemy commander similarly employed pucksters to control Red forces within OTB in an attempt to defeat the Blue forces. A model of 16 square kilometers of an actual city was utilized to provide a complex urban operational environment. The Blue simulated force was equivalent to a US company with about 30 to 35 infantry fire teams, strykers and helicopters. The Red force consisted of several kinds of insurgents with approximately 30 teams of various sizes. Four such experimental events took place in April 2005, July 2005, February 2006, and July of 2006. LG based software demonstrated intelligence by far exce eding hum an developed courses of actions as shown by the statistical analysis of the sophisticated scoring of the outcomes of each experimental run [3] 43

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Table 6 RAID Experimental Scoring Criteria Attack Mission Weight D efense Mission Weight Red Casualties 40% 10% Collateral Damage 10% 10% Blue Casualties 35% 35% Advance To Objective 5% 0% Time to Complete the Mission 10% 5% Facility Protection 0% 40% Out of the 18 paired simulation runs (2 hours each) conducted in Experiment 4, the LG assisted Blue commander outperformed the commander with a human staff 14 times (78%). In 5 out of these 14 paired runs, the human Blue team had lost to the Red team, whereas the LG assisted Blue commander had won. In many other paired runs out of those 14, while both teams had won over the Red, the LG assisted Blue team scored significantly higher. On average, f or all the 18 paired runs, the c ommander with LG software achieved score s that exceeded the score of the commander with the staff by about 10% one standard deviation. Out of the 4 paired runs where the s taff outperformed LG the difference in score for 3 of them was under 3%, and only one run had the difference of about 10%. Overall, the level of confidence in correctness of the LG generated COA s was 98% [26]. It is crucial to note that during this experiment, the Blue commander was obligated to follow the LG generated course of action, so that these scores could be used to directly judge the quality of LG solutions as compared to a team 44

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of subject matter experts (SMEs). Table 7 and Table 8 summarize the experimental results for two of the other experiments which further support the above conclusions [3] Table 7 RAID Experiment Results 45

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Tab le 8 RAID Experiment Results There was another interesting aspect o f these experiments which served as a certain variant of a Turing Test. During each of these runs, the Red team was isolated from the Blue team and was not informe d of or could in any way find out whether they were playing against Blue staff generated or LG generated courses of action. At the end of every scenario, the Red commander was asked which kind of the opponent he was playing against human team or LG assisted commander. In 16 out of 36 cases (44%), the Red Commander was wrong [26]. 46

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CHAPTER 3 APPLICABILITY AND EXTENSIBILITY OF LG 3.1 Complexity of Modeling Real World Systems as ABGs Previous chapters have introduced the theoretical foundations of Linguistic Geometry as well as some of the experimental evidence as to the high quality of the solutions provided by this methodology. LG based technology has been successfully employed in numerous government and commercial defense related projects [25]; however, each such application relies on more than just the core LG theory. In order for this approach to be applied to each particular problem domain, one must first model the re al world characteristics of such environment as LG Abstract Board Games. Furthermore, the tools and methods required to achieve such mapping from the complexity of the real wor l d to the rigid mathematical definitions of the ABGs vary depending on the parti cular domain under consideration. For example, consider the difference in requirements between naval vs ground, urban vs rural, or air vs outer space operations. In this chapter we explore and address a wide spectrum of modeling issue s required to succes sfully apply Linguistic Geometry to such real world problems The majority of these problems arise from the difficulty of mapping the vast breadth and depth of items, actions, and effects present in the real w orld into the strict mathematics of Abstract Bo ard Games required as the basis for the LG application. This research will build on and extend previous research as to the specific techniques and methodologies that allow basic theory of Linguistic Geometry to be extended so as to achieve the high 47

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levels of applicability to the wide range of problem domain s The key purposes for this work i s to present these extensions and methods in a rigidly organized and structured fashion to provide the highest benefit for community of practice attempting to apply Ling uistic Geometry to both previously explored and novel problem domains. T he first requirement for solving the problem is to model it in the form that can be solved using specific tools tools. The first step to applying LG is to adequately model the real wo rld system in discrete format of the Abstract Board Game, which can be represented as 8 tuples < X, P, R p SPACE, val, S i S t TR> Note, that this 8 tuple can be further expanded based on the particular system requirements, such as adding strikability relations for the long distance shooting and other extensions described in subsequent sections. The first 3 elements of this 8 tuple X, P, and Rp are the most critical elements. Once the operational board, the set of pieces, and the capabilities of th ese pieces are defined, the remaining game elements are easier to specify and LG can be applied to the overall system. As the purpose for modeling a real world system as an ABG is to employ LG to generate winning strategies, we must keep in mind tractabili ty as one of the primary design constraints. Larger sizes of the sets X, P, and Rp cause increased requirements for computer memory, processing power and time. A common theme through the rest of this work is to find the balance between limiting complexitie s of the ABG for practical performance considerations and achieving sufficient complexity to faithfully represent the events of a real world system. 48

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The initial development of LG was based on chess games, so unsurprisingly, chess and similar board games can be easily modeled as ABGs. The main reason is that these games have a welldefined and accepted set of rules, set of possible game elements on a specific board, and standard goals. Let us now explore the challenges of such modeling without the help o f such existing frameworks. 3.2 Concurrency Section 2.1 introduced the concept of concurrency categories as related to ABGs. The games can be Alternati ng Serial (AS) Alternating Concurrent (AC), or Totally Concurrent (TC) systems. Note, that even large AS games have significantly lower branching factors than smaller TC games. For instance, an AS game with 2 players controlling a total of 400 game pieces each of which can make 5 distinct moves has the branching factor of (400/2)*5=1000, whereas the same entities in a TC game would result in the overall branching factor equal to 5400=3.9x10279. As a matter of fact, the branching factor of 1000, for a game with each piece capable of 5 distinct moves, would be exceeded with just 5 game pieces in a TC ABG (branching factor of 55=3,125). AC games are less complex than TC; however, they are much clo se in complexity to TC than to AS games. For the same example above, the AC game branching factor would be 5200=6.2x10139. The real world rarely presents a problem where only a single agent can move at the same time typically all actors such as cars, sh ips and people, can all move simultaneously Serial games are not well suited for modeling problems and concurrent games must be employed. However, the added complexity of such games as described 49

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above, should serve as a reminder to minimize, as much as f easible the size of the ABG set of P and reduce complexity of Rp. 3.3 Spatial Discretization Let us now consider the discretization problem. Linguistic Geometry as well as other discrete game approaches requires both the space and time to be broken up into di screte segments. Spatial and temporal discretization problems have to be addressed together due to the effect that the tight interconnection between them has on the ability to model movement and actions in the resultant ABG. For example, if the game board employs cells of 1 km in size and a 1 min time step, then the smallest speed of movement that can be represented is 1 km/min. However, if the time step is increased to 2 min, or the cell size reduced to 0.5 km, then this speed is reduced to 0.5 km/min. 3.3.1 Reg ular Grids The Abstract Board in the ABG is just a finite set of points. However, it is typically more convenient to model it as a type of a regular grid due to the need of mapping the LG strategies back to the real world. This modeling requires breaking up the entire surface or volume into sub regions or cells. One such grid, akin to a chess board, is a regular rectangular grid with square cells. Another type of grid may employ hexagonal cells. Let us recall that the purpose of such discretization of spac e is to model movement and actions of entities as discrete transitions (jumps) from one cell to another during a single time step. As such, it can be observed that there are distortions of space inherent to both of such typical grids. 50

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3.3.1.1 2D Grids In the case of a rectangular grid, a distance to the adjacent cell in a diagonal direction is about 41.4% longer than a distance to the adjacent cell in the orthogonal (rank and file) directions. Using a reachability of moving one cell at a time in all 8 directions fo r a real world entity would be very misleading as such an entity would be able to move 41.4% faster in some directions leading to incorrect movement paths. Essentially, an entity could cheat by moving diagonally as much as possible in order to take advan tage of such a speed boost. This relative distortion is reduced as the radius of such intended approximated circle is increased. For example, the longest range of the edge of the dark grey area in Figure 21 is only 13% longer than the shortest range to the edge. While only a partial improvement, we could employ a technique such that all the movements utilize at least 4, 5, or more cells per single time step to minimize effects of distortions. However, such improvement comes at the cost of significant increase in overall size of the board which is usually correlated with performance or computing memory requirements. For instance, reducing the cell size by 4 in order to model movement employing 4 cells instead of 1 would effectively increase the total number of cells for the same area by a factor of 16. Using a hexagonal grid can help reduce some of such problems due to the primary property of such grids that each cell is surrounded by six equidistant neighbors. Therefore, movement of 1 cell per time step has no distortion of distance. However, such lack of distortion is misleading as it is limited to single cell neighbors. If you consider, the radius of 2 cells, the longest range is about 15% longer than the shortest. The distortion can be 51

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reduced similarly to rectangular grids by employing larger rang es to better approximate circles. Figure 21 Rectangular and Hexagonal Regular Grids The problems with grid based discretization do not stop at just linear distance distortions angular distortion and obstacle representation bo th pose a problem to discrete based modeling of real world environments and accurate movement representation. Consider representation of movement along a given vector. If such a vector is aligned with some primary directions 8 in rectangular grid or 6 in hexagonal grid the lines can be represented accurately. Yet the majority of lines would not follow one of such directions and as can be seen in Figure 22, such a line has to be rasterized which creates jagged, zigzag movement rather than a true straight line. It should be noted, that just as with distance distortions, the line rasterization distortion can be addressed by reducing the sizes of cells and modeling movement as 4 or more cells at a time, rather than 1 per game turn. 52

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Figure 22 Rectangular and Hexagonal Line Rasterization 3.3.1.2 3D Grids While 2D Abstract Boards may be sufficient for some problems, a much higher level of precision can be achieved with 3D discretization. For some Defense Systems this extra degree of realism is essential. One class of such systems is air operations. As menti oned above, some problems with air units can be approximated with a 2D game; however, that is not always the case. Consider a scenario, where a group of airplanes is flying through a mountainous area with surface to air defenses. A 3D representation will a llow us to handle complex behavior such as flying through the canyons to stay out of line of sight of surface radars. Furthermore, the reachability relations for cells at higher altitudes may allow faster flight than those at the lower altitudes above grou nd level. Some weapons can only be fired from aircraft at targets that are within a certain cone of attack, so that the plane has to adjust its position before strike. If we attempt to model these types of games with a pseudo 2D ABG, we would have to intro duce the second pseudo layer in which any position above a given cell will have identical properties. While we can model one of the aspects mentioned above (such as canyons by noting ground level at every 2D cell), we cannot model all of the potential diff erent behaviors over that 53

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cell. Using a 3D model, we can distinguish and plan our strategy to take advantage of such effects as flying through this location at 100 ft above ground level to avoid radar, moving to attack altitude, or accelerating to burst speed at 20,000 ft and keep track of line of sight visibilities at all times. Several approaches can be used to model 3 D space. One way is to use a dense packing of space by certain shapes (usually approximating spheres). The goal of such packing is to ach ieve an effect similar to hexagonal grids on a plane every cell is equidistant from all its neighbors. Such discretizations are usually complex and counterintuitive for most people. More importantly, they are unnecessary and often not appropriate for real world Defense Systems. The reason is that in real life 2 horizontal directions are very similar, while altitude is quite different. The aircraft may be able to move at 500 miles per hour in any direction at any given altitude; however, it is usually only able to change altitude at the rate of about 20 miles per hour. For the same reason, it is also common to use a different scale in the horizontal and vertical dimensions. Since motion in vertical direction is so different from horizontal directions, it ma kes more sense to model 3D Abstract Boards as a s tack of 2D B oards. If the 2D B oard is discretized as a rectangular grid, a 3D cell could be a cube or a rectangular prism. Hexagonal 2D discretization produ ces hexagonal 3D prisms ( Figure 23) [2, 3, 11, 27, 28]. Each of the cells represents a chunk of 3 D space and stores the properties of the section of space contained in that region, such as air, water, ground, etc. The reachability relations are then designed to conform to the properties of game elements an airplane can only move through cells fille d with air, while a ground unit can only exist in an air cell immediately 54

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above a ground cell. Furthermore, similarly to 2D games, the concept of Phase Space can also be extended for the 3D games, thus making them 4D (or higher). Figure 23 3D cell: hexagonal prism 3.3.1.3 Spherical Grids So far, all the discretizations have only dealt with either 2 or 3 dimensional areas above a planar surface. We have assumed that the 2D grid is covering a uniform portion of a flat surface. However, the Earth is not flat. Plan e approximations assumed above work well for relatively small area. On the other hand, some Defense Systems may require an operational district that is large enough to notice the curvature. Oth ers may require considering the entire surface of Earth as the operational district. In this case, we have to discretize the entire surface of a sphere (for large, yet not full Earth models, we can use subsets of full spherical mappings) [2, 28, 29]. There are two general classes of approaches to this problem. The first is to map the surface of a sphere onto a plane and then discretize this planar representation. The second approach is to discretize the surface of the sphere directly. The main difficulty of both methods is maintaining the uniformity of cells. As mentioned in the previous section, an important property is for the distances between n eighboring cells to be uniform. We would also like each cell to have approximately equal area and shape. A comprehensive survey of Discrete Global Grids can be found in [30]. 55

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Consider the first approach. The first step in this approach is to map the surface of the sphere onto a planar structure with as little distortions of distances, areas, and shapes as possible. This problem has been studied extensively by researchers in the area of map projections. Unfortunately, there is no projection which maintains distances, areas, and shapes. There are projections that may be able to maintain some of the properties (e.g. equal area projections); however, there is no projection that is equidistant in all directions. As a result, we may have cells, which are not spaced uniformly on the sphere. One of the better projections is the Snyder equal area projection onto a polyhedral globe [31]. The second step of this method is to discreti ze the planar structure into 2D cells. The cells can be of any shape; however, as for 2D B oards, hexagons have some of the most desired properties. The difficulty of this step is usually due to the shape of the planar structure used. For instance, consider projections onto a polyhedral globe. These structures consist of several planar faces that can be triangles, squares, pentagons, hexagons, etc. Each of these faces then needs to be broken down into cells. However, if we attempt to discretize a pentagon as a tiling of hexagons, we will encounter problems along the edges and at the corners. The easiest shape to discretize is either a square (which usually has large projection distortions), or a triangle. Furthermore, there is usually a tradeoff between the quality of the first and second steps. Snyder equal area projection onto an icosahedron (20 triangular faces) has much higher distance distortions than a projection onto a truncated icosahedron (20 hexagonal and 12 p entagonal faces). On the other hand, the icosahedron is easy to discretize consistently into hexagons with no edge 56

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problems, while pentagons and hexagons do not allow for such a consistent discretization. One of the better approaches of this type is presented in [32]. The second approach consists of discretizing the surface of the sphere directly. One popular method is to start by approximating the sphere as polyhedron (usually a platonic solid). Then each face is broken down into several smaller parts to construct a more complex polyhedron. By such successive subdivision, a sphere is approximated by a polyhedr on with a very large number of faces. The key element of this process is the subdivision of the face to produce a number of smaller faces. At each subdivision, some new vertices are produced which need to be projected back to the sphere creating the new po lyhedral structure. This process is somewhat similar to the map projections mentioned above. The difficulty lies in creating the new vertices so that the distances between neighboring vertices remain fairly close to each other The final cells are created from this collection of small faces. Either a cell can incorporate the area covered by 1 or more faces, or a cell may be centered on a vertex and contain points that are closest to it. This approach with an extra opti mization step was presented in [33, 34]. The grid may not exhibit all of the desired properties (such as equal distances and areas); however, it can be used as a starting state for a dyna mic optimization technique based on the requirements for the grid. The cells are adjusted slightly to maximize an evaluation function. 57

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Figure 24 A planet level board with spherical hexagonal cells. 3.3.1.4 Curvature of the Earth While p revious sections presented either grids in the rectangular space or the spherical whole Earth space, the reality of the modeling real world scenarios is that even the rectangular grids represent some portion of the curved Earth surface. The rectangular n ature of grids presented in Sections 3.3.1.1 and 3.3.1.2 is typically employed as an approximation of space above the surface of the planet. Consider that for an observer standing on the ground with the eye level at 1.7 meters, or 5 feet 7 inches, the horizon is only 4.7 kilometers, or 2.9 miles, away. That means that even a small flat area 10 x 10 km2 cannot be accurately represented as a truly 2 dimensional game board. This problem similarly must be considered for higher elevations such as the flight of an 58

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aircraft at various altitudes. The great circle distance (as the bird flies) between 2 locations that are 10 meters above sea level is significantly different than if those locations were 10 kilometers above sea level. Such curvature must be accounted for in the ABG through either a modification of the board structure, such as including the dropping horizon and increasing distances at various altitudes as an integral feature of the 3D board, or by appropriately accounting for it during the rea chability and line of sight calculations. Similarly, the concept of the regular 2D grid is harder to maintain on a spherical surface. Simply using latitude and longitude as the X and Y coordinates for the grid will yield cell sizes that are significantly d ifferent as one moves away from the equator. At equator, o ne degree of longitude corresponds to about 111 kilometers, while one degree of longitude at 60 degrees latitude (e.g. southern Alaska) is only 56 kilometers across. A more common approach to const ruct a grid with regularly space d and equidistant cells is to employ some sort of a map projection. The most common approach used today is Universal Transverse Mercator, or UTM; however, there are many other projections based on different optimization crit eria or applicability parameters. For instance, UTM is only applicable between 80o South and 84o North latitudes. Locations near south or north poles require a different projection system, such as Universal Polar Stereographic. Additionally, each projectio n provides various levels of area equality, distance equality, or angular equality across a certain size of an Earth surface covered. 3.3.2 Terrain Obstacles In the real world, the spaces represented by the game boards are not typically as wide open as those on the chess board. There would usually be obstacles or other 59

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features both natural and manmade that affect movement and the ability to perform other actions. Natural obstacles may include impassible mountains or rivers and lakes, trees and other vegetat ion. There could also be areas which affect movement speed for instance, swamps, dirt or rocks. Manmade features such as buildings, fences, and roads may similarly affect movement and possibility of actions. Such terrain characteristics typically need to be represented on the game board by assigning properties to individual cells for example, a cell could be marked as water and thus be impassible, or marked as road and cause faster movement [3] Unfortunately, the problem of rasterization reappears when one attempts to represent the edges of such features on the game board. This results in jagged edges of all such terrain elements. The problem is intensi fied when multiple features overlap a single cell, or some features are so small that only a small portion of the cell is covered. We will investigate multiple methodologies of various levels of complexity that can be used to address the terrain obstacles problem. Some of the key factors to consider when evaluating suitability of one of these techniques to a particular situation are size of obstacles that can be captured, accuracy of representation of an obstacle, and size of cells needed to achieve desired accuracy. The latter consideration is particularly important as it is directly correlated to the total number of cells needed to represent a given real world geographic area. For practical uses, the storage, memory and performance requirements are typic ally tightly dependent on the total number of cells, also referred to as game board size. 60

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3.3.2.1 Cell Types and Under types The simplest approach to obstacle representation is to assign one or more types to each cell. In the simplest case, each 3 dimensional grid location (such as a hexagonal prism, presented above) could be assigned one of just two types open or closed. In this manner, the cell could either represent a location through which movement is allowed, or a cell which is an obstacle and cannot be moved into or through. Such representation may be quite sufficient for certain types of operations. For instance, when modeling air operations, solid ground can be represented as closed cells, while the ai r a s open cells. Movement patterns can then be easily set up that track movement of aircraft that avoids crashing into the mountains, and LG can then be applied to find the most efficient routes and analyze the strategies for any scenarios in this model [12]. Line of sight can also b e easily computed in such space. 61

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Figure 25 Aircraft movement through using basic cell types An easy expansion to this model is to include various types of closed cells to represent diff erent types of terrain e lements such as roads, rivers, swamps, and others [3]. This would allow for representation of surface based en tities with reachability patterns that are different depending on the surface type. A boat may be allowed to move across rivers and lakes, but not any other terrain types; while a vehicle may be able to move rapidly across roads, much slower while off roads, and not able to traverse any water featur es at all. It should be noted that instead of using the type of the closed cell immediately underground to represent various types of terrain a concept of under type can be applied to the open cells immediately above ground. The open cells would 62

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then ha ve a type attribute for the volume represented by the cell, as well as one (or perhaps more) under type attributes to represent the type of the terrain under the cell. A further expansion would include introduction of various types of open cells air, forest, building, and others. Just as with under types, various entities would then have different movement patterns based on the type of the cell e.g., movement could be slower through the forest and the vehicles can be completely restricted from entering building open cells. This model allows for a very complex environment to be represented; however, the limiting factor is the size of individual grid cells relative to the size of the key terrain features to be represented. Effectively, no terrain element smaller than the cell size can be adequately modeled. For instance, consider 2 roads running parallel to a river or a canal between them. If the width of each of these roads and rivers is 10 meters, and the size of the cell is 50 meters, all 3 of these features will be represented on a single cell and it becomes impossible to set up reachability relations that faithfully represent movements of various entities in such terrain. The only solution is to reduce t he cell size until it is small enough to represen t necessary elements. Two key problems of this approach must now be considered alignment and rasterization (see Section 3.3.1.1). Even if the size of the cell is small enough for any one feature, the edges or location of these features would not necessarily coincide with the edges or area of the cell. For instance, th e edge of a large forest that spans multiple cells may fall in the middle of the cell. Similarly, even a small building or road may be located on the edge between two cells. The r asterization problem occurs when trying to represent 63

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the edges of the feature s (or the entire linear features, such as roads) on a regular grid. These problems all essentially produce the same effect a single cell that is split in half whereas o ne part is covered by one type of a feature while the other part is not. P resented an other way a particular feature partially covers one or more cells. In such cases, there is no clear method as to whether such cell is to be marked with the particular feature type. It may be just as incorrect to assign the type as not to assign it leavin g no effective resolution. It is important to note that the alignment and rasterization problems can never be completely resolved by a further reduction of the size of the regular grid cells there may always be features straddling the cell boundaries reg ardless of how small individual cells are. The effect of these problems can be significantly reduced, but could never be completely eliminated. Typically, the most troublesome features in this respect are of manmade roads and buildings. It is usually not practically feasible to reduce the size of the cells sufficiently to address rasterization and alignment of such small and geometrically precise features. Thus, most of the methods presented in the following sections are focused on the manmade elements. 3.3.2.2 Eggshell Model In the previous section we introduced the concept of under type to represent the type of the terrain under the cell in addition to any type information for the space occupied by the cell itself. In essence this is identical to assigning a type to the bottom face of the 3 dimensional prism representing the board cell such as the hexagonal prism in Figure 23. This can be further extended by assigning a type to each face of such cell 64

