REVISTING JULIUS CAESAR BATTLES WITH LINGUISTIC GEOMETRY by LUBNA AHMAD ALHINTI B.S., K ing S aud U niversity 2011 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 Master of Science Computer Science Program 2014
! ii This thesis for the Master of Science degree by Lubna Ahmad Alhinti has been approved for the Computer Science Program by Gita Alaghband, Chair Boris Stilman, Advisor Tom Altman 04/16 /2014
! iii Alhinti, Lubna Ahmad (M.S., Computer Science) Revisiting Julius Caesar Battles With Linguistic Geometry Thesis directed by Professor Boris Stilman. ABSTRACT Linguistic geometry (LG) is a kind of game theory scalable to the level of the real life sy stems including defense systems The outcomes of LG based applications have been shown great power and advanced solutions. It was suggested that "LG is a mathematical model of human thinking about armed conflict, a mental reality that existed for thousands of years." . It happened th at to discover efficient warfare methodologies, chess game was used. This game was invented 1.5 2 thousand years ago. As a way to validate the above hypothesis, power of LG software should be demonstrated on ancient wars that occurred before times when the game of chess was invented ,  An extensive LG based theoretical analysis o f these ancient wars should be done to show that the LG based software outcome is similar to the actual historical events. In this work, we present an LG based analysis of the events that happened between the armies led by Julius Caesar and Vercingetorix T hese events include the Battle of Gergovia and Siege of Alesia, the Roman dec is ive victory, during the Gallic Wars. Two versions of LG, namely, the Roman LG and the Gallic LG will be demonstrated. The form and content of this abstract are approve d. I recommend its publication. Approved: Boris Stilman
! iv ACKNOWLEDGEMENTS This research would not have been accomplished without the support and enc ouragement of many people. I am using this opportunity to express my appreciation and gratitude to my advisor Professor Boris Stilman for all the time and effort he has spent in answering my questions, reviewing my work, and encouraging me to think deeper. I also appreciate the valuable guidance, assistance, and advice he provided me with. Deepest thanks and appreciation to the committee members Professor Gita Alaghband and Professor Tom Altman for serving as my committee members even with all their other commitments. A special thanks to my beloved husband Tamim Alkadi for his tremendous support, encouragement and patience. I would never be able to finish this work without his help. I cannot express how grateful and thankful I am to my moth er, mother in law, father in law, and all my family and friends for all their constant support, help, understanding, and wishes for a successful completion of this thesis. I would like to take this opportunity also to thank my government for their generous financial support of my Master program.
! v TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ............................... 1 II. REVIEW OF THE LITERATURE ................................ ................................ ...... 2 III. APPLICABILITY OF LG TO ANCIENT WARFARE ................................ ..... 3 IV. GALLIC WARS ................................ ................................ ................................ 5 V. DISCRETE UNIVERSE OF CAESAR'S BATTLES ................................ ....... 6 VI. LG BASED ANALYSIS of CAESAR'S BATTLES ................................ ...... 10 Before the Battle of Gergovia ................................ ................................ ........... 10 Construction Of The LG Hypergames ................................ .............................. 12 Siege of Alesia ................................ ................................ ................................ .. 13 LG Hypergame Construction ................................ ................................ ............ 27 VII. CONCLUSION ................................ ................................ ................................ 30 REFERENCES ................................ ................................ ................................ .................. 3 1
! vi LIST OF FIGURES FIGURE 1 Roman Triplex Acies Deployment ....... .. 8 2 !!!!!!!!!!!!!! Legend t o Figure Battles .................... ... 9 3 LG Hypergame of t he Eve nts Happened Before the Battle o f Gergovia ................................................. 13 4 Gauls Potential Attacking Zones ... .. 14 5 Building Inward S iege 15 6 Skirmishes While Building the S iege 16 7 Network Language Interpretation for the Attacking Zone Issued by the Gaul's C avalry 17 8 Some of the Expected Attacking T rajectories from the Relief Army .. 19 9 Siege of Alesia with Inwards and Outwards Lines of D efenses . 20 10 Concurrent Attacks from both D irections ... 21 11 Network L anguage I nterpretation for the A ttacking Z one I ssued by the Roman C avalry out of the C ircumvallation Lines .. 22 12 The R elief A rmy T rajectories to A ttack the Roman C amp . 24 13 The Gallic LG D eception P lan 26 14 LG Hypergame Hierarchy with Two Levels of R esolution 29