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the bottom, the top, and any sides. For a hexagonal 3D cell, this would associate 9 types with each cell one for the volume of the cell, and 8 for all of the walls, floor and ceiling. We will refer to this met hod as the eggshell model due to the emphasis on the external perimeter of the 3 D cell. The primary benefit of this method is to improve modeling of manmade structures such as buildings, including their internals. Building hulls pose a higher degree of difficulties due to the small thickness of the walls. As we have already discussed, no obst acle smaller than a cell can be correctly represented. Thus, in order to represent a building hull we would need the cell size to be comparable with the thickness of the wall so that there would be a solid cell representing the wall, and 2 open cells o n either side representing locations just inside the building and just outside. This approach would require the size of the cell to be on the order of several centimeters exclusively due to the need to represent the walls as entire cells. In practice thi s is excessively wasteful, as such resolution is not needed for any other part of the terrain tens of meters for a single cell is typically sufficient to adequately represent all other features such as forests, roads, and others. The eggshell model allow s one to deemphasize the walls when selecting appropriate cell sizes the walls of buildings would actually be represented as walls of the board cells [3, 35]. Note, that the types associated with the various faces of the prism can ha ve multiple values such as solid wall, window, door, as well as floor and ceiling to represent various floors in the building. This allows for additional fidelity in modeling by associating different characteristics with each such type for in stance, 65

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window face type can be seen through but not traveled through, while the door fa ce type may imply the converse. The use of the cell faces is not limited to building walls alone other possible uses include other types of obstacles such as fenc es and barriers. Figure 26 Eggshell hexagonal model of an urban environment These benefits have to be considered in light of the alignment and rasterization introduced in previous sections. It is indeed easily possible to employ 20 meter cells to model the space inside and outside of buildings, while relying on cell walls to represent much thinner obstacles presented by walls, of various types. However, if a wall passes directly through the middle of a 20 meter cell, it can only be represented as a cell wall 66

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that is located 10 meters to either side due to lack of alignment. Similarly, a straight fence can only be represented as a jagged sequence of cell edges. Whether these errors in representations are acceptable depends on the p articular application of the models. In some cases, a wall misplaced by a certain distance to the nearest cell edge may be quite acceptable, while in others this discrepancy may require correction most frequently and easily by (adaptive) reduction in the overall cell size. 3.3.2.3 Density Model Every situation calls for a different level and type of discretization. Previous sections were focused on modeling obstacles by using open/closed cells and various types of walls to represent various real world objects as closely as possible In this section, we present an alternate methodology that instead attempts to capture the aggregate effect of obstacles rather than minute details of each individual one. This can be done by assigning additional characteristic of density to each cell in addition to the types presented above [36]. Completely blocked cells would be assigned density of 100%, while complete ly open ones would be c onsidered 0% dense, with anything else in between those values. The density parameter can be computed in the simplest manner by calculating the ratio of the space or volume (2D or 3 D) occupied by the obstacles to the overall area or volume of the cell. However, this simple method can be expanded to provide additional fidelity by considering types and densities of the obstacles themselves. One such example would be to consider a park or a sparse forest such an obstacle has its own density associated with it. Ther efore, when calculating the cell density, the percentage of 67

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the cell space occupied by the forest needs to be scaled by the density of the forest itself. It is interesting to note that even buildings could be considered to possess different densities. Cons ider the difference between an airy building with paper walls, floorto ceiling windows, and an open floor plan compared to a steel and concrete apartment building with dense arrangement of internal walls. These buildings could be considered to have differ ent density when the effect of these obstacles on line of sight and mobility is analyzed. This methodology can further be extended by adding additional characteristics that describe aggregated effect of the obstacles in the cell. For example, various categ ories of densities can be used to represent density of obstacles that provide cover and interfere with weapons engagement densities of objects that interfere with movement of various kinds of entities, density of line of sight obstructions, and others. The effect of the density based cells is, of course, different from the binary nature of the open or closed model. Rather than blocking the movement, visibility, or other actions, the cells of various non zero densities merely impede these actions to some degree. This impedance may be instantaneous or cumulative. For instance, a vehicle may not be able to travel at all through locations of density great er than 75%; however, it could travel through 4 cells of density 20% in a single game turn (depending on the cell size, game turn length and other parameters). Similarly, this model captures how the ability to see is negatively affected by the number and relative density of objects in all the cells along the source to target vector. 68

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The main advantage of this model is that the size of the cells no longer needs to be close to the smallest object being represented; however, some consideration must s till be given to the cell size. There must be enough board locations to represent realistic positioning and movement of entities during the game; and each such board cell should be small enough to capture local variances in the density. The usefulness of the model significantly decreases if all cell densities are approximately equal. For instance, if a typical downtown area is modeled using 500 meter cells, all cells will be very similar in value the effect of roads and other narrow open spaces will be completely diluted by the overall mass of the buildings in each such 500 meter block. The most apparent disadvantage o f this model is its relative lack of precision. Efficiency of the larger acceptable cell sizes come s at the price of replacing individual obstacles with aggregated effect of all objects within a particular cell. However, when considered in light of the alignment and rasterization problems of the previous models, this penalty is actually not as significan t, or completely annulled. Essentially, the fuzziness introduced by the density representation is not dissimilar to the imprecision of representing individual obstacles by the nearest cell or cell boundary. Consider the error of line of sight calculation b ased on the cell density of 5 0%, due to the building occupying half of the cell volume with the error due to snapping the buildi ng boundary from its position in the middle of the cell to the nearest cell boundaries. It is not possible to quantify the inaccuracy of each method in a general case, therefore each particular modeling problem needs to be considered separately. The best overall guidance is that the eggshell model is especially beneficial when cell size is muc h smaller than overall 69

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objects, e.g., representing one or just a few buildings using cells small enough to capture hallways and other internal features. On the other hand, the density model is best used when modeling an area with a very large number of small obstacles, where each individual obstacle is not as significant as their cumulative effect. It is important to note a significant weakness of this model, which may be less immediately apparent. We have focused the discussion above on two categories of obstacles small objects, such as th ose that are completely contained within a cell and take up only a portion of the cells space, and large objects, which completely cover multiple cells and partially cover other cells on the perimeter of the object. There is another type of an object that combines both of those characteristics a long and skinny one, such as a fence or another type of barrier. One such obstacle can easily span across very many cells while only occupying a tiny portion of each one. The density of each cell through which it passes would only be slightly above 0% yet for all intents and purposes, the fence must completely block the movement. An easy solution would be to mark the entire cell as impassible (100% dense); however, this would compl etely defeat the purpose of i ntroducing density based model in the first place. Furthermore, by reverting to essentially basic open vs closed cell methodology, we reintroduce the problems of alignment and rasterization, as well as the overarching requirement for small cells. Marking an entire large density based cell as 100% dense just because 2% of it is occupied by the fence would introduce too much of a modeling error e.g., by erroneously forbidding any movement along the fence. This problem can be partially addressed by cell 70

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size reduction similar to previous methods or by introducing special case for handling such obstacles outside of the overall board cell structure 3.3.2.4 Source Data Submodel This section presents a radically different approach to mapping real world environments into the ABG for using LG [37, 38]. Le t us recall that spatial discretization is not an end in itself. The primary purpose of any such effort is to enable application of LG by producing the following two components from the ABG definition ( Table 1 ): X = {xi} a finite set of points; locations of elements ; Rp(x, y), a set of bina ry relations of reachability (x and y are from X, p is from P) LG does not require any kind of orderly arran gement of locations just an abstract set of points. Any organization of these points is then achieved by the means of reachability relations which define for an entity and a start location whether or not a given target location is reachable in one game m ove As discussed in Section 3.6.4 the concept of reachability can be expanded to other relation s, such as: Visibility an entitys ability to se e (sense) from start to target location and detect an enemy piece there ; Strikability an entitys ability to shoot from start to target location and destroy the enemy piece there. Note, tha t each reachability relation defines movement from one point on the board to another in one game move over one time interval T. Given that most real world entities have a particular maximum speed S, it implies that all reachable locations are within physical distance ST of the start location. While this can be accomplished with a 71

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non uniform arrangement of points on the board, this requirement of equidistance of neighboring cells lends itself easily to fulfilling this requirement. It is also interesting to consider that a nonregular distribution of cells that satisfies the equidistance requirement would necessarily exhibit a somewhat uniform distribution similar to the regular grid. Indeed, Section 3.3.1.3 describe s a method of producing a collection a board points on the surface of the sphere which begins with an nonequidistant distribution and employs optimization techniques to shuffle such a collection until a required leve l of uniformity is achieved Thus the regularity of the grid, while not mandated by the LG ABG concept, is a common technique to provide the set of locations that can be operated on by the LG algorithms. The previous secti ons focused on mapping real life o bstacles into the structure of the board cells. However, there is actually no requirement for the obstacles to be represented within the ABGs only that the reachabilities are defined between any pair of cell s as either true or false. Representation of the real world objects on the board structure serves one function only it provides a model that can then be utilized to efficiently evaluate v arious reachability relation s. The board structures describe d above then help satisfy two key requirements of the ABGs they provide a set of locations that can be occupied by entities and a method for computing reachabilities. Let us now consider an alternative approach that decouples those two goals. We can still utilize a board structure similar to those described in the previous sections so as to produce a finite set of board positions. However, in order to ge nerate reachability relation s, we will use a separate Source Data Submodel. In order to apply any of the board 72

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discretization techniques from the previous se ctions, a continuous or pseudo continuous type of source data must exist For instance, this data could include the following elements: Ground elevation data as either regular grid posts or mesh, such as a TIN (Triangulated Irregular Network) ; Roads as either polygons or linear features (with possible additional attributes) ; Rivers as either polygons or linear features (with possible additional attributes) ; Vegetation as polygonal areas with attribution such as density and height ; Buildings as either 2D polygons with height, or 3D meshes This data can be applied directly to establish relation s between various positions on the board. For instance, given 2 board locations, relations based on the 3D line of sight can be calculated between them by performing ray intersection tests against the various polygons of the geometrical features above. Similarly, to evaluate whether a particular vehicle can reach one location from another within a single game move one can perform very short range path finding using t he full fidelity of the source geometry limiting such search to the distance that can be traveled by the vehicle within the allotted game step time. This methodology ensures that the various reachability relation s reflect the full fidelity of the source da ta provided rather than suffer from any of the previously discussed errors incurred by calculating the se relation s using the discretized board models derived from the source data. This is not to imply that the discrete model will have the full accuracy of the continuous model just that for any pair of locations on the LG 73

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board, the reachabilities are as accurate as the source data. Each board cell represents an area or volume of the real world space; however, for the purpose of source data utilization, a representative single location is chosen. Commonly the cell center is used, although more complex techniques can employ a set of positions, rather than a single location. The discretization, i.e., introduction of the finite set of locations on the board, c an still present difficulties due to the persistent rasterization and alignment problems due to the selection of t he set of locations For instance, consider a straight road, surrounded by walls, running through a regular grid of points. Once this road is rasterized, some points along the road will be located directly on the road, while other just next to it, and others possibly, a half of the cell width away. When reachability or line of sight is calculated between the points along this road, each such re lation will cause a jump on and off the actual road possibly causing breaks in the connectivity. As usual, this can be alleviated by the reduction in the grid size so as to increase the size of the board (a finite set) and provide more actual locations for the Source Data Submodel to produce the greatest benefit. Overall, this system drastically improves the fidelity offered by the discrete model and allows for the LG strategy algorithms to be based o n the reachability relations calculated at t he highest fidelity available from the source data. While additional modeling errors can be encountered due to discrepancies between source data and the real world, such issues are not related to the discreti zation needed for LG al gorithms; they are common to any Geographical Information System, and are beyond the scope of this work. 74

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As a final note on modeling terrain obstacles, let us mention that all of the methods described above can be leveraged and brought into various combina tion approaches to achieve the compromise between: Performance ; Accuracy; Ease of modeling Cell types and under types can be used to capture basic terrain properties, densities to capture overall effect of buildings and forests, while employing the Source Data S ubmodel to adjudicate any of the more troublesome cases such as fences mentioned in the previous section. 3.3.3 Dynamic Obstacles In addition to the obstacles that are based on some inherent or static characteristics of the terrain, such as mountains and oceans, one must also consider various versions of dynamic obstacles [4, 39]. Of course, the simplest example of such a need is terrain deformation, whereas certain terrain features can be destroyed ( a building or a p art of a forest) or created ( a bridge across the river). However, the more interesting uses for dynamic obstacles include additional movement constraints that are not necessarily related to the physical properties of the terrain itself. Specif ic examples would include restriction of aircraft flight to specific altitudes or corridors or avoidance of certain dangerous regions, such as those saturated with SAM (surface to air missile) systems defenses, or severe weather areas. 75

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These restrictions a re dynamic due to their ability to change during the scenario execution : dangerous weather may move from one location to another, SAM sites can be destroyed, thus changing the shape of the dangerous region, and various corridors can become available for air travel. These restrictions can apply to some entities within the ABG but not others. Additionally, different par ts of the ABG may be affected: movement (i.e. reachability), visibility, and sensor or weapon employment. Such obstacles must be well integr ated into various concepts of ABGs, such as reachabilities, to allow for the LG to account for their effect by directly affecting t he gener ation of trajectories and zones in ways similar to static obstacles described in previous sections. It is likewise important to note that these restrictions can be specifically defined by the user of the system, e.g. as the no flight zones, or automatically generated by the LG during strategy generation. For instance, a particular group of entities ( such as an army platoon) may not be restricted in its choice of a route to the goal destination. However, once that first main trajectory is chosen by LG, the rest of the group must all operate within a certain width corridor around the main path to maintain unit cohesion, which can be achieved by considering anything outside of the corridor as an obstacle. Similarly to this automatic generation of such obstacles, they can also be automatically removed if LG detects that the survivability in this corridor is too low with such restrictions. This approach allows for various common real world use cases to be represented within an ABG and reasoned upon by the LG algorithms. 76

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3.4 Mo bility 3.4.1 Temporal Discretization Another important part of ABG modeling is temporal discretization. As previously mentioned, real world Defense Systems generally have to be modeled employing Totally Concurrent ABGs ; however, any movement and actions must still occur in discrete intervals commonly referred to as a time steps. The size of the time st ep may vary and usually depends on the particular application. Temporal and Spatial discretizations are tightly interconnected. In the discussion of the spatial discretization above we have introduced t he idea of a necessary compromise: the cell must be small enough to be able to represent necessary level of detail of the real world terrain yet large enough so that the computational burden is tractable However, this spatial resolution parameter is also directly related to the time step used. Consider repr esenting a vehicle moving at 1 mile per minute. If we use a time increment of 1 minute, we may want to use 1 mile cells. On the other hand, this spatial scale may not be sufficient due to the nature of the terrain. If we decrease the size of the grid to mile per cell, we have a choice of decreasing the time scale to 30 or 15 seconds (so that the reachability is 2 or 1 cell/turn) or maintaining the spatial scale (using 4 cells/turn reachability). Conversely, if there are other events that may happen more f requently in the game, we may need the time step of 30 seconds per turn, which may require a change in the spatial scale. As the reader can see there is a tight interconnection between temporal and spatial discretization. However, the time step may be affe cted by factors other than just the relationship between cell size and agent speeds. There may be 77

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additional non movement events affecting the time scale, such as weapon effects and sensor updates. Additional consideration will be given to the various effe cts of temporal discretization, as it relates to the diversity of the modeled ent ities and their actions, in Chapter 4. T his discrete time interval must be an integral part of any discussion of movement, or other state changes within the ABG 3.4.2 Reachabilitie s The movement pattern within LG is defined employing re achability relations on the game board [2, 3, 27]. As a result, the ability to represent certain patterns is subject to our choice of spatial (and temporal) d iscretization. Conversely, the choice of the board must reflect the movement patterns that have to be represent ed within the game It is not feasible in practice to precisely represent the mobility of all entities thus, a compromise solution must be chose n. E rror is unavoidable and therefore, the priority must be given to selecting an appropriate spatial and temporal resolution that supports generation of the game strategies corresponding to the real world strategies. The focus of such analysis should be on determining which errors have tactical implications (e.g. how many game moves it would take for an entity to travel to a particular key location and what route it would take), rather than on physical ly precise modeling of the movement 78

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Figure 27 The 2D rectangular grid board and reachabilities of pieces Consider a case where we need to model jet planes moving at 20 miles per minute and cargo planes moving at 6 miles per minute using a time step of 1 minute. If we use a cell size of 1 mile, the size of the board may be larger than optimal for performance considerations. On the other hand if the cell size is 6 miles, the cargo plane can be easily modeled, but the speed of the jet can only be represented as 3 or 4 cells i .e., 18 or 24 miles per minute. Another choice might be 5 miles/cell, so that the je t has a reachability of 4 cells per move and the cargo plane of 1 cell per move corresponding to 5 miles per minute. For a particular problem, an expert in the problem domain must be consulted to determine the minimum precision necess ary to represent the real world. For example, 18 or 24 miles may not be an appropriate approximation for the jet, but 5 miles per minute may be acceptable ins tead of 6 for the cargo plane. Furthermore, consider a scenario with jet airplanes moving at 1,200 mph and cargo ships moving at 20 mph. The size of the cell cannot be larger than 20 mph x TimeStep, so that the move ment of a ship could be represented. However, this requires r eachability re lati ons for a jet of 60 cells per game move as a minimum. This corresponds 1 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 79

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to more than 10,000 target locations reachable from any start location in one game move for a full circular reachability on a 2 D board. As s uch computational burden is likely to be t oo large for practical use; these 2 types of pieces cannot be represented on the same board. Section 3.5 presents LG hypergame and variable step ABG approaches to address such situations. In addition, the movement of real objects is affected by other physical factors such as gravity and inerti a. For 3D boards we need to define reachability relations to include vertical motion as well as horizontal based on their parameters. A plane may be able to descent faster than to ascent. This may be reflected by considering cells one level above and two l evels below current posit ion reachable in one time step [40]. Figure 28 Side view of the cruise missile reachability pattern Furthermore, inertia may prevent the plane from turning 180 in one game move and the reachability must be defined based on physical properties of the agents such as minimum turning radius [27, 41]. Similar properties may include higher speed at higher altitudes for airplanes and different reachabilities over different terrain types (such as amphibious vehicles). Another property of movement for real objects is fuel limitations. However, this property may not always need to be modeled. For instance, a truck with period of operation that is less than 5 hours can be assumed to have unlimited fuel; while, a missile in cannot be in flight for that entire time period Moreover, some game CMHex cell, side view Approx. 2 n.m. 200 ft 6 n.m. 6 n.m. 80

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elements, such as an airplane in flight, may not remain on the same location for 2 consecutive game moves This may impose certain restrictions on the generation of trajectories and zone timing principles. For instance, in an LG zone a piece can arrive at the attack (interce pt) location at any time before the target piece reaches it. However, if a piece cannot hover or stay in one place, the trajectories must match up exactly both pieces must arrive at exactly the same time. This implies that more complex LG trajectories an d zones must be generated. Figure 29 Defining reachability relationships based on turning radius 81

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3.4.3 Trajectory Selection One of the common problems of modeling movement comes from the inaccuracies of spatial discretization. Conside r rectangular and hexagonal 2D boards with reachabilities of one cell in every direction. Any path for a piece in this framework will be jagged similar to a straight line being drawn on a rasterized display. A path on a discrete board is considered to pass through the centers of cells; as a result even the shortest trajectory is not a straight line on a 2D Euclidean plane but a collection of several segmented trajectories each with a number of straight segments. Clearly, on a 2 D plane, the trajectories on t he outside of the bundle appear longer, while some of the internal trajectories are extremely jagged (a turn on every step). For some applications this may not be an important issue. This is especially true if the model is used for higher level planning wh ile another system handles low level control (e.g. through the LG hypergame, Section 3.5.1 ). However, sometimes these zigzag trajectories have to be addressed One way such problems can be addressed is through Trajectory and Move Evaluation Function, which is a standard component o f LG [27]. Quality of individual moves and trajectories is constantly evaluated based on parameters such as simultaneous participation in multiple zones and avoiding interception. We can easily incorporate other criteria into this evaluation in particular, the 2D Euclidean smoothness and deviation from the true 2D shortest direction. This allows us to discard trajectories that would not be considered reasonable by human experts, while still evaluating entire bundles in case the unusual trajectories prove useful. The exact parameters of this evaluation depend on the problem domain. 82

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Figure 30 Bundles of shortest trajectories Figure 31 LG Zones with Trajectory Bundles for Aircraft Engagement 3.4.4 Direction Phase Spaces There is one more issue we need to consider with regard to modeling movement of the realistic entities on the LG boards, both 2D and 3D. An army uni t may be able to 83

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move north and southwest the same distance (e.g. 5 miles) over a given time period (e.g. 2 hours). However, consider a situation where a jet is flying north 500 miles per hour. If we have to use time increment of 0.5 minute and cell siz e of 2 miles (due to specific problem requirements), we may need to account for the fact that the plane cannot suddenly turn around in a single game turn What this implies is that instead of a circular 360o reachability we may need to use a movement pa ttern that more resembles a sector of a circle with a certain arc. However, the orientation of this sector depends on the direction of the velocity vector on the previ ous move [2, 27, 28, 41]. Figure 32 Directions with respect to a hex grid Note that the definition of ABGs and generating grammars (Sections 2.1 and 2.2 ) do not allow us to keep track of such additional inf ormation during trajectory generation Consider the sets SUM, MOVEi, STk, and function MAPx,p from Section 2.2.1, which must be generated assuming a stateless piece. However, the concept of the direction of an entity movement can be easily incorporated into the design of the board itself by extending the 2D or 3D board into Phase Space. For instance, let us use a hexagonal 2D grid and discretize direction into 6 values 0 thru 5. 6 directions which is a natural choice for hexagonal grids, although different number could be used based on the specific d omain A 2D cell corresponding to the LG point (x,y) turns into 6 different points (x,y,d), 84

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where direction d = {0, 1, 2, 3, 4, 5} ( Figure 32). This allows us to model the current location of the piece with its direction as the LG point, on which it is located. Furthermore, the reachabilities (and therefore trajectories) are considered from current point (x1, y1, d1) in the Phase Space to destination (x2, y2, d2). This allows us to account for situations described above. For instance, co nsider reachability relations shown in Figure 33. If a piece p is at location (x,y) with velocity vector 0 (N) and wants to move to location (x,y+2), we can only arrive th ere with velocity vector 0 (N). This implies that Rp((x,y,0),(x,y+2,d))=True only for d=0. On the other hand Rp((x,y,0),(x,y1,d))=False for any d, since we cannot fly directly back. Clearly, if this piece needs to reverse its direction of flight it has to do so over several moves gradually changing its direction (x1,y1,0) (x2,y2,5) (x3,y3,4) (x4,y4,3). Figure 33 Trajectories of the aircraft changing direction Phase S pace allows us to model realworld situations with a lot more precision than a simple 2D grid. Phase Space can be used with other types of basic boards (such as 3D), as well as other types of phase dimensions (e.g. fuel or acceleration see below ). 85

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Howev er, increasing the number of phase dimensions and the level of their discretization increases the board size. Total size of the board becomes: |X|=|2D Space| |Phase Dimension1| |Phase Dimension2| |Phase DimensionN|. For a 2D board of 1,000 cells with 6 possible velocity directions and 3 values of acceleration, the board size becomes 18,000. As with the size of 2D cells, one of the goals is to use the minimum number of phase dimensions with the most coarse discretization levels that can represent the world in sufficient detail. 3.4.5 Speed Phase Space The notion of the Phase Space presented above applies to any kind of information that can influence units movement and, therefore, must be included as part of the LG MAP and trajectory calculation. Another common example of such a requirement is modeling changes in speed, or acceleration. Just as most aircraft cannot fly backwards, their ability to turn and the specific turning radius are dependent on their rate of movement. C onsider a car driving down a road. As it approaches an intersection, it must slowdown in order to make a sharp right or left turn. This inability to execute a 90o turn at high speed could be reflected employing the reachability relations in the Phase Sapce as follow: Rp((location1,fast),(location2,fast)) = False ; Rp((location1,slow),(location2,slow)) = True. Additional reachabilities would then also be specified to allow transition between various speed states, e.g. Rp((location1,fast),(location2,slow)) = True. Various levels of 86

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fidelity can be achieved using different number of speed levels (compare with the 6 directions in the previous section). Use of P hase S paces allow for the ABG model to capture various differences in reachability related to the sta te transitions. As with any modeling choice, one must consider the tradeoffs between fidelity and computational requirements. In some cases, the value of the additional computational complexity introduced by the Phase Space may not justify the cost. Other simpler techniques can often be used if the lower level of fidelity is sufficient for the needs of a particular scenario. For instance, a trajectory can be broken down into distinct acceleration, travel, and deceleration segments using techniques presented in [18]. An even simpler method would be to decrease the model spatial or temporal resolution until the acceleration is no longer relevant. This would be equivalent to modeling the movement of the car in the example above using 5 minute time steps ; in such case the car could accelerate and decelerate many times within a single game move removing the need to track this parameter explicitly. 3.4.6 Ballistic and Orbital Traj ectories A p articular difficulty is presented by the motion of entities with vastly and rapidly changing reac hability relations such as g ames involving ballistic missile deployment [29]. A trajectory for such an entity is very complex including a boost phase characterized by extremely high acceleration, mid course phase, and reentry phase. It is exceedingly difficult to model such movement using discrete techniques presented above including speed P hase S paces due to the very large dist ances covered by the movem ents of such 87

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ballistic vehicles and the high fidelity of spatial and velocity modeling required for accurate representation of the movement characteristics of such an entity. This is further complicated by the nature of the equati ons of motion required to compute the trajectories of such game pieces in the 3D space [42]. Complex differential equations must be employed to calculate the interdependent effect of the boosters thrust, the decrease in the weight due to ex pended fuel, and the velocity. Gravitational pull of Earth further complicates the matter by introducing extra force acting on the missile and which can be harvested for intentional maneuvers, such as gravitational turn [42]. Figure 34 Ballistic trajectories on hexagonal shperical ABG In such cases, a special technique needs to be employed to extend LG to achieve applicability to this domain. LG is based on a hierarchy of formal languages presented in 88

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Section 2.2 For ballistic trajectories, and other similarly complex movement paths, the Language of Trajectories can be overridden with a different diale ct. Instead of following the grammar of Section 2.2.1 to generate trajectories, the movement path can be calculated using other methods, such as by numerically solving the differential equations of the ballistic missile flight, and then mapped into the discrete space of the ABG and the Language of Trajectories. These resultant trajectories should take the same form as an LG Trajectory generated using the grammar: to = a (x)a (x1) a (xl). Rather than discretizing the space and movement properties as required for an ABG and using this discrete model to generate trajectories, we first generate the trajectories using an approximate continuous model of ballis tic movement of entities in the 3D space, and only then discretize the final result. Since these final trajectories belong to the Language of Trajectories within the ABG, they match well the hierarchy of formal languages that make up Linguistic Geometry. Most importantly, LG Zones can still be constructed to analyze the tactical impacts of the trajectories and determine which additional trajectories need to be generated. Key aspect of this method is that these trajectories become just as much a part of the LG Zones as traditional LG trajectories and they can be used together with the standard trajectories generated by the appropriate LG grammars This in turn allows for modeling of scenarios that include both complex moving objects that require these advanced generation methods, as well as entities whose trajectories could be generated u si ng conventional LG approaches Ballistic trajectories are not the only trajectories that are better ge nerated using nondiscrete methods. 89

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Another common use case includes o rbital satellite trajectories [43], which included geostationary, geosynchronous, low orbit, and many other s. Figure 35 Orbital trajectories 3.5 Heterogeneous Systems The key difficulty that surfaces in many different forms in modeling co nflicts is related to model ing vastly varied entity types with a single game. One can typically optimize the construction of the Abstract Board Game based on a small set of similar entities, for instance fighter aircraft or tanks. However, the best choices of the board resolution and time intervals for one such group are often far from the best for the other. Unfortunately, in real world defense problems, it is far more common to have combined arms operations with disparate sets of assets involved in joint missions. As such, it is 90

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impo rtant for the simulation and battle command communities to be able to model such scenarios in an integrated fashion so that good strategies can be found for all participants of such interdependent operations A common approach to solving this problem is to model heterogeneous entities by simply separating them o ut into common groups and creating separate games for each group. To use the above example, the airborne component of an operation can be modeled separately from the ground component. Such decomposition is in line with the overall concept of the decomposition of complex problem into simpler sub problems as discussed in Section 1.4 However, as discussed above, typically such individual sub problems c annot be solved independently, e.g., the airborne and ground operations are often tightly intertwined and interdependent. The solution for each problem must be found in the context of the overall pro blem, while also taking care of any effects that the actions in one game have on the other. It is also worth mentioning that within the domain of joint military operations, the missions can involve a lot more diverse components [29]. Consider a n operation involving an insertion of ground troops from an ocean onto an island. In addition to the obvious navy landing ships and ground assault force s there would be supply bridges by cargo aircraft and ships. Such supply routes must be protected by joint operation of naval and air forces. The air support may be launched from land based airstrips or from naval aircraft carriers. Such carriers themselves may need to relocate to the target area which would typically include movement of an entire aircraf t carrier group that includes a large set of ships, submarines, and support assets. Satellite based communications and 91

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surveillance would also be employed and they themselves may be attacked. There may even be a ballistic missile involvement (or other very long range delivery vehicles). Even just the ground operation component would include armored vehicles, missile and artillery support, missile defense components, special operations teams, and other dissimilar assets. These kinds of problems are comp licated because they simultaneously involve entities that move at 2 miles per hour and 17,000 miles per hour, time scales that cover 30 minutes or several days, terrain areas that cover several kilometers or those that cover half the surface of Earth [29]. This is a very large scale campaign of several tightly interconnected smaller tactical operations. The scenario covers a large portion of the Earth s surface due to remote supply routes. On the other hand, some of the tactical components take place over areas of the island, which may be only 100 miles across, and require high level of detail for accurate planning. The duration of individual tactical maneuvers varies from 30 minutes for some air missions to 12 or more hours for supply trans port bridges. The entire campaign may last several days. The resources involved in the game range from ground troops to ships and submarines to cargo and fighter planes to satellites. This represents distribution of speeds from 3 miles per hour to 30 miles per hour to 1,200 miles per hour to 17,000 miles per hour respectively. Compared to aircraft, the navy elements are 40 times slower, while ground troops are 400 times slower. The types of individual missions may include air, navy, and ground combat, reso urce transportation, and surveillance. There may also be potential effects on politics and economy. The total number of units involved may be on the order of thousands, while some missions only require dozens. 92

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However, even though the tasks are so widely d issimilar there is a tight coupling between the individual missions [11, 44 46]. If resource transfer is interrupted by the enemys combat units, the outcome of other combat mission s may be affected due to lack of necessary supplies. Similarly, a combat operation that cannot be won may be saved by launching a different operation earlier to cut enemys resupply chain. All of the missions can be influenced by satellite surveillance incomplete information, or even deceptive behavior by each side 3.5.1 LG Hypergames There are several difficulties of representing such a complex campaig n as an abstract board game. Some operations require representation of the entire planet Earth with low resolution, yet high level of detail is required for others. This implies that a very large high resolution board is needed. However, the majority of th at representation will be wasted since high level of detail is only needed in specific areas. The game agents include objects with speeds that differ by the factor of 20 or 200 (even disregarding the satellites). Representing such wide range of motion patterns on the same board requires very large reachabilities. The time scales necessary to handle different missions would also be extremely diverse Air combat missions may require 30 second intervals, while the movement of ground units within this time step is negligible. In addition, the entire campaign involves thousands of agents, out of which only a small portion would be used for any one given mission. Yet, we cannot consider one task at a time, as they are very tightly interconnected. These are just so me of the issues, which motivate the development and application of the LG hypergame method [23]. 93

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LG hypergame app roach addresses the requirement of model ing such large scale complex campaigns by providing a framework for modeling and solving them as a set of tightly interconnected ABGs [2, 28, 46]. Formally, an LG hypergame consists of a set of ABGs and a set of inter linking mappings (ILMs) to maintain interconnections between the games. The original problem is decomposed into multiple ABGs such that each ABG on ly needs to include a fairly homogeneous set of entities with common spatial and temporal discretization requirements. The ILMs are then used to map state space from each of the sub games to all other sub games. Such mappings include translation of locatio ns, time steps, entities, and any other data. This way when an entity moves or perform an action within one game, such actions would be automatically mapped and correctly reflected within the space and time of the other game. Consider one such example. A special force team may acquire targeting information (such as laser guidance) on a tank within a high resolution ground game. This information can be translated into a laser guidance acquired action on the air force game, followed by a missile engagement f rom a nearby aircraft. This destruction can then be linked to a tank game and ground supply games, so that supply vehicles can proceed safely and the tank, which may have been heading to engage the enemy tank, can be redirected to better targets. A key ben efit of such ILMs linking ABGs is that they can employed during the strategy construction stage pro actively rather than only during the simulation stage reactively. In other words, ILMs can be used to construct strategies within each ABG that take into account planned actions in other ABGs, thus producing a combined, integrated 94

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plan for actors within all ABGs at the same time as if they were a part of a single board game. This prevents the traditional disadvantage of problem decomposition producing individual strategies for su b problems that may not together form a solution to the overall problem. In essence, ILMs eliminate decomposition into several sub problems rather they allow one to maintain the problem in it is entirety, and instead to decompose the board and state spac es within which various entities perform their action s in the ABGs that provide the best representation to all their activities. 3.5.1.1 From ABGs to Hypergames There is a natural decomposition for such complex systems as the one described above. This is due to th e way human experts in the field approach planning of such campaigns. The goals are broken down into sub goals, creating tasks and supporting sub tasks necessary for each higher level task. It is natural to model the situation in a similar manner by modeling each of the tasks as a game. However, as for any realworld problem, this decomposition usually does not produce completely independent subsystems. Therefore, each individual game may not be solved by itself in isolation. Effects of the interactions with the other sub problems must be taken into account. Attempting to solve separate games may produce ineffective, incomplete, or incorrect solutions. LG hypergames are introduced for handling these types of complex systems. A hypergame is precisely such a collection of several interlinked concurrent ABGs It may include a number of different types of ABGs (military, resource allocation, transport, political, economic) with different boards of various Spatial and Temporal resolutions [3, 46 48] A move in one of these ABGs may change the state of the rest of the ABG s included in the 95

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hypergame. Intuitivel y, a hypergame is a collection of smaller sub games that are synchronized in time and space as well as through game agents. Formally, a hypergame consists of a set of abstract board games (ABGs) and a set of inter linking mappings (ILMs) for maintaining in ter connection between them. ILMs map all the ABG components from each of the ABGs to all other ABGs. These mappings may include translation of locations, time, game pieces, and any additional data. Figure 36 Air (left) and Land (right) ABGs of an Air Land hypergame The closeness of connections created by these mappings can also vary. Subsystems can be either connected serially corresponding to cause and effect relationships, or concurrently to simulate interweaved effects. The serial 96

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interconnections occur for two systems when the outcome of the first system affects the second system. For instance, the result of resource resupply mission ( game 1) affects the outcome of a combat mission ( game 2) that requires these resources. In this case, the first mission takes place before the second mission. However, at the end of the supply mission, the game objects from game 1 (resources), are transferred to the game 2 through an inter linking mapping of game agents. The outcome of the comba t mission is therefore dependent on whether or not those resources were received and introduced into game 2 or destroyed and lost in game 1. The second type of interconnection allows for even closer concurrent interweaved coupling between systems. For in stance, consider a naval battle and an air battle occurring simultaneously. Due to the difference in spatial and temporal scales and the speed of the objects, these conflicts should be represented as different ABGs However, the airplanes must be present i n the naval game just as ships must be present in the air game. In this case, ILM link the representations of the same objects in two or more simultaneous games. These objects must be synchronized in space and time, so that movement of a ship on the naval game is immediately reflected in the air game and a destruction of a ship by an aircraft in the air game is reflected in the naval game [49]. This is clearly a more dif ficult type of interconnection due to a high degree of coupling and the requirement for ongoing synchronization. 97

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Figure 37 Air (left) and Navy (right) ABGs of an Air Navy hypergame Another important property of hypergames to be i nvestigated is a possibility for dynamic generation of new games. For a given collection of ABGs, ILMs can provide the necessary interconnections. However, this collection may not be fully known in advance. In this case, it is desirable for the new ABGs (of a hypergame) to be generated when the need arises. Consider a navy aircraft carrier battle group relocating to the site of conflict and encountering enemy air patrol. Since in real world Defense System such information may not be available ahead of time, there may be no air combat game set up over the specific region. We could resolve this skirmish within the naval game; however, such resolution would not provide very accurate solutions due to lack of sufficient detail. To handle such s ituations effectively and precisely, there must be a possibility for a new game of required type to be created and linked to the existing games with ILMs. Such generation of a new ABG can be an event triggered and represented as another item in 98

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the set of actions (within transitions in an ABG) similar to movement or shooting. A shooting action uses a precondition target is within weapon range of a game agent to change the state of the system by removing the target. Similarly, an ABG generation action coul d use a precondition a game object is present that requires a different ABG In this case, when an enemy plane appears in the naval ABG (from a different game through an ILM), a new ABG is created automatically. Another benefit of the hypergames is a po ssibility for using ILM for sharing information other than just game state data (based on inter linking the ABG components) In particular, strategy information may be interlinked as well. Consider the previous example of the simultaneous naval and air co mbat games. State linking ILMs e nsure that both games can see and interact with all of the present objects including the ones that are controlled by the other game. The strategy is then generated within each of the ABGs separately based on this complete da ta set. However, the strategies must be planned collaboratively Introducing ILM for strategy information, such as LG zones and trajectories, allows for cooperative planning. For instance, let us assume that the enemy force consists of 3 ships out of whic h the air patrol can only destroy 2 in the available time. If the friendly battle ships can destroy only one specific enemy ship, this information is essen tial for the air game to decide which 2 enemy elements to attack A strategy with ILM s allows for an interchange of zones between the two games. Therefore each game has information on both zones. This combined information c an then be used to construct a global hyper strategy that will be used for both ABGs 99

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3.5.1.2 Common Inter Linking Mappings As mentioned above there are several types of inter linking mappings [46, 47]. The ILMs needed for a given problem domain depend on the particular problem specifications. However, there are several types that are required for most Defense Systems hypergames spatial, temporal, agent, and strategy ILMs. Spatial ILM is used to translate between locations of different ABGs. Observe that this is not necessarily a one to one correspondence because several cells from a high resolut ion board may be mapped to the same location on the low resolution board. The mapping can also be established between the boards of different types. For instance, a satellite may be represented on a board of orbiting positions, yet it may have to be mapped to a location on a 2D board. In such cases the translation is based on the particular systems requirements, e .g., it can be mapped to a cell ove r which it is currently located Furthermore, some locations may be impossible to map onto another games board. This means that the objects associated with such locations at the moment (pieces, obstacles) are isolated within one ABG of a hypergame and not inter linked with other ABGs. Temporal ILM is another common type of mappings, which is used to synchronize t he games in time. Different sub games may have different time steps and may start at different times. As a result we need to e nsure that the data is updated in all of the inter linked games at the correct turns. For instance, if an air game uses 30 second intervals and a navy game uses 4 minute intervals, then the positions of ships in the air game must be updated every 8 game turns, while positions of airplanes are updated in the navy game on every turn. Spatial and temporal ILMs may be used together to a chieve more robust 100

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synchronization. If the board of the air game uses resolution 8 times higher than in the ship game, then a ship moving one cell in the ship game actually moves 8 cells in the air game. With more accurate spatial and temporal ILMs, we can avoid the ship jumping 8 cells every 8 game turns in the air game by moving it m ore smoothly one cell per turn. Agent ILM takes advantage of the previous two ILMs to synchronize the game pieces between subgames. Spatial ILM is used to link the objects location to the right locations on all of the linked game boards, while temporal ILM insures that the locations are updated on the correct game turns. Agent ILMs usually contain filtering so that only the necessary objects are inter linked. In addition, mis cellaneous game piece data may be interchanged such as weapons load, fuel, and health status. Finally, even the type of the object may be changed or the object may be replaced by other objects (e.g. supply ship arriving at the destination in the resourc e game, may be transferred over to the combat game as the actual supplies transferred several aircraft, tanks, etc). The other common type of ILM strategy ILM has already been presented in the previous section. The particular information being exchan ged depends on the particular problem domain and the amount of details required. For instance, instead of inter linking actual zones, only a summary or analysis can be transferred. This topic requires further investigation. 3.5.1.3 Benefits of Hypergames The major benefit of using hypergames is the ability to handle very complex tightly interconnected systems. Consider the campaign presented in Section 3.5.1 Each of the sub tasks can be represented as a separate ABG Red navy bridge, Red air bridge, Blue air bridge, Disruption of Red navy bridge navy combat, Disruption of Red navy bridge 101

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air combat, Disruption of Red air bridge, island ground co mbat, island air support, satellite reconnaissance, etc. Additional subgames may also be generated as needed. Each of the individual ABGs can use the type of spatial and temporal discretization that is most suited for the game [47]. Spatial discretizations may include 2D, 3D, Full Spherical, and Orbital boards of different resolutions, while the best time step is selected based on the size of cells and objects to be represented. Som e of the games may be present through the entire campaign, whereas others are only active for the duration of a particular mission. Whenever several especially different types of objects need to be represented within the same mission (such as ships and airplanes), the game can be split into two or more inter linked subgames of the most appropriate spatial and temporal scales. Appropriate agent ILMs can insure that only the necessary objects are present in each of the ABGs thus maintaining computational complexity of individual games at the appropriate level. All through this decomposition, required interconnections between subgames can be maintained using inter linking mappings within a single global hypergame. Therefore, both serial and concurrent interwe aved effects can be observed and used to generate cooperative strategies that can be combin ed to form the global strategy. 3.5.2 Variable Step ABGs The key difficulty that surfaces in many different forms is the conflicts inherent in trying to model vastly varied entity types with a single game. One can typically optimize the construction of the Abstract Board Game based on a small set of similar enti ties for instance fighter aircraft or tanks. However, the best choices of the board resolution and time intervals for one such group are often far less than optimal for the other. 102

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Unfortunately, in real world problems, it is far more common to have combi ned operations with vary disparate sets of assets involved in joint missions. As such, it is important for the simulation and command communities to be able to models such scenarios in integrated fashion so that winning strategies can be found for all participants of such interdependent exercise. Section 3.5.1 offered one solution to such problem employing LG h ypergames. However, this solution can be computationally expensive due to the need for multiple ABGs and ILMs. In this section, we present an alternate method of modeling such situations by employing the hybrid method of Variable Time Step Abstract Board Games based on some of the key features o f h ypergames applied to a specific subset of problems that allow for a more light weight solutions. A typical hypergame segregates the entire problem into several ABGs which differ mainly in three parameters: Spatial discretization Temporal discretization Types of interactions While there would be various other differences such as movement and action patterns and entity properties they typically arise as an effect of different spatial and temporal resolution. The choice of particular spatial and tempor al scales for each ABG is defined by the properties of the entities that are to be modeled within such a game (e.g. speed and range of sensors or weapons) as well as terrain features that need to be modeled (e.g. size of buildings or width of roads). The 3rd main parameter type of interaction is of special importance. In most of the discussion s we have focused on 103

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military games that take place in the geographic space and involve physical actions such as movement. However, there may be other ABGs use d which are categorically different. Some examples would include modeling interactions on a social network, defense of cyber security networks, and politico economic interactions. For the current research the focus is going to be on ABGs that involve phys ical actors interacting on a physical geographic battlefield. We should also note that in any likely computer software implementation of a game model, each new Abstract Board Game would require additional computational resource such as memory. This is du e to the basic need to maintain game model structures in memory game board structures, mobility and action patterns, as well as any runtime cached data. Due to such overhead, it may be of benefit for practical applications to consider some hybrid solutio ns an approach that incorporates some aspects and advantages of the hyper game method with the basic structure of a single ABG which we will investigate. 3.5.2.1 Cell Sizes, Time Steps and Entity Movement Speed To this end, let us consider two hypergames that have differ in only one of the key parameter. Specifically, we will focus on a concept of games that share geo physical nature and spatial resolution and differ in temporal aspect only. This is a common scenario because the spatial resolution is usually d riven by the needs to correctly model the terrain features and by the speed of entities that need to be represented, while the temporal resolution is determined by the computation of the speed of entities and chosen 104

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spatial resolution. We will use the cont ext of military operations in urban terrain (MOUT) [50] for the following discussion. In the MOUT domain, the spatial resolution is largely dictated by the desired resolution of terrain. The desired winning strategies may include movement down specific roads, hiding behind or inside buildings, moving through alleys and bridges. In order to be able to produce such strategies, the game board should employ resolution sufficient to capture such features. For i nstance, if the width of most roads is 10 meters, using 100 meter cells is out of the questions since a single cell would then likely contain several roads and buildings. It is feasible to employ cells that are somewhat larger than the sizes of individual features employing advanced techniques such as eggshell cell models and density based cells (discussed in Section 3.3.2 ); however, even with such techniques the cell size cannot significantly exceed the size of typical terrain features. Thus, the spatial resolution can be thought of relatively static for the problems within MOUT domain we shall assume it to be 20 meters as a representative number, while the actual best size may be somewhat different for different urban areas. We must now consider the temporal resolution. Given spatial resolution and the speed of an entity we can calculate the desired temporal resolution as a function of the desired number of game board cells traversed in a single game time step. The reason that the number of cells representing a single game turn jump for a piece (cells per turn) is treated as an independent variable and the game step as t he dependent variable is due to the optimization concerns. The cells per turn is directly related to the branching factor in fact, the branching factor may be proportional to the square of the cell per turn value. By 105