! 1 CHAPTER I INTRODUCTION Linguistic Geometry (LG) is a game theoretic approach that is capable of not only solving board games problems, it is also capable of solving practical complex problems including planning a mission and managing a battle in real time. A significant increase in size of problems that could be solved in real time manner has been demonstrated using LG   LG predicts future states based on the current state and suggests proper courses of actions leading to victory. Upon receiving new information during a simulated combat, LG is capable of dynamically regenerating actions in real time. Bright and bril liant ideas could be suggested by LG applications, which lead to the suggestion that LG is a superintelligent mathematical mode l of human conflict resolution As a step to validating this hypothesis, the power of LG will be demonstrated on the major battle s of Julius Caesar in the Gallic Wars, which happened before the invention of game of chess. We are going to present two versions of LG, the Roman LG and Gallic LG, where both will generate advanced strategies in real time. The purpose of this ana lysis is to show that all the actions and events that happened during these historical battles have its rational and the theory of LG explains this rational very well.
! 2 CHAPTER !! REVIEW OF THE LITERATURE The generalization of the most advanced chess players' experiences in the 70s led to the development of LG. The PIONEER, a computer chess program was developed in the 70s and 80s as a result of this generalization. LG that solves problems employing differ ent paradigm "From Search to Construction" was developed as a result of subsequent generalization. In [1 ] and [ 2 ], applicability of LG to the major battles of Alexander the G reat and the Hannibal have been demonstrated. An extensive LG based analysis has b een provided. The main tool used by the Hannibal to win was applying deception, which was effectively modeled in terms of LG. To confirm a deception plan, the LG software will gamble on the inferiority of the enemy's LG software. Static and dynamic schemes will be generated to strength en the deception plan. The Alexander the Great battles analysis demonstrated that his victories were ach ieved by turning the enemy into a vulnerable target usually via outflanking his troops The idea of vulnerable target is well grounded in LG. In the LG based analysis of these historical battles, LG has demons trated its power and capability to produce courses of actions similar to what has been reported by the historians. There were no mysterious actions or events th at LG could not model.
! 3 CHAPTER !!! APPLICABILITY OF LG TO ANCIENT WARFARE LG structure is composed of two layers: game construction and game solving. In the game construction layer a hierarchy of abstract board games (ABGs) called the LG hypergame is constructed to model the real world , , ,  This hierarchy consists of the board itself, the players and the game pieces, the game constraints, etc., Hypergames are very useful in modeling and combining operations that apply in dif ferent areas, different spaces, and different levels of resolution. In this hierarchy, levels of resolution increase as we go down. The most general view is obtained from the top level where the resolution is in its lowest level. T he global operations are controlled and managed at the top level. As the resolution increases, the ABG grids will cover smaller areas showing more details and permitting deeper control. An LG model is considered to be in its highest resolution level when it permits controlling pi eces at the level of an individual soldier. The second layer in the LG structure is the game solving where the resources are allocated and strategies guiding the players to achieve their goals are generated dynamically. Having advantageous initial state fo r the game depends on how good the resource allocation is. Each language has an alphabet. The LG alphabet is different types of trajectories. A trajectory is a planning sequence of steps leading to a destination. Trajectories are either main trajec tories or negation trajectories of various degrees. A main trajectory could be an attack trajectory, a relocation trajectory, a domination trajectory, a retreat trajectory, and an unblock trajectory. A zone is a network of trajectories. There are differen t types of zones, for example: an attack zone and an unblock zone. The LG strategies are sets of
! 4 actions performed along the trajectories of zones. Moving along a trajectory is an example of an action that could be performed What is important to indicate is that the LG generated strategies are constructed and not resulted from search. Each state has a different set of zones that uniquely represents the state. A piece could be part of several zones at the same time. It also could be involved in many trajec tories within a specific zone. To generate strategies, LG uses a quality function that evaluates the quality of trajectories and zones. LG uses only trajectories with high qualities. A trajectory is indicated to be safe based on the probability of the piec e passing through it safely This probability is defined for all entities and affects the quality of trajectories. In ancient wars, attrition rates are also identified in cases where hand to hand fighting and massive shooting occur and affect the quality of trajectories and zones. A ttacking from the rear always causes high attrition rates while attacking from the front causes low attrition rates. A target is considered vulnerable if it has no way to retreat. Trajectories with vulnerable targets have very high quality.