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reducing the branching factor the gr owth of the game tree can be somewhat limited, while in additional providing more optimal model for implementation in software solutions. Another reason for maintaining the smaller of th ese value s is to avoid the large discrete jumps which lose any information of what occurred during the jump. For instance, if an entity can jump over 15 cells in one game turn, there is no knowledge in the model about how those 300 meters were traversed and no actions are possible along that 300 meter portion of travel. On the other hand, as we have mentioned in Section 3.4 there are certain benefits to choosing a slightly higher value for the radius of each game mov e jump the distance and angular distortions of discretization can be easier corrected for. Due to these concerns, the empirical evidence has lead us to consider the optimal value for cells per turn to be in the range of 2 to 7, with the preference to v alues in the lower half of the range 2 4. Consider the following table that demonstrates the relationship between time steps and cells per turn values for the board cells size of 20 meters. To illustrate the usefulness of such a table, let us consider two examples from it. First, we will start with modeling movement of a squad (a group of soldiers about 9 people) on foot. A typical defensive movement speed may be 6km/h. The table then suggests multiple possible combinations of tuples 25s/turn with 2ce lls/turn, 35s/turn with 3 cells/turn, 50sec/turn with 4cells/turn, 60 seconds with 5 cells/turn (as well as some other). For the second example, we will use the movement of a vehicle with a typical speed of 100km/hr. As you can see, the possible combinatio ns within the table are quite limited 5 s/turn with 7 cells/turn and 10s/turn with 14 cells/turn. 106

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Cell size 20 m Cells per turn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Time step (s) 5 14. 4 28. 8 43. 2 57. 6 72. 0 86. 4 100. 8 115. 2 129. 6 144. 0 158. 4 172. 8 187. 2 201. 6 1 0 7.2 14. 4 21. 6 28. 8 36. 0 43. 2 50.4 57.6 64.8 72.0 79.2 86.4 93.6 100. 8 1 5 4.8 9.6 14. 4 19. 2 24. 0 28. 8 33.6 38.4 43.2 48.0 52.8 57.6 62.4 67.2 2 0 3.6 7.2 10. 8 14. 4 18. 0 21. 6 25.2 28.8 32.4 36.0 39.6 43.2 46.8 50.4 2 5 2.9 5.8 8.6 11. 5 14. 4 17. 3 20.2 23.0 25.9 28.8 31.7 34.6 37.4 40.3 3 0 2.4 4.8 7.2 9.6 12. 0 14. 4 16.8 19.2 21.6 24.0 26.4 28.8 31.2 33.6 3 5 2.1 4.1 6.2 8.2 10. 3 12. 3 14.4 16.5 18.5 20.6 22.6 24.7 26.7 28.8 4 0 1.8 3.6 5.4 7.2 9.0 10. 8 12.6 14.4 16.2 18.0 19.8 21.6 23.4 25.2 4 5 1.6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 16.0 17.6 19.2 20.8 22.4 5 0 1.4 2.9 4.3 5.8 7.2 8.6 10.1 11.5 13.0 14.4 15.8 17.3 18.7 20.2 5 5 1.3 2.6 3.9 5.2 6.5 7.9 9.2 10.5 11.8 13.1 14.4 15.7 17.0 18.3 6 0 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 12.0 13.2 14.4 15.6 16.8 Figure 38 Cells per turn and Time step vs Speed in km 3.5.2.2 Variable Time Step T hese examples illustrate the typical problem of modeling entities with vastly different movement patterns within the same game, or on the same board. As yo u can see, there is no overlap for these two types of game pieces. A speed of 6km/hr cannot be represented at even one game per turn, if the game turn is 15 seconds or below, while the vehicular motion of 100 km/hr cannot be represented with less than 15 c ells per turn at a game step of more than 10 seconds. The hypergame approach would suggest that 107

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this problem can be resolved by employing 2 different games with different spatial and/or temporal resolutions which are then linked via ILMs. However, due to the terrain features being one of the primary driving forces behind the choice of the spatial resolution, these two games can actually employ the same game board with 20 meter cells, while the game steps will be different e.g. 25 seconds for dismounted forces and 5 seconds for mounted forces. However, as mentioned at the start of this section, we would like to introduce a hybrid approach that combines two (or more) of such ABGs into a single variable time step ABG. The game board data structures can th en be shared and then the need for any location ILMs is eliminated. Of course, there still remains the need for sequencing ILMs related to different temporal resolutions. Such mappings would be employed to control the sequencing of the various entities t o maintain movement of each entity at its desired time step and control the non movement actions to be executed on either common or individual time scales. As with hyper games, interlinking mappings are employed both during execution of the steps and the ir planning so as to produce a common cooperative, integrated winning strategy for all game pieces involved. There is an additional benefit to this approach besides ability to represented entities with vastly different movement properties. This method can also be employed to represent pieces with very similar, yet slightly different, movement properties. Let us consider an example of a game with two types of aircraft. The first one is modeled by representing its movement as 3 board cells per game turn. The second aircraft is 15% faster Our choice is to represent it s movement as either 3 cells/turn or as 4 cells/turn. In 108

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the first case, such a model would miss the critical data that the second type of the aircraft is faster and can overtake or outrun the first type; while the second case overestimates just how much faster this new aircraft is by a factor of 2 (33% versus intended 15%). Employing variable time step model, one could simply represent such new game pieces as moving at the same 3 cell/turn, and shorten the time step for such entities by 15%. Even though both types of aircraft would then move at the same 3 cells per discrete jump, one of them will be able to make such a jump 15% more frequently and thus result in the overall movement speed of 15% higher. 3.5.2.3 Practical Implications In order for this methodology to be truly beneficial to the practice community it is also important to define and formalize solutions to the various problems that arise when one attempts to discretize the real world problem s into the Variable Time Step Abstract Board Games. The common military model is necessarily vastly more complex than the typical board games such as game of chess. Let us compare simple movement and capture of game pieces one at a time with complex real w orld simultaneous movements, passive and active sensor detection, weapon engagements, shooterto target pairing, integrated weapon guidance, weapon cycle times, reload times, ammunition loads, re supply and re distribution of assets. In addition to such modeling concerns, it would be of benefit to analyze the performance benefits of this approach compared to more traditional modeling methods. While performance is an important factor, accuracy must be analyzed in conjunction with it as well, since one of t he benefits of this approach is to model more accurate ly the real 109

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world parameters of entities rather than just their approximations. The length of the trajectories and the depth of the game trees can be calculated for various trade offs of accuracy vs. pe rformance for both Variable Time Step ABG and conventional approaches. 3.6 Incomplete Information 3.6.1 World Views Incomplete or false information is another non standard aspect of real world s ystems (economic, military social, and other ) It turns out that some o f the expert principles discussed above are most useful in the presence of incomplete information. If everything is known about the enemy, such advice may not be needed. For instance, when the group is attacked by one enemy aircraft, but we know that there are another 3 on the way, standard LG methods can be used. There are two general directions of handling incomplete information. First is to use visibility flags for each piece. Under certain conditions (e.g. distance to target, line of sight, etc) a piec e can become visible. Then LG ability to update strategy on every step together with expert principles can be used. This takes advantage of the natural properties of LG methods, which include dynamic decomposition and updating of strategies as the game situation changes. This way, while the information is unavailable expert defined behavior can be used as described above. However, as soon as the new game agents become visible they are automatically and immediately incorporated into LG zones and included in the strategy. The second approach to incomplete information is to solve this issue more thoroughly by introducing several worldviews. 110

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The real state of the system is reflected in the True Worldview. That is where all movement, engagement, and other actions are actually executed and adjudicated based on the actual system state. This execution is often referred to as the simulation component of LG based tools, as contrasted with strategy construction portion. In addition to this true state, each player has its own worldview Player1 Worldview and Player2 Worldview [11, 37, 38, 47]. Each of these worldviews contains correct information about its players elements; however, the information about the other player may be incomplete or false. This allows for a significant advantage over the simple visibility model, as it allows for a player to have some inaccurate data. This may reflect a situation where an object was visible before, but has moved out of line of sight. In this case, the opposing player does know of the existence of that agent, although not its exact position. Conversely, inaccurate data may be due to a decoy an element that is seen by the opposing player as an object of a different type. The strategy is planned based on the available information. As the information becomes more accurate and complete, the strategy is constantly updated as in the visibility model. Such approach is sometimes referred to as reactive if the discrepancies between the Player and True Worldviews are discovered during simulation, the player would react by adjusting the strategy in accordance with the new information. This method can also be expanded from purely reactive approach to a proactive one. While 2 different worldviews allow us to react to incomplete, outdated, and decoy data, it does not provide a way to plan for the enemys reaction to it. For instance, we can launch a decoy, yet in our worldview we see it as a decoy since we have accurate 111

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information about our forces. Therefore, a strategy based on our worldview will not account for the enemy considering the decoy as a more important object. To allow for this consideration, we can provide an extra worldview within each players worldview. Thus, each player has its own w orldview with a nested worldview, which reflects what this player thinks the other player can see Figure 39. This allows a player to develop strategi es with intentional disinformation taking into account enemys reaction to the false data [2, 11, 27, 47]. True Worldview P1 Worldview P1 pieces P2 pieces Environment P2 Worldview P1 pieces P2 pieces Environment LG P2 Moves LG P1 Moves Filtered LG P2 Moves and Filtered Discrepancies P1 pieces P2 pieces Environment P1 Reality Filter P2 Reality Filter Settings Settings Filtered LG P1 Moves and Filtered Discrepancies P2 View P1 View Figure 39 Incomplete/false information 3.6.2 Deception Employment and Discovery Methodology of nested worldviews allows for modeling of intentional deception employment, as well as discovery of the deceptio n [51]. When planning a deception, such as a decoy or a feint using a small force to mislead the enemy forces, the Blue player can spawn additional virtual worldviews that reflec t how these actions will appear to the Red player. By generating strategies within those virtual state spaces, Blue player can evaluate the likely responses of the Red player to how Blue players action will appear to the Red 112

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player, rather than to what tho se actions are (faithfully represented in the Blues primary worldview). Discovery of deception is similarly performed by using additional virtual worldviews to test various hypotheses of enemys true actions. Blue player can populate hypothetical enemy pi eces in locations of critical importance and calculate strategies based on such placement. Besides testing entity placement, various avenues of approach or courses of actions can be similarly explored. Such m ultiple configurations can be used to generate a range of possible ptions the enemy may employ. These various strategies can then influence the Blue players own strategy. For instance, Blue player may choose to direct game pieces in a way that allows confirming or denying particular versions of Red tact ics. This would allow for the deception to be detected through targeted discovery of additional information. This closely correlates with the way military commanders consider multiple enemy Courses of Action and define Priority Intelligence Requirements that would be required at key decision points in order to direct the friendly forces in a way that counters that COA that the enemy is actually following. 3.6.3 Communication Groups In the real world, the information is even more incomplete as what had been discus sed in the previous sections. Rather than each player possessing a completely accurate picture of his or her own forces, the friendly pieces may actually consist of several disjoint groups with instances of current information in t h em. The team communicat ion groups was chosen to refer to these groups to highlight that the defining characteristic of such segregation is the communication requirement to keep the 113

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current state information synchronized between all sub groups of a single player. Common use case s for this are based on an organization where the force consists of several teams which have efficient communications, command and control inside the group, but have imperfect and delayed information sharing between the groups. LG based tools can address this requirement by extending the concept of worldviews even further [36 38] In addition to each players worldview, each communication group can have its own view which would reflect accurate information about the groups members, but possibly outdated information about any fri endly forces from the other groups. Since these state representations are different, each such group must also undertake its own independent LG strategy generation based on the local information knowledge. Just as information about the enemy can improve ov er time through new discover, so can information about the friendly forces within each communication groups worldview. This additional data can be based on sensors (friendly units observing other friendly units), or based on the data shared between groups To properly represent the real world, various types of communication links between these groups can be considered in terms of the information shared, such as unit positions, states or strategy information, as well as efficiency, e.g. communication delay s and ranges. By modeling such critical parts of the infrastructure, LG methods can then be used to analyze effects of various communication configurations. This infrastructure can itself be attacked during the scenario and dynamically affect which links are and are not available to share friendly to friendly information. 114

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Even though there is not likely to be intentional deception between various groups of a single players forces, incomplete and outdated information can cause the same effect of error in st rategic choices. Consider two such groups which have very outdate d information about each other. When an enemy appears, group one may believe that they are the only ones able to intercept the enemy and copy their forces to that task. Similarly, group two may do the same due to lack of information about the first group. This would result in an overcommitted forces and potentially opening up a weakness to other enemy attacks. Just as reflected worldviews were employed to address deception and lack of precise information about the enemy, reflected worldviews can be introduced within each communication group to reflect the best estimate of the current state and actions of other friendly assets using LG based extrapolation based on their last known state and plans. 3.6.4 Sensors All of the incomplete information techniques rely on modeling of realistic sensors [2, 3, 36 38] Sensor simulation always takes place within the True Worldview by applying sensor characteristics to the actual position and state of the enemy (or friendly) entity. However, the sensor planning can be performed within any worldview as part of either generation of own acti ons to find out information about the enemy or of assessment of enemys likely actions during their own information gathering mission. 115

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Figure 40 A cruise missile illuminated by the aircraft radar. Sensors can be defined simply as just detection ranges with a Boolean output if the target is within range it is detected, otherwise it is not. However, realistic sensor would include far more detailed characteristics. Various types of detection can be i ntroduced to distinguish detection an entitys location, velocity, affiliation, type, armament, command hierarchy and other data. This would allow for modeling of simple radar that detect an object at particular coordinates but provides no information as to the identity of that object. A different sensor could be able to distinguish between a tank or truck thus providing the type of an object. Yet another sensor could, through signals intelligence (or SIGINT), determine that actual identity of that entity within the enemys command hierarchy. This could be further extended to provide Detected, Tracked, Recognized, and Identified states frequently used in the real world to defined rules of 116

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engagement for missions or particular weapon deployment. Various further categorizations of these states can be introduced for different types of active sensors used to track targets for guided weapons use (e.g., laser and radar guided weapons). In addition to multifaceted, rather than binary, information prov ided by such models, the detection event itself needs to be probabilistic to reflect real world constraints. Each sensor can include a set of Probability of Detection functions which provides likelihood of detecting particular piece of information about an entity based on that entitys type and state and range from sensor to target state of the sensors simulated by an LG based system the user could introduce PD (Probability of Detection) functions. The same sensor could provide location of an entity with high probability at long range, but unable to provide additional type and affiliation data until the target gets closer Two types of sensor contacts can be considered positive and negative. Positive sensor contact occurs when one players sensor is chec ked against a particular target and the detection event of any type takes place based on the Probability of Detection. The detected information is then used to update the appropriate worldviews with this new data. However, let us consider what happens to s tale or outdated information. A simple decay function over time can be added to signify the lack of reliability of older information. Negative sensor contacts can be used to confirm that the information is no longer valid and speed up its decay or complete ly remove it. An entity could approach a location where the enemy was last believed to be based on the entitys worldview. Sensor detection could then be evaluated against that last known location and if the enemy is 117

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not detected at that location, the reli ability of that old data can be lowered based on the sensors probability of detection. This can be thought of as using the sensor in reverse. I f the entity was actually there, the player would have detected it with a particular probability of detection (Pd). Therefore, if the entity was not detected then with probability of (1 Pd) we can say that the entity is not there. Multiple negative sensor contacts can bring the confidence level of that old information below some threshold and completely eliminate tha t data point from the worldview. 3.7 Weapon Systems 3.7.1 Strikabilities Another important property of agents in Defense Systems is the ability to attack. In chess and similar board games, a piece is captured when an opposing piece moves to the same location. Unfortunately, it is not always quite so simple in the real world. A piece may be destroyed from a remote cell; or when a piece moves to a location occupied by t he opposing piece, it may be that the piece attempting the attack is the one that ends up being destroyed. The first aspect of attacks is the strikability relationship [2, 3, 27, 28] These relationships can depend both on the type of the game agent and the type of the weapon used. Thus, an airplane with bombs and airto air missile s can have two different relations of strikability. The first relation may include all cells directly below current position, while the second one includes all cells within the cone of attack. Some of these strikabilities may be more complex than other. For instance, consider a plane that can fire a missile at any target within 10 cells and the cone of attack of 6 0 in the horizontal 118

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direction and 10 in the vertical direction relative to its current velocity vector. To compute this relationship we first need to find all cells in the specified cone within the range. However, we still need to take into account line of sight requirement by checking for obstacles and removing any locations behind them from the set of strikable locations [2, 27, 40]. 6 4,3 6,3 Figure 41 Strikability with obstacles In addition to strikabilities, attacks in Defense Systems may need to be modeled as creation of a new game piece. Consider an aircraft firing a sub sonic cruise missile at a target 10 0 miles away. In this case, it may take several game turns for the missile to reach its destination. During this time it should exist physically within the game as a game piece so that other objects can interact with it (e.g. intercept it). Regardless of whether the 119

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weapon is modeled as a strikability relationship or generated piece, the weapon load is usually very limited and that fact must be considered during strategy generation. 3.7.2 Weapon Effects Another aspect of attacks for realworld Defense Systems is that not every attack results in destruction one, several, or none of the pieces involved in the attack may be destroyed. If one object shoots another, the second object may be able to shoot back at the same time (if allowed by the strikability) since t he game is Totally Concurrent. More complex cases may include more than two pieces shooting at each other. There are two primary methods to evaluate the outcome of any single attack Probability of Kill (PK) or Attrition Rate (AR) based models. There PK model is usually employed for an engagement between game pieces that represent singular real world entities such as an aircraft firing a missile at another airplane. In such cases, the result is usually probabilistic there is a Probability of Kill (PK) t hat determines the outcome. This probability depends on the type of weapon and the type of target. Other factors may affect this probability, such as position of the target relative to the attacker (consider firing at the plane in flight or on the airstrip ). In simulations, a pseudo random number is usually generated and compared to the PK to determine the outcome. The importance of PK lies in the analysis of the situation. In LG this analysis is performed by dynamic decomposition using the Hierarchy of Formal Languages. Consider the zone in Figure 19. If all PKs are equal to 1 (unconditional destruction), the Black side will be able to destroy the Gray bomber p0, and q2 and q3 are not necessary. However, if PK for each plane is equal to 0.5, the red side may be able to 1 20

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get through. The probabilities of kill can be incorporated at the zone analysis level to determine best possible behavior. It is frequent ly convenient to decompose the PK values into two elements Probability of Hit (PH) and Probability of Kill (PK) [36, 37]. PH would then be a function that is characterized by the accuracy of the weapon itself, while the PK would be related to that weapons ability to destroy a target of a particular type if it does indeed hit it. Further fidelity can be achieved by modeling destruction of various subsystems of an entity rather than a complete destruction. Such uses commonly include destruction of the ability to move (mobility kill or Mkill) and ability to shoot (firepower kill or Fkill) as distin ct from complete or catastrophic destruction (K kill). On the other hand, Attrition Rate model are typically employed for ABGs where each game piece represents a group of real world entities at a certain level of aggregation. An example of such model would include higher echelon operations. E ach piece could represent an entire battalion of infantry or vehicles. Probability of Kill would not be well suited to represent what happens during an engagement between such aggregated entities. Attrition Rate on the other hand would define what ratio of enemy forces is destroyed based on the size of the attackers, the size of defenders, and the ARs of the various weapon systems employed. After each such engagement the size of each aggregated entity is decremented by t he appropriate amount and once a certain threshold is reach that entity is considered completely destroyed. A benefit of AR versus PK based model is its determinism. Each attack would always result in the same overall outcome rather than depend on a pseudo random number generator. 121

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3.7.3 Paired/Prerequisite Trajectories Weapon engagements frequently involve more than a single attacker and a single target. Indeed, in certain cases the weapon employment requires the presence of one or more additional friendly entity to provide sensor guidance. The most common examples are laser or radar guided missiles; however, this concept is easily extended to weapons that, while not physically needing such guidance, can be require to ensure that appropriate detection state has be en achieved. This could even require independent confirmation of target from more than a single sensor. This principle is referred to in the military simulation community as kill chain or sensorweapon pairing In order for Linguistic Geometry strategy generation to properly account for such employment of assets, a new concept of paired or prerequisite trajectories needs to be introduced [4, 37, 39, 40]. Generating grammar for Language of Zones can be extended to restrict generation of negation trajectories for entities that use weapons with additional sensor guidance requirements to only those target trajectories that have already had another negation trajectory generated against them using the appropriate sensor. Each such interception would essentially require two trajectories to be generated one to guide the sensor to the target and one to guide the weapon. T he timing parameters of the trajectories within the zone further need to be calculated so as to ensure that the sensor illumination occurs prior to and including the moment of the actual attack. If the ground teams trajectory to illuminate the target usin g a laser designator intercepts the target at time T1, then the negation trajectory for the missile must intercept the target at time T2 > T1 assuming that the team can maintain laser illumination from 122

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T1 to T2. This timing calculation ensures that any negation trajectory connects to the target trajectory during the window when the target is illuminated. In some cases, this window must apply to the entire movement of the entity rather than only the final engagement step. This would correspond to the requ irement to have the target illuminated as a precondition to even launching the strike against it, rather than just to the final destruction. As described in Section 3.6.4 multiple sensor guidance type can be employed to denote various weapon guidance or weapon use authorization criteria. The prerequisite trajectory requirements within the zone can then become more and more intricate to reflect all su ch combinations. A single weapon could require a trajectory from an authorization sensor as well as from a guidance sensor, while the same sensor trajectory could be satisfying the prerequisite role for multiple weapons at the same time. These factors must be accounted for not only in the generation of Zones, but also in Z one analysis and trajectory selection to ensure strategy consistency with these principles. If the same sensor asset is required in two different places to support two different weapon assets, then only one of those weapon engagements will actually be able to take place LG strategy has to correctly account for this fact and find alternate negation options for the other target. P aired/prerequisite trajectories concept has even further reach ing implications. It provides a method to automatically generate chains of actions in an LG Zone whenever one action requires another to take place first. For example, for operations in urban terrain, this capability permits establishing a suppressive fire or overwatch trajectory 123

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by one entity before another entity is allowed to perform the main desired action, such as cross ing a street or entering a building This could be further extended and generalized into a longer chain of connected trajectories. 3.7.4 Sy nchronized T rajectories A related subject to prerequisite trajectories is one of synchronized trajectories [4, 39]. This principle provides a strict method to include interdependent and synchronous actions within LG Zones and LG strategies. A common principle in warfare is achieving local superiority and massing of fire on the enemy target. In LG terms this would correspond to generating multiple negation trajectories for multiple friendly assets to the same enemy target trajectory, and more specifically at the same interception point so as to engage the target at the same time. These additional timing requirements can be easily accommodated within t he LG Zone generation algorithm similar to those described in the previous section. These principles extend past just the generation of LG Zones. The overarching idea of maintaining s ynchronization of the various units applies to all stages of the strategies based on those Zone s Specific entities may need to slow down or stop to wait for those entities that have fallen behind or simply have longer to go. Arrival at the end of the traj ectories must be synchronized to maximize firepower. In addition to the synchronized negation or interception trajectories, this method can be used for movement along related sets of trajectories such as multiple units moving to the same target area or entities escorting other essential assets. Applicability of this technique is quite far reaching and depends primarily on the organization of these synchronized groups and the desired behavioral characteristics It can be used for aircraft 124

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strike packages synchronizing an attack against multiple time critical t argets. At the same time it can represent multiple groups of infantry maintaining unit cohesion within each group while also ensuring the groups themselves stay abreast of each other. Particular details of how closely this synchronization needs to be enfor ces can also be easily controlled within the timings of the LG Zone and the subsequent strategies. 3.8 Mission Concepts 3.8.1 Goals and Missions The purpose of any LG based application is to generate the winning strategy for a particular real world situation. The winning conditions, or goals of the scenarios, are most typically represented by a set of missions for one or both sides of the engagement. Utilizing a highly flexible and sophisticated structure for the definition of these missions extends the power of Ling uistic Geometry by leveraging the basic strategy generation principles to address specific and complex objectives of military or civilian operations [36 38]. Typically a scenario would consist of multiple interconnected missions. Each mission would be assigned to be executed by a particular group or set of groups of entities, to allow for cooperative execution. The LG can assist with choosing appropriate allocation of assets to individual goals. Various types of missions can be directly translated into corresponding LG Zones, suc h as Attack, Defend, Relocate, Cordon, and Search. Missions can be terrain oriented (e.g. occupy an area) or force oriented (destroy a particular enemy force). There can also be additional Targeting Criteria to further define 125

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the intended mission targets by parameters such as entity type, detection states, and force strength. Dependencies between missions can be easily defined in terms of Start, Pass, and Fail Mission Criteria. This allows for specification of a combination of events or parameters that must be met before a mission can start, be considered successful or failed. Each such criterion can be a complex logical proposition of variables that include simulation time intervals status es of other missions, friendly or enemy force strengths with particular areas, and other factors. Missions can be further restricted by specific instruction as to how the goal is to be accomplished rather than simply the goal itself. Such additional information could include way points to be passed through on the way to the main objectives speed limit for the entities performing the mission, and desired level of unit cohesion. The complex set of generic interconnected missions described above provides a very powerful method for definition of objectives within a scenario. However, these methodologies can be made more accessible to the end users within a specific operational domain by introducing an additional translation layer to c onvert their particular mission sets and mission structures to the LG mission constructs. For instance, a single Seize mission within a particular domain could be translated to several Relocate missions to staging grounds near the target area, followed by an Attack mission to displace the enemy from that area, and c ulminating in a Cordon mission to provide security and prevent the area from being retaken. Various armed forces branches employ a concept of an Execution Matrix to describe the objectives of the units across multiples stages of the overall mission [37 39, 52]. This approach provides an easy and straightforward way to 126

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define the order in which tasks will be e xecuted within a mission; however, it lacks the flexibility provided by the more comprehensive and flexible LG mission model. A well defined transformation exists to convert from the Execution Model to this more flexible approach; however, this conversion may not always be possible in the other direction. Figure 42 Mission Execution Matrix for Operations in Urban Terrain 3.8.2 Prescribed Behaviors Another aspect of real world Defense Systems is that they may combine long periods of inactivity between conflict stages. Consider an aircraft group flying to the target 100 miles distant. The first 50 miles may be completely uneventful, yet we still have to consider them as the opponent may appear even then. Then, there may be a short period of combat (e.g. the outer ring of defenses), followed by another stretch of inactivity until the next level of defenses or target is reached. The fight at the target will then be followed by the group returning back to base. The flight back still needs to be 127

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modeled since there may be more conflicts. This presents the case of soft (inactive) and hard (active) stages. Most of the standard games contain only active stage s because the soft periods are considered too simple. While soft phases are easier to handle, a complete model for a Defense System has to include both due to their interconnection. Proper strategy at the inactive stage may increase the chances of success during combat. Consider the above scenario when we know only the location of the target, but not the composition or location of the enemy defense. The strike group must fly to the target together and in a formation that provides the maximum variability of responses when the enemy is discovered [11] For instance, such a mission may be flown with bomber in the tail of the group, while fighter aircraft and electronic countermeasure aircraft are leading the package. Suc h extra level of complex behavior has to be defined by the human experts in the field and then introduced into the strategy generation algorithms. In this case, extra level of expert defined behavior and standard LG methods are employed together by the LG strategy generation approach to find the best course of actions in both soft and hard stages. Another example of expert defined behavior that can be used at combat stage is commitment of resources to a threat. When a package is attacked by one enemy plane it may not make sense to send all defenses to intercept it and leave the bombe r unprotected. 128

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CHAPTER 4 OPERATIONAL DOMAINS While each real world problem is unique in its own right, there are d efinite patterns and commonalities of systems that can be observed. As vario us techniques for ABG modeling were presented in Chapter 3 it would be now beneficial to consider a set of specific real world operational domains to allow us to highlight and analyze the specific applicability of each particular method. We will review specific use cases across each of the categories of the previous chapter and present recommendations for which combinations of methods are best suited to achieve high quality solutions for each domain. This presentation can ser ve as a blueprint for other researchers seeking to apply LG to various real world systems by highlighting key decision points that drive adoption of some methods over others for each particular problem. While it is impossible to capture every single type of possible real world mission s, it is important to make sure that the chosen set of domains is representative of the varied problems one may encounter in practical use. The following domains will be considered as a characteristic set of real world systems so as to present sufficient breadth of the LG applicability considerations Operations in Urban Terrain (UT) Air and naval operations (AN) Ballistic and orbital operations (BO) Joint Forces operations (JF) 129

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The first of these domains will be presented in the most detailed and stepby step manner following the outline of Chapter 3 However, in the interest of conciseness, subsequent domains will not reiterate any generally applicable discussions and instead will strive to highlight the key differences and un ique considerations for those particular problem areas. 4.1 Operations in Urban Terrain Let us begin this analysis with one of the most challenging domains operations in urban terrain. The reason for this inherent complexity is in the domino effect created b y the intricate terrain features requirement s. A city presents a very dense and varied environment requiring the use of sophisticated techniques to properly capture, which in turn affects the complexity of the reachability relations, trajectory selection, weapon and sensors models, and other ABG components [3, 37, 38]. T his section wi ll be based on Military Operations in Urban Terrain (MOUT), so as to allow us to discuss weapons and other combative aspects; however, the majority of the concepts apply to civilian adversarial use as well, e.g., police, fire, rescue, and other emergency services. Let us then consider the spectrum of the operations covered by the concept of MOUT. They can take place in the downtown of a modern city with skyscraper dominating the landscape along with urban parks or on the outskirts of small villages in otherwise rural regions. The game pieces would have to represent infantry soldiers civilians, or other dismounted personnel, transport vehicles, infantry combat vehicles (ICVs), tanks, unmanned ground vehicles (UAVs), unmanned air vehicles (UAVs), and pos sibly extend to close air support winged and rotary aircraft. The actions that can be undertaken, in addition to simply 130

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moving through the streets and buildings, would include direct line of sight (LOS) weapon engagements, indirect weapon fire ( e.g., artillery), observation, smoke deployment, barricades deployment, breaching structures or obstacles, evacuation of wounded, recovery of damaged equipment, resupply of fuel or ammunition, searching of buildings, and many others. Tactical considerations are further complicated due to the heterogeneous city environments such as alternating open plazas or wide streets, dense city blocks with alleyways, city parks, bridges, overpasses, markets, and rivers. Such varied terrain complicates even such simple tasks as fin ding the best route between two points due to the considerations of relative safety. Inability to tell friend from foe due to the civilian population further confounds this domain. The overarching goals of MOUT operations are also more complex than more cl ear cut typical Army missions. As stated in Section 3.2 only one choice is easy Totally Concurrent games are certainly required to capture the b usy city streets that are so familiar to us all. We will now systematically address the more difficult decisions of this problem domain. 4.1.1 Spatial Discretization Consid er the layout of a typical city, such as Figure 43. The key features that immediately come to mind can be roughly categorized into roads, water, vegetation, and buildings. The underlying terrain elevation must also be considered as hills an d valleys can drastically affect tactical implications within a city. Each of the feature types can also be further differentiated. The roads can be characterized by their centerline, width, height (in the case of overpasses), surface type ( e.g., gravel, d irt or asphalt), and further extended to cover other drivable surfaces such as parking lots. The water features can be 131

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defined by their overall outline, depth, trafficability ( e.g., can it be crossed in a car, on foot or by swimming), seasonality ( e.g., c reeks that are dry in certain seasons), and type ( e.g., lakes, marshes). Vegetation can be approached in two distinctly different ways individual or aggregated. Individual approach can be well suited for sparse growths such as the few trees in a suburban region each individual tree can be tr eated as a single object with its position, height, width ( e.g., width of the crown and width of the trunk), and type ( e.g., deciduous or ever green). However, in more dense areas, such as parks, it may be easier to think of the vegetation as an area characterized by a particular height, overall density, and type. This can also be more suited for bushes and other smaller undergrowth. Figure 43 Map data: Google, Bluesky Buildings offer even more options on the aggregate vs detailed spectrum. For the high fidelity, each building can be modeled as a precise external shell, internal walls, external and internal doors and w indows, staircases, elevators, ventilation, and other minute details. This level of detail may be appropriate for the operations concerning a 132

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single or just a few key buildings, such as hostage rescue or embassy defense. If we instead consider a larger ope ration, involving movement across several kilometers, such internal details are not as critical. In this case, the hulls of buildings (footprint, height, number of floors) are the main tactically relevant features in terms of LOS, cover, and mobility. As t he mission is not focused on actions taking place inside of buildings, an aggregated description of the building internals can be employed to distinguish between wide open warehouse, office cubical farms, residential apartments, or other layouts. These cat egories can in turn be used to model the amount of time, danger, or other costs of certain actions within the buildings, such as clearing, searching, simply traversing. Aggregation can be taken another step further by limiting the resolution to city blocks or built up regions rather than specific building. This level is appropriate for the scenarios where only the broad strokes of an operation involving higher echelon forces on a la rge urban area are of interest. It is important to realize that in such case s the d etailed LOS and movement routes cannot be analyzed. Under ideal circumstances, the choice of the resolution of the model is driven by the needs of particular missions. However, in practical use the other limiting factor can be availability of the machine readable data on the urban environment. Modeling internal layout of a building requires source data that includes internal building floorplans. Even ext ernal windows and doors require specific data that may be hard to procure. On the other hand, simp le building footprints or even aggregated urban neighborhood data is more readily available as evidenced by commercial maps applications online and modern GPS navigation devices. 133

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4.1.1.1 Grids With the picture of the environment that needs to be captured in mind, we can now consider the appropriate techniques for capturing them as an LG ABG board set of points. As discussed in Section 3.3.1 a common approach is to use a regula r grid. Simpler scenarios where lines of sight and movement inside of buildings are not of interest ( e.g., emergency vehicles routing) could be sufficiently addressed with a 2D board. Another alternative for such routing problems is to employ a mobility graph representing the road network as such graph is more sparse than a full 2D grid. However, for military operations in a city, the height of the buildings plays a large part in the tactical analysis of the terrain and therefore the 3D grid is typically the best choice. While a mobility network could be employed in this case as well, the density of such network would need to approach a regular grid due to necessity of allowing tactical movement through the buildings and open space s in addition to basic routing through the streets. The vertical dimension of 3D grid can be addressed by stacking 3D prisms (see Section 3.3.1.2 ) as this approach most naturally mimics the vertical walls and building floors stacked one upon the other. The horizontal dimensions of the grid can typically employ either rectangular or hexagonal cells. Section 3.3 .1.1 presented the comparative analysis of the benefits of each approach and in a general case the hexagonal grid provides a better solution due to uniformity of distances in more directio ns. A possible exception in the urban environment is modeling a very grid planned city where most, if not all, street s intersect at 90 degree s angle. In such cases a rectangular grid helps alleviate rasterization, and to a lesser degree, alignment proble ms (see Section 3.3.1.1 ). 134

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The alignment is only partially improved as the distances between roads and buildings are often non uniform Furthermore, both of these benefits are usually fairly limited due to cities not sticking to the same plan for their entire area. Consider that following figures depict rectangular and hexagonal grids for 2 areas of the same city (Salt Lake, UT). As you can see, the rectangular grid is somewhat better aligned with the grid planned portion ( Figure 44 and Figure 45 ) ; however, that benefit seems completely lost just a few miles away ( Figure 46 and Figure 47) In the latter case, it is hard to judge as to which method provides the best fit and, a s we will see, cell sizes and various obstacle repre sentations have a much stronger overall effect. Figure 44 Rectangular 2D Grid for Grid Plan Urban Area 135

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Figure 45 Hexagonal 2D Grid for Grid Plan Urban Area Figure 46 Rectangular 2D Grid for Non Grid Plan Urban Area Figure 47 Hexagonal 2D Grid for Non Grid Plan Urban Area 136

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4.1.1.2 Obstacles Regardless of the shape of the under lying grid, we need to determine how to appropriately represent various terrain features on a game board. Two interrelated choices are required for this as presented in Section 3.3.2 we must choose the appropriate method to represent the obstacles and select the appropriate cell size for such obstacles. Whereas in some problem domains the cell size is selected based on the desired fidelity of movement and strategies given a particular temporal resolution, the urban domain presents terrain features that are so small as to dominate the cell size consideration. Depending on the problem at hand and the source data available, different urban elements will be the smallest feature of interest. We have previously introduced the spec trum of aggregation of the structures in a city. At one extreme, only entire city blocks, or built up areas (BUA), are represented rather than individual buildings. Each such area may be 300 500 meters across initially leading someone to consider similarly large board cells. However, these BUAs are commonly separated by streets which are quite important to capture on the board for the appropriate strategy generation. These roads are frequently 10 to 30 meters wide and therefore require a comparable size board cell. On the other extreme of the building aggregation spectrum, the particular problem may require representation of the detailed internals of particular buildings. In such cases, the smallest desired feature to be represented can be a door or a window. As residential doors are typically approximately 1 meter wide (36 in United States), the board resolution may need to be close to this size as well. Technically, the smallest feature could 137

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be even smaller, e.g., about 1215 centimeters, if each wall had to be represented as a cell. However, as discussed in Section 3.3.2.2 eggshell model can be used to avoid this and still repre sent the environment in sufficient detail [35]. Figure 48 LG Trajectories on Hexagonal Eggshell Urban Board Above examples illustrate the common range of urban use cases. If internals of buildings are required or external doors and windows are critical, eggshell model can be employed with cell sizes of about 0 .5 5 meters depending on building types ( e.g., residential vs industrial) and desired fidelity. Otherwise larger cells can be employed comparable in size to the smallest building or narrowest street that is needed to be explicitly represented, e.g., 10 30 meters. In order to alleviate the rasterizat ion and alignment problems density models from Section 3.3.2.3 can be employed. This method 138

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also allows the user to employ slightly larger cell size s. For instance, consider the 3 discretizations of the same urban area using 10, 20, and 30 meter hexagonal grids show in, respectively, Figure 49 Figure 50 and Figure 52. The vertical height of each cell is easiest to choose based on the height of the floors in the region as it provides a convenient convention of one cell per floor and is usually sufficient to appropriately model relative heights of buildings in an urban environment. We will use 3m for all of these examples. Building outlines, roads and rivers are displayed on top of the discretize d space where each cell is highlighted based on its dominant type using the following color scheme Note that the brightness of the orange color representing building depends on that cells density, i.e. percentage of the cell occupied by buildings. Hexagons that are entirely within a structure (100% dense) are shown with a bright shade, while those with small densities are darker. Cell Type Open Space Road River Building Eliminated Color Grey Yellow Blue Orange Green 139

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Figure 49 Urban area using 10m hexagonal grid The high resolution of 10 meter cells allows us to capture both small buildings and accurately represent the outlines of the larger buildings As you can see in the bottom right hand corner, the small huts are typically represented faithfully by a single hexagon of high density and very low density neighboring cells to capture the small overhang. As buildings get somewhat bigger the coverage grows from one high density to two and more. Large buildings, as those to the north east of the traffic circle, have a solid center of multiple full dens ity cells and an antialiased border around the edge. The most noticeable items that are smaller than a single hexagon are tightly arranged shipping containers just north of the river which are represented in aggregate as medium to high density cells. This board can easily be used for movement and indivisibility analysis, subject to the normal errors of this approach discussed in Section 3.3.2.3. H owever a small area of roughly 3.5 by 2.9 kilometers requires 333x349 2D cells. Considering the hills 140

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and buildings elevations in this area, 140 layers of 3m tall hexagonal prisms are also required bringing the total number of cells to 16,270,380. By filtering out unnecessary cells such as those underground or up in the air the total number can be brought down to 130,518 ( reduction by approximately 125 times). Figure 50 Urban area using 20m hexagonal grid Let us consider the effect of increasing the cell size from 10m to 20m. As you can see in Figure 50, th e small buildings in the south west are no longer represented individually but the density variations are sufficient to distinguish between tighter and sparser groupings. The shipping crates north of the river are modeled very similarly to 10m case. The larger buildings are also represented reasonably well, albeit with a fuzzier outline. The roads still standout sufficiently as well. While this board is clearly less precise then the previous example, it can still be used for modeling movement. However, depe nding on the required fidelity, this resolution may not be sufficient for LOS 141

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calculations by itself requiring additional techniques such as using Source Data Submodel shown in Figure 51, as discussed in Section 3.3.2.4 The board size for the same area is reduced to 167x175x140, i.e., 4,091,500 total, or only 35,009 after unnecessary locations are eliminated Figure 51 Urban Area Source Data Submodel Figure 52 shows what happens when the cell size is further increased to 30m. At this resolution, there is much less differentiati on in areas with smaller buildings such as in the south the density is almost monotonous across that entire region. As the cell size is now bigger than the cross section of a lot of the larger buildings as well, those structures are not represented as co nsistently. The density is equally affected by the alignment of such buildings within the cell as by the size of the building itself, with frequently wide fluctuations of the density of the hexagons representing single building s This model could 142

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possibly be used for a low fidelity simulation; however, even Source Data Submodel (SDS) would not be of much help in this case as it only enhances fidelity of LOS and movement between cell centers, and there are simply not enough cells to capture key locations relevant to a tactical operations. The board size for the same area is now reduced even further to 111x117x140 cells i.e., 1,818,180 total, or only 16,434 cells after unnecessary locations are eliminated Figure 52 Urban area u sing 30m hexagonal grid The following table summarizes the relative benefits of these 3 resolutions. Note that this analysis must be performed for each specific urban area of interest, as it is directly correlated to the sizes of structures that are presen t. An area with only larger buildings may allow use of larger cells than a village with primarily small huts. The final choice of resolution is thus driven by the specific terrain features as well as the necessary fidelity of the ABG to achieve the desired tactical modeling accuracy. 143

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Hexagon S ize Total Cells Filtered Cells Movement Quality LOS Quality LOS w/SDS Quality 10m 16,270,380 130,518 High High High 20m 4,091,500 35,009 Good Good High 30m 1,818,180 16,434 Low Low Low 4.1.1.3 Dynamic Obstacles Previous section covered obstacles that exist as part of the terrain itself and can be directly incorporated into the board structure. Military operations frequently involve dynamic obstacles. Some common examples include concertina wire, road blocks, rubb le, mine fields or improvised explosive devices, destroyed bridges, and smoke obscuration. These obstacles can exist prior to start of the mission, be created as part of the mission, or destroyed in the course of the operation. In addition to physical obstacles, no go areas or no fire areas can be defined for political, social and cultural reasons. For instance, it may be politically undesirable to drive a heavy combat vehicle through a particular neighborhood. Similarly, areas around certain key buildings can be off limits for both movement and weapon use [36, 39]. Due to the dynamic nature of such o bstacles, they must be incorporated into the board structure or removed at various points of the scenarios analysis. LG strategies generation can take them into account in the same exact way as static terrain elements, whenever they are present on the boar d. Its useful to consider that restrictions often involve linear features such as barbed wire, boundary of a minefield or a perimeter 144

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around a church. As discussed in Section 3.3.2.3 such objects are difficult to represent within the density model and they are better addressed as static fence like obstacles by the Source Data Submodel presented in Section 3.3.2.4 4.1.2 Mobility Now that the board structure has been selected, let us consider how to properly represent movement of various game pieces. In Linguistic Geometry, this is accomplished e mploying relations of reachability, as presented in Sections 2.1 and 3.4.2. Typically, reachabilities need to be discussed in conjunction with choosing appropriate temporal discretization for each game step. However, we will defer those considerations until Section 4.1.3 4.1.2.1 Reachabilities The game pieces for an operation in urban terrain typically fall into two general categories people and ground vehicles. These groups have both common and unique requirements that must be considered. For both of these groups, we need to take into account the type of the terrain the movement occurs on e.g., entities can move faster on roads than off road. Similarly, cell density considerations are important as both vehicles and people on foot would move slower through a more dense area, whether due to ve getation density tightly clustered shipping containers or other small manmade structures. On the other hand, vehicular motion must also account for the effects of moment of a fast moving object inability to rapidly change direction or speed which are not as 145

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much of a challenge for pedestrians. These challenges can be accounted for by expanding the board structure to include additional phase spaces for direction (Section 3.4.4) and speed (Section 3.4.5 ). Note, that the importance of these additional dimensions is directly related to how small the game step is. With a sufficiently large game step (greater than the amount of time it takes to accelerate to full speed), speed dimension is no longer required as full acceleration or deceleration takes less than a game step and thus does not need to be represented in the discrete world of an ABG. Similarly, if a game step and board cell are both large enough for a vehicle to completely turn around within a single board cell during a small fraction of a single game turn, the direction is of less imp ortance. However, the typical cell sizes and time steps for a MOUT game do necessitate the use of at least the direction phase space. The speed phase space is of a less importance for common urban cases if one consider s that the acceleration and decelerati on represent a very small portion of an overall movement trajectory and that driving is commonly done at rates far below maximum possible vehicle speed. This allows us to simply represent the vehicle reachability using its most average or most common rang e of speeds, and avoid the cost of the speed phase spaces unless such fidelity is required. Such utility comes at the cost of the increased board size as the total number of 3D cells must be multiplied by the number of discrete directions and speeds. Due t o these reasons, we will use 6 discrete directions (to match the hexagonal cells) and avoid the speed phase space altogether for the rest of this section. Dismounted personnel also has one distinct characteristic the ability to travel through buildings both horizontally and vertically. Cars can also perform such movements 146

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through multi story garage structures, but in a much more limited manner. By modeling the space as stack s of hexagonal prisms representing individual floors, vertical movement via elevators or stairs can be easily captured. The horizontal indoor movement can be differentiated from outdoors movement in terms of its speed by using different reachability relations in indoor vs outdoor cells, or by leveraging cell densities. The latter approach is particularly interesting as it allows modeling different ease and speed of passage through wide open structures, e.g., an aircraft hangar, or dense ones, such as a floor with office cubicles. Figure 53 Dismounted outdoor reachability pattern By putting all these concepts together, a sample reachability for a human game piece can be constructed using 2 patterns. The fir st outdoor pattern ( Figure 53) allows the entity to move to any cells 2 hexagons away with a ny orientation to capture persons 147

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ability to rapidly ch ange direction to look, walk, or run in any direction. It also allows for a movement up or down a single board layer to represent the ability to climb up or descend down various slopes. The maximum slope ratio allowed by this reachability, assuming 20 mete r by 3 meter hexagonal prisms, is 3 in 40 or 1 in 13.33. This can be easily adjusted by the vertical size of the cells or including additional layers in the pattern. As this pattern represents outdoor walking, the applicability is specified as originating from a land, road, or forest cell type, but not indoors or water, and can reach any destination cells other than water. Asymmetric applicability allows a piece to use this relation to enter buildings, but not exit or move within one. The 2nd indoor patte rn ( Figure 54 ), can then be used to complete the reachability. Movement on a single layer is defined the same way as in the previous example, as two hexagon s in any direction, but movement up or down a slope is no longer present as buildings tend to have flat floors. However, movement up and down 1 or 2 layers in a single column is allowed instead to represent usage of stairwells or elevators to chang e floors. This pattern is only applicable inside, b ut allows exiting buildings. The last missing element is to account for the density effect on the speed. Two limits can be specified to properly capture this requirement: maximum point density and maximum accumulated density. For instance, maximum point density can be set to 100% while maximum accumulated density can be set to a lower value of 30%. Within a 100% dense building, a person could then move 1 cell per game turn but no more, as maximum density w ould be already exceeded. However, outside moving through a sparsely grown park or urban clutter of only 10% density, the piece could move 2 hexagons. Once the local density exceeded 15%, the person would have to slow down 148

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to a single hexagon per game turn as in a building. Different speed of movement in a building can also be captured by setting the density of an open aircraft hangar to a low value of 10%, while setting the office building to a higher value, e.g., 60%. Figure 54 Dismounted indoor reachability pattern Both of these indoor and outdoor patterns are excessively complex and correspond to a very high branching factor. For instance, the indoor pattern has up to 23 cell reachable in one game turn (assuming large open building), with 6 directions each, providing the game piece with 23*6 or 128 different moves. However, most of these are very atypical to human motion such as moving 4 0 meters forward while turning around 180 degrees within a span of 30 seconds. To simplify the computational complexity of an ABG, this reachability can be trimmed to eliminate such uncommon cases and instead included only the bare minimum required to capture the tactical effects of the entities 149

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movement. The guiding principle of such analysis is to consider how entire trajectories would be composed of such individual moves rather than focusing on accuracy of each specific move. Figure 55 Vehicle road reachability pattern Let us explore this using the example of the r eachability relation pattern for a vehicle shown in Figure 55. It represents the entity being able to move on roads up to 7 hexagons per single game turn, yet seemingly unable to move 4 or 6 hexagons in a single game turn. However, this does not negativ ely affect the ability to generate desired trajectories if the trajectory is at least 3 moves long. Any desired distance up to 14 cells away can be easily constructed out of 1, 2, or 3 individual movements each precisely 1, 2, 3, 5, or 7 cells long. Similarly, some of the destination hexagons are shown reachable with a very limited set of directions which when combined into a trajectory are still 150

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sufficient to represent any desired movement. Such reduced, or optimized, reachability relation would not be a ble to capture all possible physical movements of the vehicle, but at a significant branch factor reduction the error would not be more than 1 cell or 1 move. Considering the small cell sizes and short game intervals, e.g., 10 seconds, these errors are neg ligible for tactical validity of the generated strategies. This reachability also allows movement up or down a slope of rati o 6 in 20, or 1 in 3.3, forward or backward For completeness, another reachability pattern for off road movement should be included to fully define the reachability relation. As s hown in Figure 56, it is simply a reduced range version of the previous pattern to simulate slower maximum speed of a vehicular movement on terrain other than paved surfaces. Figure 56 Vehicle off road reachability pattern 151

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4.1.2.2 Trajectory Selection The purpose of the reachability relation is to capture the range of possible movements f or an entity so that tactically advantageous LG trajectories, zones and strategies can be generated. As discussed in Section 3.4.3 and shown in Figure 30 for any starting and destination points multiple trajectories can usually be generated of the same length, due to discretization artifacts. LG Trajectory and Move Evaluation Function can be employed to select the preferred trajectories for such trajectory bundles. Typically two principles are involved physical criteria, such as preferring straight path to a zig zag, and tactical criteria, such as threat avoidance. In an urban environment, physical consideration s typically center on minimizing transitions between different terrain types. For instance, all otherwise the same a vehicle would prefer to drive on the road rather than off. Similarly, a person would typically move outdoors until reaching the target house, rather than constantly cutting through buildings Shortcuts through buildings can, of course, become p referred if they significantly re duce travel duration or improve the trajectory in other way. As usual, such factors are secondary to tactical consideration s Typical tactical consideration s involve balance of dual requirements of safety and observation. A bility to observe enemy positions or possible avenues of approach is often key to a mission. Additionally, a combat vehicle with modern optics and fire control systems can effectively engage targets at several kilometers range. Even dismounted soldiers can attack targets several hundred meters away. Thus occupying positions that provide long ranges of sight and, therefore of fire, is advantageous for tactical control of the terrain. However, movement through such positions exposes friendly forces to the 152

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s ame possibility of observation and engagement from the enemy. Locations that cannot be easily seen and fired upon are more desirable for safe passage. In some cases, asymmetry between capabilities of various forces can be further exploited in this analysis. For instance, modern combat vehicles with long engagement ranges moving through an urban area occupied by enemy forces with only short range rocket propelled grenades (RPG), can improve their survivability by avoiding coming within RPG range of any buil dings where an enemy could hide. Conversely, if an enemy is armed with long range anti armor missiles, the safety may be found close to the buildings to make targeting harder for the enemy. Figure 57 Vehicle threat map 153

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Figure 58 Dismounted threat map All of these considerations can be formalized under the general category of LG Terrain Threat Analysis. Using considerations listed above, such as safety distance from buildings and long range observation o r engagement potential a threat & quality level can be assigned to each board location. These values can be different for each particular unit type depending on its particular capabilities and vulnerabilities and can evolve over time. Figure 57 demonstrates the vehic le threat map for the same urban region, highlighting the dangers of being close to or in between buildings. Conversely, terrain analysis in Figure 58 is tuned to show the safety provided by buildings and other cover to the dismounted troops while accentuating the dangers of open spaces. I nitial static Terrain Threat Analysis values can be computed based on static terrain information, such as terrain types, lines of sight, and likely enemy composition. These values can then be augmented with the specific knowledge of any friendly and enemy position or armament data, as such knowledge is discovered during the mission, to evaluate the dynamic Terrain 154

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Threat. For instance, the static analysis can highlight one particular road as less dangerous than another, but if an enemy ambush is discovered along that route, its threat levels will significantly rise in the dynamic analysis. The benefit of incorporating these values into the formal threat maps correlated to the LG board positions is in selecting the safest, or otherwise advantageous, trajectories during LG strategy generation. As an example, consider the trajectory for a dismounted game piece shown in Figure 59. The more direct path would have to follow the roads Yet, as can be seen from the shading of the hexagons, the total threat along the path is minimized by ducking in between buildings and minimizing road crossings. Figure 59 Dismounted threat avoidance trajectory 155

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4.1.3 Heterogeneous Systems 4.1.3.1 Variable Step ABGs The game pieces for an operation in urban terrain typically fall into two general categories people and ground vehicles which e xhibit a very wide range of mov ement speeds. Y et their actual movements have to be represented on the same board with relatively small cells so that effects of b uildings, roads and other terrain features can be captured. As speed, board cell size, and game turn intervals are interrelated as discussed in Section 3.4.1 this leads to a conflict between the best time interval to choose for all vehicles and the one for pedestrians A person on foot moving at 5 km/h on a board with 20m cells using the reachability relation in Figure 53, would correspond to a time interval of about 2 9 seconds. A vehicle driving at 100 km/h on the same board using a reachability relation in Figure 55 has to use a time interval of about 5 seconds. In order to accommodate this, we must either adjust the reachability of the vehicles from 7 cells to 40 or more, which is computationally expensive, or switch to a Variable Step ABGs. As we have thoroughly covered in Section 3.5.2 using just this example, Variable S tep ABGs pro vide a lo t more efficient solution for urban operations and allow for precise adjustment of the individual time steps to suit specific speeds of each entity type. 4.1.3.2 LG Hypergames In addition to ground units air assets such as helicopters and close air support, can be considered as a natural extension of this domain. Maximum speed of an Apache attack helicopter is near 300 km/h, or 2.5 to 3 times faster than a ground c ar, which can 156

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certainly be represented using a Variable Step ABG. However, in order to implement this solution the game board must include the locations required for movement of the helicopter above ground level. These cells are not needed for the vehicular and dismounted movement and, as discussed in Section 4.1.1.2 can be filtered out in order to minimize the total number of locations. This reduction can be quite significant, e.g., from 4,091,500 to 35,009 cells for the 20m example from that section. On the other hand, the high resolution of the 20m hexagons is excessive for the helicopters as they would not typically try to fly through narrow streets and in between buildings. These reasons combined lead to the conclusion that the board requirements of these two sets of pieces are distinctly different, and, therefore, Variable Step ABG is not an appropriate solution. The better approach is to model helicopter actions in their own ABG that is best suited for such operation a nd to employ LG hypergames to link such air and urban operations together so that strategies can be found for both ABGs at the same time (see Section 3.5.1 ). This method would then be just as applicable to modeling the close air support by using an appropriate ABG to capture a larger geographic area at a lower resolution more appropriate for the very fast movements of aircraft. Details of such modeli ng of air operations are presented in Section 4.2 LG hypergames provide an exhaustive solution to representing varied assets such as dismounted in fantry and jet aircraft. However, this solution does come at the added computational cost of additional ABGs, the overhead of hypergame synchronization, and the complexity of setting up such a system. It is always useful to ensure that the fidelity of such approach is justified. Air support in urban operations presents an interesting 157

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example of a problem where a much simpler solution may be sufficient. From the perspective of the ground forces, any movement routes, targeting solutions, and angles of attack of the aircraft are irrelevant. The only considerations are the effects that the air support brings to the battle. This provides an alternative method of modeling the aircraft as just a universal weapon that can destroy any locations on the board, of course after a suitable delay to represent the time required to scramble the jets and reach the target. 4.1.4 Incomplete Information Urban operations, like most others, require modeling of incomplete information. The two key factors to consider are the complexities of the terrain environment and the population density. The most common sensor employed by each game piece is simply eyesight, possibly augmented by binoculars and other optics such as those found in the combat vehicles targeting systems. Visibility in a city is hindered by the various structures and obstacles, severely limiting the ability of one player to observe movements and actions of the other player using such devices. However, such restricted information can be augmented by the civilian population or civilian sensor assets. Depending on the political alignment of the citys inhabitants, they could form a sensor network that can collectively observe almost any location within the city and report this information to the enemy using simple landline and cellular telephones. With a large enough portion of the civilians on the ir side, one of the opponents could achieve almost complete picture of their adversarys forces positions. This is a common problem for urban counterinsurgency operations. Civilian population provides another benefit to insurgents the ability to blend i n and hide in plain sight, which further reduces the ability of their enemy 158

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to pinpoint their positions due to being unable to tell a combatant apart from a peaceful citizen. This effect further complicates the incomplete information problem as the opponents forces can then be detected only when they perform an obviously hostile action s, such as shooting. The other global sensor asset that can be present is a network closed circuit television cameras, or CCTVs. While such systems can provide significant su rveillance benefit they may be vulnerable to destruction. Both CCTVs and observation provided by the overall populace can be modeled as a global sensor covering the whole city to avoid the overhead of capturing each individual citizen or camera. However, o ptics on the friendly units and dedicated sensor assets, such as unmanned aerial vehicles (UA V s), can be modeled individually using line of sight visibility relations especially when Source Data Submodel is employed for added fidelity. Figure 60 shows the line of sight from two observers on an urban game board using 20 meter hexagonal cells and Source Data Submodel. Highlighted cells are visible from at least one of the 2 game pieces, while a few cells with a brighter color on the north side of t he area by the river are visible from both. More modern equipment can also currently be employed in the cities including acoustic, seismic, and see throughwall sensors. While wave propagation of such devices is more complex than visible light, they can be modeled using the same concepts of probabilistic sensor detections contributing to the incomplete information worldview described in Section 3.6 159

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Figure 60 O verlapping Source Data Submodel LOS from 2 observers, 20m cells Depending on the size of each force and available communications, more than one worldview per player may be required. For instance, two platoons operating a couple kilometers away from each other in a dense city may not be able to exchange information effectively enough to be considered a single communication group (see Section 3.6.3) with a combined worldview. Rather, each platoon should have its own view of reality and receive information from the other group with a delay that represents the difficulty of conveying and transmitting knowledge in a complex urban environment. It can also be beneficial to include modeling the limitations of the communication devices since radio waves can be hindered by the buildings and other structures and periodically leave the two groups out of touch with each other. On the other hand, modern devices such as Blue Force Tracker (BFT) can be employed to effectively share the units location information between these groups effectively joining them into a s ingle worldview. 160

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4.1.5 Weapon Systems 4.1.5.1 Strikabilities Common weapons in urban environment include both direct fire and indirect fire systems. Direct fire weapons can only engage targets that are wit hin line of sight (LOS) of the shooter and include such c ommon ex amples as assault rifles, machine guns and rocket propelled grenades (RPGs). While the actual flight of the projectile does not occur along a straight path over a long engagement range, the ballistic trajectories of such weapons are flat enough to be prac tically considered as a direct line between the shooter and the target. These systems can then be modeled similarly to LOS based sensors described in the previous sections. However, in addition to the LOS requirement, there are typically additional imitati ons including minimum and maximum ranges, eleva tion, and traverse rotation. For example, a chain gun mounted on a Bradley can only engage targets within approximately 10 to +60 degrees elevation which prevents it from being able to fire on enemies in a t all nearby building. RPGs and most missile systems cannot be used to lock on to or destroy targets that are closer than a minimum engagement range specific to each system. The maximum ranges of some of these weapons are quite large, e.g., in excess of 3km for vehicle mounted chain guns or tank guns, which corresponds to more than 150 20m hexagonal cells on a game board. Such long strikability relations are hard to represent as patterns on a hexagonal grid similar to reachability patterns in in Figure 55 Instead, they are easier to define simply as parametric strikabilities defined by their minimum and maximum ranges, as well as horizontal and vertical angles defining the fields of fire. The strikability relations can then be evaluated 161

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between any pair of board location by verifying that the vector from source to target comp lies with these weapon parameters and LOS requirement. Additional surface danger zone (SDZ) restrictions can be added to further restrict employment of such weapons to avoid accidental killing of friendly forces. The strikability relation is considered blo cked if any friendly game pieces are located within the SDZ of the weapon, typically described as a dispersion cone along the firing line as shown in Figure 61 [53]. Figure 61 Cone SDZ for firing small arms direct fire weapons I ndirect fire systems, e.g., howitzers and mortars, can effectively destroy enemies along ballistic or other non line of sight trajectories and can be employed from out of direct line of sight of the target and from long distance s Strikability relations of such weapons in the simple st form can be defined simply by the minimum and maximum range at which they can engage targets, assuming that effects of terrain obstacles are insignificant. However, the fidelity and complexity can be increased by considering that the actual ballistic t rajectories can still be blocked by intervening structures, such as 162

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buildings, in close proximity of the targets. Utilizing specific weapon characteristics, the strikability can be determined by verifying that an unobstructed ballistic path exists between any pair of points. As with direct fire weapons, additional friendly safety considerations can further limit employment using risk estimate distances (REDs). In order to avoid casualties by prohibiting fire upon enemy units located too close to friendly fo rces, REDs take into account inherent inaccuracy of the weapon platform, as well as fragmentation patterns of the projectile. As these systems can fire from beyond line of sight, targeting aids, such as spotters, must also be considered. A spotter who can visually observe the target, provides information to the indirect weapon controller that allows for the fire to be precisely adjusted to hit the desire d enemy target. Another characteristic of these systems, is the time required to deploy, setup, and aim s uch complex weapons. A soldier can typically fire an automatic rifle almost instantaneously at any discovered enemy. However, a three person mortar team would require additional time to setup, load, aim and fire their weapon. For higher echelon indirect fire assets, this delay would further incorporate the communication inefficiencies of requesting fire support up the chain of command. Lastly, close air support (CAS) comprises a special type of both direct and indirect fire weapons. CAS can include both fixed and rotary winged aircraft engaging enemy with missiles, bombs, or machine guns so as to assist friendly ground forces. The strikability relations of the s pecific weapons used by CAS can be modeled using direct and indirect fire strikabilities described above; however, they can be easier represented as a simple universal relation simulating the fact that almost any location on the board can typically 163

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be enga ged by such assets. REDs and the time required for the air support effects to be delivered on target must still be consider ed in the same manner as for indirect weapon s 4.1.5.2 Weapon Effects Section 3.7.2 presented two primary methods for adjudicating the outcome of weapon engagements Probability of Kill (PK) and Attrition Rate (AR). PK model is better suited for ABGs where each game piece represents a single real world object, e.g., a tank or an airplane. On the other hand, AR model is appropriate when game pieces are aggregated to represent collections of real world objects, such as a platoon of soldiers. Operations in urban domain straddle both of these cases, as game pieces can represent both individual vehicles and small groups of soldiers. For this reason, either model can be employed depending on the preference for stochastic or deterministic game. PK model can b e applied to small group game pieces by evaluating a probability that each individual member of the group is eliminated using random number generator, and then adjusting the size of the group represented by the game piece accordingly. On the other hand, th e AR model can be applied to single entity game pieces by reducing its size during every engagement based on the calculated attrition until it reaches a threshold which indicates that the entity is destroyed. For instance, a vehicle of size 1 could have its size decremented by 0.5 every time it is engaged by an RPG. With a destruction threshold set at 0, the vehicle will be considered destroyed after 2 RPG hits. ARs can be adjusted to correspond to the PK model, so that the destruction occurs after the ex pected number of hits required based on the desired PK values. For the above example, a PK of 50% would also correspond to 2 RPGs, on average, required to destroy the vehicle same as the AR 164

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model. The key difference between these two approaches is that attrition model leads to a deterministic ABG where every game leads to the same outcome if the players execute the same sequence of moves. This can be desired for a game used for tactical analysis to produce repeatable results. On the other hand, the PK mo del introduced a probabilistic element that makes each execution potentially different from another, which is more suited for a war gaming rather than the analytical use case. The w eapons used in these scenarios typically exhibit different lethality based on range due to the difference in accuracy. This concept can be formalized as a Probability of Hit curve that can be used with both AR and PK based models by simply multiplying the base values by the appropriate PH value determined by the range. Figure 62 illustrates an example of a PH curve with 100% probability of hit at short ranges and sharp drop off towards a 10% probability at medium to maximum range. These curves are frequently determined largely by the weapon itself, the accuracy could be affected by the target. For convenience, any target specific effects could be considered part of the ARs and PKs so as to allow use of a single PH curve per we apon. On the other hand, probabilities of kill and attrition rates are commonly different for each particular target type. For instance, an assault rifle can have a reasonably high attrition rate against enemy infantry, but near zero rate against an armore d vehicle. PH, PK, and AR values are further affected by the competency of the shooter. This is especially true in urban environments where combatants can span the range between professional army and untrained insurgents. Depending on desired fidelity, additional considerations that affect PH such as the posture of the target and shooter to distinguish between the high accuracy of a prone 165

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soldier engaging a stationary enemy, and the low accuracy of a running s oldier engaging a moving target, can also be in cluded. Figure 62 Probability of Hit curve with rapid drop off Close quarters of the urban environment require additional consideration for the blast radius of the explosive effects of some weapon. An RPG fired at a game piece representing a vehicle would also cause significant casualties to any infantry in close proximity, which could be represented by other game pieces. In this case, th e attrition effects should be applied to the intended target as well as any other units within the blast radius of the weapon. As lethality decreases with increased range from the epicenter, a sliding scale can be used to reduce the attrition for pieces lo cated further away from the center of the explosion. 4.1.5.3 Paired and Synchronized Trajectories While most of the weapons in urban environment are aimed by the shooter themselves, some weapons do require guidance from a different unit. Typical examples include c alling in an airstrike which may require a soldier on the ground to illuminate the 166

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target, or a spotter who must acquire line of sight on the target so as to direct long range artillery fire. Using paired trajectories concept from Section 3.7.3 LG Zones can be generated that take into account this dependency and always include pairs of trajectories one for the shooter and one for the observer provi ding fire guidance. Even more common is the need for very close coordination between multiple combatants to achieve local fire superiority. This can be achieved by generating multiple LG negation trajectories from several game pieces against a single e nemy piece and then control movement along them so as to engage the target at the same time. This concept was presented in Section 3.7.4 as Synchronized Trajectories [36]. An example of a schematic LG Zone employing this concept is shown in Figure 63. The main trajectory represents the movement of the lead Blue fire team through a city, while the first negation trajectory for the Red piece shows a likely interception location. In response to this Red action, 3 prerequisite second negation trajectories are generated for friendly teams to occupy positions from where they can engage the enemy location. Finally, a second negation trajectory is generated for another Blue piece to flank and ente r the building occupied by the Red unit, once the supporting units are in position to provide covering fire. 167

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Figure 63 LG Zone with synchronized negation trajectories 4.1.6 Mission Concepts 4.1.6.1 Goals and Missions Military operations in urb an terrain can involve a wide spectrum of missions as demonstrated by the Army tactical doctrinal taxonomy in Figure 64 [54]. The goals can range from offensive and defensive combat missions to engineering and stability operations. Furthermore, each mission typically consists of multiple intertwined subtasks required in order to achieve the overarching primary objective. Execution Matrix, as shown in Figure 42 is commonly used by ground forces to represent tasks that must be performed by each subordinate unit during each phase of the operation. For instance, an engineering section may need to clear a n obstacle, while 1st and 2nd platoons provide security. In 2nd phase, the 1st platoon can then assume fire support position, while 2nd platoon prepares for the main attack. And finally, in 3rd phase, the 1st platoon can attack the enemy by fire so as to suppress them, as the 2nd platoon assaults the target objective. 168

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Figure 64 Army tactical doctrinal taxonomy Each task in Figure 64 has very distinct meaning to a military commander. However, from an LG perspective these missions can be reduced to their core building block s moving to a particular location in order to perform a particular action, e.g., destroying an enemy unit or deploying a minefield. These target locations and actions can then be used to generate the main trajectories in LG Zones, followed by the appropr iate negation trajectories, so that the winning strategies for these ABGs can be found [36 38]. Translat ion between military task definitions and the corresponding target locations and 169

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actions can be very simple ( e.g., road march) or very complex ( e.g., cordon and search), and can be different for various branches of armed forces ( e.g., Army vs Marines) [52]. In all cases, detailed cooperation of military subject matter expert is required to properly cap ture the mapping of such tasks into LG missions discussed in Section 3 .8.1 4.1.6.2 Prescribed Behaviors Just as Army tactical doctrinal taxonomy above, the armed forces have codified the appropriate tactical prescribed behaviors in a variety of field manuals such as [50, 55 58]. The goal of these documents is to provide consistent information for training large number of soldiers in the doctrinally correct principles and behavio rs. These manuals can similarly serve as guides to ensure that the units within the LG ABGs follow the same rules. As described in Section 3.8.2 prescribed behaviors can be used side by side with the LG Zones to generate the overall strategy for both active and inactive periods during the mission. One example would be various techniques and formations for vehicular movement, such as bounding ove r wat ch shown in Figure 65 [59]. Employing these techniques even when enemy is not yet detected, allows the military unit to be in the best defensive or offensive posture in case the enemy is discovered. The key consideration for implementation of such doctrinal tactics, techniques, and procedures (TTPs) with an ABG is the interac tion between these principles and LG based reasoning. The TTPs are meant as guiding principles that ensure reasonably successful outcome across the wide range of missions; however, these principles are generic and may not be 100% applicable in every situat ion. On the other hand, LG based strategies generate winning solutions for each particular situation within an ABG, yet some of th ese strategies 170

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could be difficult to execute for a military force if they do not comply with the TTPs that the soldiers are trained to follow. Thus a compromise must be achieved, whereas the LG generation is restricted by some doctrinal principles while remaining free to break others if the situation demands it. As with mission tasks, military subject matter experts must be cons ulted to ensure that appropriate TTPs are respected so that the resultant strategies are usable by the actual human soldiers represented by the game pieces within an ABG. Figure 65 Bounding O verwatch 4.2 Air and Naval O perations While air and naval domains may seem very dissimilar, they share a lot of common elements. Both of these operation types typically cover large geographic regions and involve relative ly fast moving motorized platforms, rather than individual people. In addition, modern naval combat include s large numbers of aerial elements, such as planes, helicopters and missiles launched from aircraft carriers, missile cruisers, and other vessels. To a lesser degree, these operations must also account for ground units and their 171

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effects and interactions with air and naval forces. These environments are quite different from the urban operations covered by the previous Section 4.1 Rather than small high resolution regions, very large areas must be modeled. The technological platform, weapon and sensor capabilities are also more advanced. The velocities of the aircraft and cruise missiles far exceed those of any ground based vehicles. Even modern optics and main gun targeting systems of a battle tank can only detect and engage an enemy up to 4 8 kilometers ahead further restricted by terrain LOS effec ts. On the other hand, radars and long range missile s provide the aircraft with capability to engage one another at ranges in excess of 100 km. Cruise missiles extend such ranges even further to thousands of kilometers. Furthermore, sophisticated counterme asures can be deployed to avoid detection or targeting. This section will analyze applicability of particular modeling approaches from Chapter 3 to various aspects of such air and naval operations [2, 11, 27, 49] focusing on highlighting the key differences and specifics of these domains as compared to the previously discussed urban operations. 4.2.1 Spatial Discretization The terrain environment that must be modeled is significantly less complex than a city the only key consideration is ground elevation, albeit over a much larger area [11, 12, 49]. Most of the game board would cover open terrain of various elevations hills, valleys and plains. It may be prudent to consider some of the ground fe ature elements such as fore sts and urban centers in order to better account for the interaction of ground forces with the air elements, e.g., best hiding locations for Surface to Air Missile (SAM) 172

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sites. However, due to the large size of the operations these terrain elements can usu ally be considered in aggregate rather than as each individual building or tree. The lack of abundant manmade objects simultaneously removes the need for high fidelity resolution and the problems of alignment and rasterization. As individual buildings or s treets are not of high importance, the size of the cells is no longer tied to the smallest terrain feature of interest. Instead, the resolution can be chosen based on the overall size of the target area, desired temporal discretization, and speeds of vario us agents. 4.2.1.1 Grids Under these conditions, either rectangular or hexagonal grids can be employed. However, without any need for alignment with orthogonal manmade features, hexagonal grids would present a better choice due to more equidistant cell centers (se e Sections 3.3.1.1 and 4.1.1.1 ). Purely naval exercise could be well served with just a 2D game board. Even air assets can be modeled without adding a 3rd dimension, if the scale of the operation is such that relative altitude of the entities does not significantly affect movement or engagement characteristics. Naturally, for full fidelity modeling of air combat, a 3D board can provide significant benefits. Maximum speed, vulnerability to ground launched missiles, performan ce characteristics of radar and weapons all depend on absolute altitude of the aircraft as well as relative altitude between two entities. The easiest method for extending hexagonal grids into 3 dimensions is to use hexagonal prisms, just as in the urban e nvironment [2, 11]. This solution captures well the distinctly different nature of vertical movement of aerial platforms as compared to the horizontal movement. 173

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Air and naval operations span large geograp hic areas and are affected by the curvature of the Earth problem discussed in Section 3.3.1.4 Geographic projections, such as UTM, can be employed to map the spherical ground surface into a 2D grid of hexagons with minimal distortions. Unfortunately, such flattening creates an impression that the line of sight exists between two low altitude points even if in reality it is blocked by the horizon. Ev en from the top of the aircraft carrier, roughly 60 meters above water, the horizon is only about 28 km away. Rather than trying to incorporate this curvature into the grid structure of the board, the easier solution is to simply account for the limitation in line of sight range by incorporating it into the definition of maximum range of a sensor. 4.2.1.2 Obstacles Just as visibility limit s that are imposed by the horizon, the rest of the physical obstacles for aerial and naval movement are due to the terrain elevation. Various mountains and valleys directly affect movement, detection, and weapon engagement of aircraft, while shoreline and water depth similarly constrain actions of the seafaring vessels. Due to the scale of these engagements, forests and manmade str uctures are typically of much less concern. Terrain elevation can be easily captured on a 3D board using techniques from Section 3.3.2.1 Open and closed cells can be used to distinguish between air locations that can be moved through and ground locations that block movement and line of sight. We can then add various types of closed cells to differentiate between ground and water, as well as other terrain types such as forests or urban areas. To reduce board size, the closed cells can be eliminated, as no entities can ever occupy those locations, and under types used instead to store the type of underlying solid layer 174

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in each open cell. Using such board structure, various reachabilities can be defined for naval vessels to only travel in open cells over terrain of type water ( Figure 66 ) and to block aircraft from flying into mountains ( Figure 25). Figure 66 Coast terrain as hexagonal grid board The primary downsid e of these models i s the staircase effect where any sloped terrain, e.g., mountain or hill, is represented as a jagged line when viewed from the side (see Figure 22 ) due to alternating solid and open cells. This directly affects the ability to correctly analyze mutual visibility for any locations located close to the ground in such areas as the edges of the closed cells would erroneously block LOS along the slope. Due to the size of the overall terrain regions needed to be represented, it is not computationally practical to increase the spatial resolution to significantly affe ct this problem. Instead, two other solutions to the rasterization problem can be used density model (see Section 3.3.2.3) and Source Data Submod el (see Section 3.3.2.4 ). Of these two, Source Data Submodel is the best solution to ensure the line of sight accuracy. However, 175

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for best fidelity of 3D space representation, these two methods can even be used in conjunction, as in urban terrain. 4.2.1.3 Dynamic Obstacles There are typically no physical dynamic obstacle s in naval and air operations with possible exception of river locks and draw bridges. Ho wever, there are numerous restrictions due to the geopolitical and tactical considerations that can effectively achieve the same effect. Common geopolitical examples include restricted airspace, no fly zones, and territorial waters which could affect both the game pieces permission to move through or engage targets in certain regions. Other restrictive zones can be generated, either by the user or automatically by the LG threat analysis, based on the threat presented by Surface to Air Missile sites, naval minefields, and other static defensive measures [39]. All of these types of dynamic obstacles are not abso lute they can be overridden by the strategy generation if mission requirements or rules of engagement change, e.g., an aircraft being given permission to enter restricted airspace of a foreign nation. Threat based restrictions can also be removed by elimi nating the enemy units that are the source of the threat. 4.2.2 Mobility 4.2.2.1 Reachability Movement of aircraft and ships has a lot of similarities with ground based vehicles discussed in Section 4.1.2.1, such as inability to rapidly change direction or speed. These effects can be modeled in much the same way, by utilizing direction (Section 3.4.4 ) and 176

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speed (Section 3.4.5 ) [2, 27, 40, 41]. The particular number of discrete speed levels and directions depend on the properties of the entities, the size of the board cells, and temporal resolution. As the overall resolutio n of the ABG is increased with smaller cells and time intervals between game turns the importance of these elements becomes significantly higher. For example, modeling intricacies of close range aerial battle between two fighter aircraft would necessitate very detailed spatial and temporal resolution s as well as high number of speed levels and directions. On the other hand, a strategic operation taking place over hours and hundreds of kilometers, can be properly modeled at lower resolution. In this cas e, the speed and direction spaces may not be required at all, as acceleration and turning radius have significantly less effect on the overall outcome of the missions. In practical use, modeling fighter jet dogfighting is of less benefit then larger scale aerial and naval warfare where the outcome is based on planning and executing winning strategies, rather than skills of individual pilots. In such cases, a reasonable compromise between higher fidelity modeling and lower game board size can be to employ 6 directional phase space and avoid speed levels, same as presented in Section 4.1.2.1 However, aerial flight does require some modeling of the vertical aspects of motion. The aircraft may be able to dive faster, even gaining speed, compared to climbing altitude which may slow down its speed. These effects can be captured using 3D reachability patterns, similarly to modeling maximum slope that can be c limbed or descended by a ground vehicle. An example of a reachability relations for an aircraft is 177

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shown in Figure 67 A naval vessel can utilize an even simpler pattern shown in Figure 68 representing just the speed and turning radius as the vertical dimension is of no concern Figure 67 Aircraft reachability at same altitude, climbing, and diving Figure 68 Basic ship reachability 178

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4.2.2.2 Trajectory Selection Trajectory selection considerations for this domain are very similar to that of ground operations discussed in Section 4.1.2.2 They can be roughly categorized as physical and tactical In addition to the physical considerations common with ground vehicles, such as preference for smoother trajectories ( Figure 30 and Figure 31), aerial movement can incorporate additional factors such as fuel consumption and flight characteristics at various altitudes. Trajectory selection can then try to optimize the balance between fuel consumption and flight time depending on mission requirements long distance flight or urgent combat operation. Similarly, the altitude can play a role in tactical considerations. For instance, it can be beneficial to fly relatively higher than likely target aircraft, or to fly lower so as to make detection more difficult due to radar ground clutter. Just as with urban environments, such static preferences can be captured in the LG Terrain Threat Analysis. Even more important for air missions is the dynamic or positional component of this analysis. Known or likely positions of key enemy weapon systems, along with their engagement ranges, can be used to construct a heat map of more and less dangerous regions based on likelihood of threat. Trajectories for friendly forces can then be generated so as to minimize traversal of such dangerous locations and then updated as threat map evolves due to destruction of enemy threat ( Figure 69) These same concepts apply equally to naval tactics. 179

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Figure 69 Air threat avoidance trajectories before and after SAM site destruction 4.2.3 Heterogeneous Systems Air and naval engagements involve varied set of platforms. There are fast moving jet aircraft that can exceed Mach 3 (3 times the speed of sound), or 3675 km/ h. On the other hand, missile cruisers and aircraft carriers typically top out at 3060 knots (act ual value is classified), or 55 111km/h. Of course, typical cruising speed would be significantly lower for both domains e.g., 1000km/h for jets and 35km/h for ships This corresponds to speed ratios between air and naval forces of at least 30 or more wh ich is a larger discrepancy than between human on foot and a ground vehicles. As discussed in Sections 3.5 and 4.1.3 there are two approaches to efficiently represent such disparate game pieces in LG hypergames and variable step ABGs. The main determinant for choosing one method o ver the other is whether the game board requirements are the same or different. If a common game board can be utilized, variable step ABGs provide a more 180

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efficient practical solution, as per Section 3.5.2. Otherwise, LG hypergames can provide a solution for a more diverse environment. For operations in urban terrain, the board structure was largely dictated by the inherent terrain features of the city resulting in a common game board for all entities. However, air and naval operations do not share this characteristic. Due to the simpler terrain environments and larger overall areas, game board requirements can be significantly different for various ga me elements. For example, naval littoral operation close to the shore could span an area of just 100 by 10 0 km and require game cells no larger than 300 meters across and 60 meters tall to properly capture ground elevation effects ( Figure 66). Vessels moving at 36km/h can then be represented using the 2 cells per turn reachability in Figure 68, using a game step of 1 minute. The same board could be employed, using variable step ABG, to capture movement of an unmanned aerial vehicle (UAV) or other air assets moving at 150 km/h, using a 4 cells per turn reachability and a game step of 30 seconds ( Figure 70) By varying the length of the reachability and the game step, other speeds within this range can be precisely captured as well. 181

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Figure 70 Littoral ABG with ships and UAVs LG Zone Let us consider the eff ects of introducing jet aircraft moving at 1000 km/h. First, a 4 cell per turn reachability would require a very short g ame turn just 4.3 seconds. Alternatively, longer 8 cell reachability could be used with an 8.6 second time step. However, the jet woul d be able to cross the entire 100 km coastal region in approximately 6 minutes. It could also have long range, e.g., 30km, weapons that can fire across a third of the entire game board. Such operational area is simply unrealistically small for this type of aircraft platform. The 300 meter horizontal resolution has also unnecessarily high fidelity for capturing effects of such a fast moving vehicle with long range precision guided weapons. It would be m ore appropriate to employ a large game board with larger cells, for example 1000x1000 km area with 3 km hexagons. Movement of the jet aircraft can then be efficiently represented across this entire geographic area using a 2 cells per turn 182

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reachability with a 22 seconds game step Just as the smaller higher fid elity board could be used for both ships and UAVs, so can this larger board be used for both fast moving aircraft and slow moving long range naval vessels that need to travel long distances. For instance, an aircraft carrier relocating across a 1000km dist ance could be modeled with 1 cell per turn reachability and 5 minute game step Figure 71 Large scale air combat ABG with LG Zones Rather than segregating entities by their air or naval domain, the hypergames can be created to match the sizes of the geographic regions that need to be traversed during the mission, at the appropriate resolutions. The game pieces, both aircraft and ships, can then be assigned to whichever ABG that provides the best representation for its movement spe ed, operational ranges, and weapon systems. The entities could even switch from one ABG to another during the mission, for example when a missile cruiser approaches the shore and can benefit from the increased resolution of the coastal game board [29]. 183

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4.2.4 Incomplete Information Mo dern Air Force and Navy employ s a vast array of sensors to provide situational awareness. These sensors can vary in terms of the information that they provide some can simply detect the presence of an object, while o thers can also provide speed, type, and other characteristics of a target. Cooperation between multiple sensors can also be required to track the target as it moves through the battlefield. Detailed wave propagation models [60 64] and cross section analysis [65, 66] for various types of radars, infrared (IR) systems, sonars, and other equipment are exceedingly complex However, for tactical simulations much simpler models can often be sufficient [36 38]. A probability of detection (PD) curve can be defined to represent the probability that the target would be detected at various distances up to the maximum effective range of the sensors possibly subject to line of sight restrictions A different curve can be specified for each detection states such as position or type, as well as for each particular target type ( Figure 72) Thus details of the underlying physical sensors can be abstracted away while capturing any needed effective capabilities of these systems. These probabilities of detection values can then be utilized for positive and negative sensor contacts within an ABG to construct the world view for each particular player as described in Section 3.6.4. World views approach allows modeling of various types of incomplete information including outdated, incorrect, or missing data about enemy forces [11]. 184

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Figure 72 PD Curve for Ground Radar against Aerial Targets In addition to sensors, air and naval engagements are characterized by effective communication systems that can disseminate situation awareness data across widely distributed friendly forces. However, the l imitations of such systems must still be considered as some information may not be shared due to physical limitations of radio networks ( e.g., range or bandwidth) and organizational constraints ( e.g., need to know and chain of command). These consideration s can effectively segregate the full set of game piece s into subsets or Communication Groups as described in Section 3.6.3 Limitations can then be imposed, as needed, to restrict or delay exchange of information between these groups [36]. Finally, countermeasures must be considered such as radar jammers, communication jammers, or chaff. These devices are intended to interfere with the operation of detection sensors, fire control systems, and communication networks. They can be modeled similarly to the systems they are trying to defeat. For instance, a directional jammer can be defined in terms of the maximum range and horizontal and vertical cone of sensor suppression that it creates. The effectiveness of the jammer can then be defined in terms of the reduction in probability of detection that it achieves at 185

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various rang es against various sensor types. Other jammers could be omnidirectional and defined just in terms of their maximum range and effectiveness. Communication jammers operate in a similar fashion and can either slow or completely deny exchange of information wh en in range of target communication links. 4.2.5 Weapon Systems 4.2.5.1 Strikabilities Similarly to weapons employed in ground operation, air and naval platforms employ both Line of Sight (LOS) and Non Line of Sight (NLOS) weapons [2, 27, 40]. However, in addition to the considerations discussed in Section 4.1.5.1, effects of orientation momentum and relative altitude must be included. Consider an aircraft flying at high speed and dropping a bomb. Even using guided bomb technology to adjust the trajectory in flight, the velocity of the aircraft at the time of release plays a critical role in limiting the possible ground locations that can be hit. Using directional phase space, this can be represented by a strikability offset in the direction of travel as shown in Figure 73 Similarly, a short range airto air missile strikability could be specified to only be able to target locations in a c one in front of the aircraft ( Figure 73, right). This strikability is also offset vertically, to represent that the missile is more effective against t argets below the shooters current altitude, than those higher up, due to the effect of gravity. These strikabilities can be defined to take into account LOS; however, due to the maneuvering capabilities of the missiles and other guided munitions, this res triction is not always 186

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appropriate. Obstacle sizes must be considered to determine whether the projectile can successfully reach the target. Figure 73 Air to ground and air to air strikabilities 4.2.5.2 Weapon Effects Air and naval operations are centered on combat between motorized vessels and aircraft rather than groups of dismounted personnel. As the number of these entities is typically relatively small, it is common for each game piece to represent a single real wo rld object. As discussed in Section 4.1.5.2 such ABGs lend themselves to easier modeling using PK, rather than attrition based modeling of weapons effects [49]. Probability of Hit curves can be used to capture various lethality characteristics at different ranges, just as for ground forces. Due to complexity of e ach underlying entity, it may be beneficial to introduce damage states rather than a single Boolean destroyed state. For instance, a missile cruiser could sustain multiple hits that could destroy its propulsion systems, 187

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individual weapon platforms, or se nsors. This can be modeled by tracking a damage state for each key component, and evaluating the probability of destruction for each component rather than a single PK. F or significantly larger strategic operations, aggregation could be employed so that a g ame piece would correspond to multiple aircraft or vessels. In those case s the attrition rate model would be a better choice. While many of the weapons employed in this domain are explosive, the blast radius can frequently be disregarded simply due to the size of game cells. Even in the smaller littoral ABG described in the previous section, each hexagonal cell was defined to be 300 meters across which is larger than conventional explosive ranges Additionally, as these weapons are frequently aimed and guided directly onto the target object, even other entities within a single cell are unlikely to be damaged in such attacks. 4.2.5.3 Prerequisite Trajectories In order to reach their intended targets, v arious modern weapo ns require fire control guidance This can ap ply to missiles, smart bombs, precision attack munitions, and other long range non LOS systems. In some cases, the fire control guidance can be provided by the same entity that is launching the weapons. The easiest method for modeling this situation is to simply incorporate the sensor parameters directly into the weapon strikability. For instance, if a missile launched from an airplane has a range of 20km, but the fire control radar only has a range of 10km, the strikability could be limited to 10km so as t o represent the effective engagement range of the combined sensor weapon system. In other cases, targeting guidance could be provided by stationary or moving sensor platforms distinct from the shooter. LG strategies can incorporate such 188

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requirements into L G Zones by using prerequisite trajectories described in Section 3.7.3 for just such use case [40] Sensor negation trajectories can be generated from any game pieces that can provide various types of fire control guidance first, followed by computing weapon attack negation trajectories that take into acc ount the duration and time periods when the target is illuminated by the appropriate sensor s These trajectories must always be treated as prerequisite and executed in pairs, as there is no benefit in committing a weapon platform unless the corresponding s ensor is also available. Larger groups of trajectories can also be considered jointly to account for sensor handoff between multiple, typically stationary, platforms. Figure 74 LG Zone for defense against cruise missiles with integrated fire control 4.2.6 Mission Concepts The key to producing an artificial intelligence system that provides useful strategy generation capabilities for any military operation al domain is ensuring that the mission 189

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and the doctrinal concepts for the specific forces are properly captured. The proper approach for this task for ground forces was presented in Section 4.1.6 and can be applied in just the same way to any other domain through appropriate use of military manuals e.g., [67 69] and subject matter experts. Due to the wide range of possible operations that can be undertaken by each branch of the military capturing the entire breadth of missions is a prohibitively large undertaking. Instead, close collaboration with military experts can be used to model specific needs of any particular subsets of operations in sufficient detail to provide tactically correct winning strategies. Consider using LG ABGs to model and analyze Suppression of Enemy Air Defenses (SEAD) missions [67]. Relevant mission concepts would include doctrinally correct techniques for flying multiple aircraft in a strike package through enemy territory, employment of electronic warfare planes airborne early warning and control systems, rules o f engagement, prioritized targeting of enemy ground sensors, surface to air missile launchers, and fighter aircraft, and many others [11, 28, 36]. A s varied as these principles may be, the LG approach is still the same. Specific mission definitions must be mapped to LG missions and goals, described in Sections 3.8.1 and 4.1.6.1 while any doctrinally mandated behaviors need to be abstracted as constraints fo r generation and execution of the LG Trajectories and Zones, as discussed in Sections 3.8.2 and 4.1.6.2 4.3 Ballistic and Orbital Operations (BO) Ballistic and orbital operations are significantly differ ent from ground, air and naval operations in terms of the global operational space and extreme performance characteristics of the entities. Yet, in addition to various ballistic missiles and satellites, 190

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this domain must also includ e a variety of terrestrial assets. Missile launch and sensor sites, strategic targets, and other infrastructure components both stationary and mobile are essential for complete modeling of strategies. All of these elements mu st still be mapped into the same ABG format in order to benefit from LG strategy generation. This section will focus on how unique challenges of this domain can be solved u sing various techniques from Chapter 3 4.3.1 Spatial Discretization Interc ontinental Ballistic Missiles (ICBM) can reach targets up to 16,000 km away, while satellites can orbit completely around the planet. M odeling these operations requires board space that covers the entire surface of the Earth. As presented in Section 3.3.1.3 Discrete Global Grids can be used for this purpose ( Figure 24 ). As with 2D and 3D grids, the size of the cells in the spherical grids can be tailored as needed for particular scenario. Section 4.1.1 has illustrated how the appropriate board resolution can be determined based on the sizes of relevant terrain obstacles. However, terrain features do not play as large of a role for operations on planetary scale. Section 4.2.1 has instead demonstrated how cell sizes can be based on the overall area covered and speeds of entities to be represented. Ballistic missile speed does not yield itself to such analysis du e to extrem ely large variance at different portions of a ballistic trajectory from 0 to around 7km/s, or over 25,000km/hr. This necessitates use of different types of reachabilities that are not as tightly coupled to the size of board cells (Sections 3.4.6 and 4.3.2). On the other hand, terrestrial entities do exhibit more consistent movement characteristics and can drive the choice of ABG spatial resolution. Due to the more strategic nature of these 191

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ope rations, the board cells can be fairly large. For instance, ICBM equipped strategic bomber or submarine relocating between continents and oceans can significantly affect the outcome of a global engagement, but movement of 100 kilometers would likely have l ittle effect and does not need to be precisely tracked A reasonable compromise may be to employ cells of 500 km across, as shown in Figure 75 3D sta cks of cells can also be employed, although due to spherical nature of the game board each cell would represent a truncated hexagonal pyramid. However, a simpler approach would be to avoid the 3rd dimension in the ABG board entirely, and capture the elevat ion as the property of the ballistic and orbital game piece instead. Relative altitude of terrestrial entities by comparison is so close to the surface of the Earth as to be negligible for computing movement o ver such large game cells. Figure 75 Orbital trajectories on hexagonal planetary grid 4.3.2 Mobility Movement of terrestrial entities on global boards can be modeled just as ground, air, and naval movement in Sections 4.1.2 and 4.2.2 At this scale, the time step for these 192

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game pieces must be adjusted acco rdingly For example, a jet moving at 1000km/h using 1 cell per turn would employ 30 minute game steps A cceleration and direction space are not needed at this temporal and spatial resolution as any such changes in velocity happen over much shorter time durations than a single game step. Figure 76 LG Trajectories for terrestrial pieces on a planetary grid Ballistic and orbital motions present different challenges [43]. Maneuverability of such entities is significantly restricted as they are essentially locked into chosen trajectories with only small variations possible. As previously mentioned, ballistic trajectories also require a wide range of velocity variations. This type of movement does not yield itself to discrete movement reachability patterns of previous domains, and therefore the LG Language of Trajectories cannot be easily applied. As discussed in Section 3.4.6 an alternative approach can be used to generate trajectories for such 193

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entities. The movement can be modeled using appropriate equations of orbital or ballistic motion to achieve accurate representation [42, 70]. The resulting paths are then discretized into the form of the LG Trajectories by sampling their coordinates at each game step interval, such as t hose shown in Figure 75, Figure 77 and Figure 78. Note the small dots displayed on top of the continuous ballistic paths used to represent these discrete trajectories. Most importantly, these trajectories can participate in the LG Zones as main or negation trajectories. Since these final discretized versions are from the same Language of Trajectories, they can be interming led with those generated using standard LG generating grammars. Thus, a main trajectory, generated using continuous methods and then discretized, can be negated by a trajectory using standard discrete reachability relations, and vice versa. Figure 77 Initial LG Zone for ballistic missiles 194

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Figure 78 LG Zone during ballistic missile engagement, with radar illumination 4.3.3 Heterogeneous Systems Using the combination of reachability based and equations based trajectory generation allows for modeling of widely different entity types in a single ABG For any strategic level operations, such planetary board grid would be sufficient by utilizing Variable Step ABGs (Section 3.5.2 ). However, ballistic missile operations can also be interdependent with smaller, tactical operations centered around their terrestrial components launch sites, sensors sites, and ta rget locations. For instance, interception of ICBMs can be done prior to their launch using an air strike or a Special Forces ground operation. Similarly, ground or ship based sensors, used to track ballistic missiles in flight, as well as interceptor miss ile launch sites can come under attack and need to be defended. These detailed missions around key locations can be modeled using air, naval, and ground combat ABGs from Sections 4.1 and 4.2 and linked together with the 195

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planetary level ABG to form a single hypergame. This method allows for LG s trategies to provide a common solution to this entire collective operation. 4.3.4 Incomplete Information Ballistic and orbital operations can benefit from the same techniques for modeling of incomplete world view, sensors and countermeasures covered in Section 4.2.4 The main distinction s are sensor ranges, types and resolution s involved. Geostationary satellites, in orbit over the equator approximately 36, 000 km above ground, can provide early recognition of ballistic missiles launches or tracking of moving aircraft or other vehicles even at such extreme distance. However, the cone of their field of view can hinder their ability to provide such detection ov er a large area. Another example is the X band sea or ground based Radar (XBR) that c an provide fire control sensor functions to enable interception of ballistic missiles during the midcourse phase of their trajectory ( Figure 78) For modeling purposes, these platforms can be represented by the type of provided information ( e.g., detection, classification, or tracking), range, fields of view, and Probability of Detection curves for various target types. Various countermeasures, such as decoy reentry vehicles, can be parameterized in terms of their PD reduction. Combined information from all such sensors can then be used to construct appropr iate worldview for each player or subgroup, based on their communication constraints. 4.3.5 Weapon Systems At this operational scale, weapon effects can still be modeled as Probability of Kill or Attrition effects. As previously discussed, the choice of PK or AR model is largely driven 196

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by whether the target game piece represents a single entity or an aggregated collection. Consider the operation of the Ground Based Midcourse Defense (GMD). When an ICBM is detected and tracke d in flight, a GroundBased Interceptor (GBI) can be launched to boost an Exoatmospheric Kill Vehicle (EKV) onto an intercept trajectory with the target. EKV can then maneuver, in a limited capacity, to collide with and destroy the ICBM. For modeling purposes, all of this complexity can be redu ced to a single PK values for the ballistic interceptor to destroy its target based on the target type. Due to the large size of cells used on a planetary grid, a discrete cellbased strikability for EKV is not going to be effective. Instead, a range can be defined to represent EKVs ability to maneuver onto the target. Probability of Hit can be used to adjust for any additional considerations such as range, angle of approach, or the quality of fire control guidance provided by various sensor systems. The s ame considerations can be employed for attacks against satellites in orbit. On the other hand, an ICMB that reaches the end of its trajectory will cause immense destruction that cannot be captured as a single PK. Massive number of casualties and structural destruction have to be modeled as various attrition rates in concentric circles around the epicenter of the explosion. These patterns represent the strikability relation of such weapons of mass destructions, and can be further affected by the missile payload ( e.g., multiple independently targetable reentry vehicles), type of warheads ( e.g., nuclear or conventional) and detonation method ( e.g., surface or airburst). 197

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4.3.6 Mission Concepts Satellites can provide a wide range of sensor and communication capabilities for both civilian and military use. Effects of these platforms on local operations can often be captured in the smaller ABGs described in previous sections. The use of planetary level ABGs is typically necessitated by the strategic scale of the operation, such as those involving long range ballistic missiles. At this scale, geopolitical considerations play a major role. For instance, placement of the X band Radars and Ground Base Interceptors required for GMD is a controversial topic in international politics. This places severe constraints on possible locatio ns of these sites that override the military consideration s of best deployment of these systems. Similarly, potential targets of ballistic missiles can be chosen based on strategic value, such as critical defensive infrastructure, as well as sociopolitical effect of inflicting large civilian casualties. Political considerations also place rigid limits on which kinds of weapons can be employed in respo nse to different enemy actions. Such decision criteria and restrictions must be mapped to LG missions and constraints that are used to guide generation of LG Trajecto ries and Zones as described in Sections 3.8.1 and 3.8.2. 4.4 Joint Forces Operations (JF) 4.4.1 Heterogeneous Systems Real world operations rarely happen in isolation. Even a small infantry mission, whether urban or otherwise, is executed with full backing of large support chain such as artillery, Unmann ed Aerial Vehicles and satellite surveillance, Close Air Support, medical 198

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evacuation helicopters, logistics resupply elements, combat engineers, military intelligence and communication networks, and others. This can be traced even further to civilian infrastructure shipping li nes, and international trade. As with any modeling decisions, balance must be struck between capturing any elements relevant to a particular operation and including too much detail that the problem becomes intractable. The same exact mission can require different scope based on the surroundi ng circumstances. Section 4.1.5 described how Close Air Support could be included in an infantry ABG as only weapon effects that can be delivered to any locations on the board, subject to a delay. This is perfectly sufficient if the surrounding airspace is safe. On the other hand, if en em y has air defense capability the n more d etailed modeling is in order to allow the appropriat e air tactics to be generated. This would in turn have direct implications for the ground mission as it could force additional consideration of whether the risk t o the aircraft is worth the benefit that it would provide for the ground forces. The difficulties of these interdependent operations is that they involve entities with a wide range of diffe rent characteristics and that they oc cur over large geographic areas while simultaneously requiring high resolutions in some sub regions [2, 23] The key techniques for modeling such complex games have been thoroughly covered these are the LG hypergames consisting of Variable Step ABGs. The main lesson from Sections 4.1.3 4.2.3 and 4.3.3 is that the major factor for determining how to segregate the operation into ABGs is board requirements the size and the resolution. The entities can be distributed across these games depending on their geographic position and required resolution, while the time step can be adjusted for each game piece based on the cell 199

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sizes to properly represent its movement characteristics. The entities are also not restricted to a single ABG, but can tran sition between them as the mission unfolds and they move from one board region to another. 4.4.2 Air Ground Hypergame This approach provides a mental framework for generating a set of ABGs for any complex Joint Operation. Figure 79 illustrates this concept using an Air Ground hypergame [3, 41, 48]. The larger game board (top) represents the geographical area where air strike against enemy targets is carried out. This ABG can include friendly fighters, bomber, and sensor aircraft, enemy fighters, SAM sites, and other air defense platforms. Simultaneously, multiple ground operations could be c onducted within this overall area. A smaller tank game is shown at the bottom of Figure 79. Note, that a black rectangle is displayed in the Air game to depict the area of this small ABG. This tank game contains higher resolution terrain model and any assets that require that addition fidelity. These games are not disjoint as events in one game can directly affect the other. The aircraft can provide Close Air Support using the LG Zones displayed in this figure. Conversely, the tanks could destroy ground based enemy air defense assets to protect the airborne strike package. LG Inter Linking Mappings can be utilized to solve these interconnected ABGs as a single hypergame. 200

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Figure 79 Air (top) and Ground (bottom) hypergame 4.4.3 Litoral Air Naval Hypergame Similar techniques can be applied to a littoral operation executed by aircraft and naval vessels [43, 48, 49]. A s illustrated in Figure 80 one board can be utilized to represent a large coastal area (top), while another provides a higher resolution view of a key area right next to shore (bottom). T he smaller board is shown as a black rectangle in the larger game. Additionally, some forces can transition between these two ABGs during the 201

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mission. It is likely that all air and naval forces begin the mission in the larger game, e.g., as part of an aircraft carrier battle group. During the mission, littoral craft can approach the shore using the lower resolution large ABG to model this movement. However, as they get close to their targets and enter the space represented by the higher resolution board, they can transi tion to that ABG in order to increase modeling fidelity. This in turn, allows for the higher accuracy strategies that take into account detailed coastal terrain and intricacies of the littoral close in combat due to smaller board cells and shorter game turns In the meantime, the aircraft can operate within the larger ABG to provide air support or counter enemy airborne capabilities. There is no need for control of these game pieces to shift to the smaller ABG as the extra resolution would not add any valu e. Furthermore, the air assets cannot be well represented on such small game board due to their long sensor and weapon ranges. At the end of the mission, as the ships return back to the carrier battle group, they can transition back from the high resolutio n to the lower resolution game. 202

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Figure 80 Littoral hypergame 4.4.4 Joint Amphibious/Air Assault Operations The hypergame framework is not limited to just two games. Consider Joint Amphibious/Air Assault operations [71] employing potentially 4 branc hes of the military Navy, Airforce, Army, and the Marines. These operations are exceptionally complex, including, for example the following elements: Suppression of enemy air defenses by air forces 203

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Air strikes against enemy ground forces Paratroopers dropped by airplanes behind the shoreline First wave of Marines inserted by helicopter to secure beachhead Second wave of Marines with assault vehicles inserted by fast moving aircushioned landing craft to secure the beachhead Naval artillery and guided missiles fire support for landing troo ps Fixed wing and helicopter close air support for landing troo ps Third wave of mechanised Marines and armor inserted by larger ships to expand the occupied area Larger Army unit, e.g., armor brigade, coming ashore to p ass through the Marines to begin the fight deeper inland Paratroopers, Marines and Army units engaging enemy ground forces This chain of events can be extended both forward and back in time to include the logistics of setting up such a complex operation, prepositioning large Marine and Army units afloat in the right formations, resupply and sustainment of landed forces, fortifying newly aquired positions by combat engineers and so on. However, construction of the ABGs and the overall hypergame follows the same basic steps : Identifying appropriate, nested areas of interest and necessary resolution for each area ; Deciding which units need to be represented at each of these resolutions and areas ; Modeling the unit characteristics for each required board resolution 204

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Figure 81 shows an example of how the space could be separated into a collection of overlapping game boards. The first, w hite board is a lo w resolution large board that can be used by any air assets such as those performing air strikes against inland targets, delivering paratroopers to their drop zones, providing close air support, and helicopters inserting Marines. The second board, displaye d in blue, provides somewhat higher resolution for naval forces traversing from their offshore position to the landing zones. This ABG could include both the initial fast moving air cushioned craft and large vessels carrying the next wave of troops. It cou ld also be used to model the helicopter operations in more detail if the 1st game is not sufficient. The third board, displayed in red, models the large scale ground operation both around the landing zone and two follow on directions of attack further in land. The final set of green areas, numbered 4 8, represent higher resolution inserts around key objectives where detailed ground combat modeling is desired for paratrooper, Marine, or Army units. Overlap between all these areas allows for units to transition between these game boards, as well as for game pieces from different games to support each other through this complex joint operation. 205

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Figure 81 Joint Amphibious/Air Assault Hypergame The U shape of the red board is intentional to illustrate that the LG game boards need not be rectangular or even convex polygons. Since a board in an ABG is just a set of locations any shape of an area can be used. This concept can even be extended further, so as to consider a collection all 5 green areas as one disjoint abstract board, rather than 5 separate ones. The final step of modeling characteristics of the game pieces on these ABGs can then utilize all the methods presented in the previous sections for air, naval, and ground forces. 206

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CHAPTER 5 CONCLUSION Linguistic Geometry has been successfully employed in a wide range of real world applications that require ability to generate strategies with faithful modeling of an intelligent enemy [3, 2325]. However, such usage is predicated on modeling all aspects of the target system as a formal Abstract Board Game. T his thesis seeks to provide a practical guide that would be of benefit to those in the Modeling & Simulation and Artificial Intelligence communities seeking to apply LG to both new and previously explored domains. Chapter 3 provided a fr amework for examining any such problem along several specific dimensions of analysis. Specific benefits and costs of each approach were considered in order to allow selection of most appropriate representation that strikes the balance between fidelity of modeling and practical computational tractability. Chapter 4 illustrated how such analysis can be performed using 4 domains that span the breadth of common real world operations. While this is of particular benefit to other researchers working on strategy generation and course of action analysis for land, naval, aerospace operations such detailed examination of pitfalls and solutions for modeling these specific domains should serve as a springboard for work in other areas, not yet explored. LG has already been applied to a wide range of applications across diverse domains in the past [23, 25]. However, there are many more types of real world problems that could be explored. Some of these may be successfully modeled and solved using the various approaches presented in this dissertation, while others could bring new 207

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theoretical problems in applicability and extensibility of LG. Future researchers applying LG to an ever increasing range of real world systems can serve to discover these new extensions and, therefore, further increase the applicability of the entire LG approach. Most of the domains analyzed in this dissertation are related to military operations as they lend themselves more directly to modeling as adversarial games. On the other hand, there are other syst ems with a clear and distinct enemy. For instance, consider modeling of American football. The players game pieces, and the board can represent respectively, the opposing teams, individual athletes, and the field. Offensive and defensive strategies can then be generated and analyzed. While sports can be simulated in this manner, video games can benefit from this approach even more. LG can be employed to provide an intelligent opponent to the human player in any game that can be modeled as an ABG. Similarly, recent research has explored applicability to analysis of ancient historical battles [7274]. Even when there is no explicit enemy, LG could be applicable by considering an implicit adversary. For example, in disaster relief operations the actions of emergency vehicles, food and water distribution, evacuation, and rescue efforts are opposed by traffic jams, destroyed road ways and buildings, riots, severe weather, and other events. Same principles can apply to air or ground traffic contro l and emergency vehicle routing [75, 76]. Even problems without a natural conflict c ould be modeled as ABGs by introducing an artificial enemy. This technique has been demonstrated for scheduling of power plant maintenance in [18] and can be explore d across other domains. 208

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In addition to continued e xpansion of applicability of LG, f uture research could also be carried out to extend the definition of ABG and the Hierarchy of Formal Languages ( Figure 3 ) to formalize various extensions of LG presented in Chapter 3. These mathematical models could then be employed to further develop the theoretical foundations of Linguistic Geometry along multiple directions of interest to the research communities. The most promising extensions of LG are related to reducing the gap between the discrete nature of ABGs and the continuity of the natural world. As the difference between the model and the real world obje ct is reduced the accuracy and quality of the resultant strategies is increased. Simply increasing the resolution of the discrete model can be computationally prohibitive Several techniques in this research have taken an alternative route of combining di screte and pseudo continuous elements in ABGs as follows: Source Data Submodel for modeling terrain obstacles for sub cell accuracy of Line of Sight. Generation of LG Trajectories using continuous modeling of motion, such as ballistic paths Use of Variable Step ABGs to accurately represent full range and minute differences of movemen t speeds of various game pieces. This direction of research can be further extended to continue shrinking the gap between the real world and its computer representation by inco rporating more continuous elements into the discrete model. For instance, Variable Step ABGs could be 209

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extended to allow not only different game step duration for each piece but also for different phases of the mission or, perhaps, even for different segmen ts of a single trajectory. Source D ata S ubmodel can be further extended to improve the accuracy of reachabilities while integraldifferential equations of motion could be used in conjunction with the standard LG discrete methods to ensure that the resulta nt LG trajectories conform to the full fidelity of physical movement characteristics of the real world entities. Such continued efforts will allow for increased faithfulness of the solutions provided by LG to an ever wider range of real world problems, wit hout compromising the practical tractability of such software applications. 210

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REFERENCES [1] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior : Princeton University Press, 1947. [2] B. Stilman, Yakhnis, V., Umanskiy, O., "Knowledge Acquisition and Strategy Generation with LG Wargaming Tools," Int. J. of Computational Intelligence and Applications, vol. 2, pp. 385 410, 2002. [3] B. Stilman, Yakhnis, V., Umanskiy, O., "Strategies in Large Scale Problems," in Adversarial Reasoning: Computational Approaches to Reading the Opponent's Mind, A. K. D. a. W. M. U.S. Diego), Ed., ed: Chapman & Hall/CRC, 2007, pp. 251 285. [4] B. Stilman, Yakhnis, V., Umanskiy, O., "Linguistic Geometry Paradigm: From Fighting Wars To Computing Them," in Proc. of the IX Argentine Symposium on Artificial Intelligence (ASAI)), 36 JAIIO (36th Int. Conference on Computer Science) Mar del Plata, Argentina, 20 07, p. 12 [5] R. Isaacs, Differential Games New York, NY: Wiley, 1965. [6] Y. Lirov, Rodin, E.Y., McElhaney, B.G., Wilbur, L.W., "Artificial Intelligence Modeling of Control Systems," Simulation, pp. 12 24, 1988. [7] A. GarciaOrtiz, et al, "Application of Semantic Control to a Class of Pursue Evader Problems," Intl. J. Computers and Mathematics with Applications, vol. 26, pp. 97 124, 1993. [8] M. Osborn, Rubinstein, A., A Course in Game Theory Cambridge, MA: MIT Press, 1994. [9] F. h. Hsu, Ananthar aman, T.S., Campbell M.S., Nowatzyk, A., "Deep Thought," in Computers, Chess, and Cognition, T. A. Marsland, Schaeffer, J., Ed., ed New York: Springer Verlag, 1990, pp. 55 78. [10] J. Lee, Chen, Y L., Yakhnis, V., Stilman, B., and Lin, F., "Discrete Event Control and Hostile Counteraction Strategies," Rockwell Science CenterFeb. 2001 2001. [11] B. Stilman, Yakhnis, V., Umanskiy, O., "Winning Strategies for Robotic Wars: Defense Applications of Linguistic Geometry," Artificial Life and Robotics, vol. 4, pp. 148 155, 2000. 211

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[25] "Linguistic Geometry Tools: LGPACKAGE," STILMAN, Ed., ed, 2007, p. 50. [26] B. Stilman, Yakhnis, V., Umanskiy, O., "Linguistic Geometry: Theory and Experiments," in Proc. of the 3d Int. Workshop on Artificial Intelligence in Science and Technology AISAT2009, Hobart, Tasmania, Australia, 2009, p. 6. [27] B. Stilman, Yakhnis, V., Umanskiy, O., "Introduction to LG Hypergames for Practical Wargaming," in Proc. of SPIE Conf. Enabling Technologies for Simul. Sci. VI Orlando, FL, USA, 2002, pp. 386 394. [28] B. Stilman, Yakhnis, V., Umanskiy, O., "LG CONSTRUCTOR for Rapid Deployment of LG Wargaming Tools," in Proc. of the SPIE Conference Enabling Technologies for Simulation Science VI Orlando, FL, USA, 2002, pp. 415 423. [29] B. Stilman, Yakhnis, V., Umanskiy, O., "LG GUARD f or Missile Defense and Offence," in Proc. of the SPIE Conference Enabling Technologies for Simulation Science VII Orlando, FL, USA, 2003. [30] K. Sahr, White, D., "Discrete Global Grid Systems," Computing Science and Statistics, 1998. [31] J. P. Snyder, "An equal area map projection for polyhedral globes.," Cartographica vol. 29, pp. 10 21, 1992. [32] D. B. Carr, R. Kahn, K. Sahr, and A. R. Olsen, "ISEA Discrete Global Grids," Statistical Computing & Graphics Newsletter, vol. 8, pp. 3139, 1997. [33] R. Heikes, Randarll, D.A., "Numerical integration of the shallow water equations on a twisted icosahedral grid. Part I: Basic design and results of tests. ," Monthly Weather Review, vol. 123, pp. 18621880, 1995. [34] R. Heikes, Randarll, D.A. "Numerical integration of the shallow water equations on a twisted icosahedral grid. Part II: A detailed description of the grid and an analysis of numerical accuracy," Monthly Weather Review, vol. 123, pp. 18811887, 1995. [35] S. Kass, Bjorklund, B., Harrison, D., Castaneda, M., Yakhnis, V., Stilman, B., Umanskiy, O., "Linguistic Approach to Training Combat Strategies," in Proc. of the 50th Meeting of Human Factors and Ergonomics Society San Francisco, CA, USA, 2006, p. 5. [36] B. Stilman, Yakhnis, V ., Umanskiy, O., Boyd, R., Pugachev, V., Hagen, L, "LG PACKAGE: Advanced Features," in Proc. of the 3d Int. Workshop on Artificial 213

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