! 5 CHAPTER V GALLIC WARS Julius Caesar, the great Roman Leader, has led the Gallic Wars, a series of military campaigns that lasted from 58 BC to 50 BC , ,  These campaigns affect ed both the Roman and the Gaul in many ways. As a result of Roman Victory the Roman Republic expanded over the whole Gaul. These campaign started when the Transalpine Gaul, what is known now as southern France, governorship was added to Caesar command. The first campaign was against the Helvetii one of the Galli c tribes, who planned a mass migration, starting from what is known now as Switzerland. Their destination was to settle on the borders of the Santones, which is near one of the province tribes called the Tolosates. This migration could have put the provinc e under threat. Caesar decided to fight the Helvetii to protect the Roman province. This threat was a great opportunity for Caesar to start his military campaigns in Gaul and prove his military skills. From a Roman point of view, conquering the Gau l was a must to protect the Roman province. Also, the productivity of the north land was one of the motives for this conquest. The Gaul was conquered after the Roman great victory at Alesia.
! 6 CHAPTER V DISCRETE UNIVERSE OF CAESAR'S BATTLES Class es of abstract board games (ABGs) and LG hypergames representing Caesar's Gallic wars are needed so we will be able to apply LG to these wars. For Caesar battles, "hexes", rectangular hexagonal prisms of 30 meters across the foundation and 3 meters height board will be chosen to present the terrain, military formations and movements, etc.; Generic pieces will be used to represent different military units such as infantry and cavalry. o a syntagma, a squadron in the Roman Legion, which consisted of multiple syntagmas "usually ten" of square shape of about 480 infantrymen called "Cohort". These cohorts as well consisted of six syntagmas of square shape of about 80 men called "Centuries". Furthermore, each of the centuries consisted of ten "groups" syntagmas pr oviding a squad level of organization. However, the first cohort was a little different in its formation. It consisted of 800 men and only f ive centuries. We are going to represent syntagma with a square [ 8 ]. o an ilia, a squadron in the Roman light armed troops and cavalry of different sizes. We are going to represent ilia with a triangle. 30 seconds interval is the time that we are going to consider here to represent the physical time required for the real world system to move between two consecutive sta tes in the ABG
! 7 Movement reachabilities for all pieces and their advancements will be represented within the indicated time interval. All probabilities of kill and all combinations of piece weapon together with their related outcome will be considered. At trition rates will be defined depending on the attack direction. In Caesar's battles, we are going to calculate the strength of each entity during the engagement. The level of the strength is affected by the attrition rate. This attrition calculation depen ds mainly on the effectiveness of the attack unit against the target unit. W e are going to consider an entity destroyed if its strength level drops below a specified threshold. Weapon reachabilities should be defined Communication and logistic constrain ts should be considered in the simulation as well, as they affect the course of the battle. Roman armies were usually arranged in a special order called the "triplex acies" where legions were drawn up in triple line ,  This kind of formatio n as shown in Figure 1 leaves gaps in the first line through which the cohorts stood in the second line could advance to be in the first line relieving those who were in the front. Deployment is as follows. In the first line, four cohorts usually stood leaving gaps between each cohort. Three cohorts stood about 160 feet behind the first line lining up behind the gaps. In the same manner, the third line consisting of three cohorts stood behind the gaps left in the second line.
! 8 The first line consisted of 2400 men, divided in four cohorts. The second line and the third line, each consisted of 1800 men. When the army faced the enemy, the light armed troops and cavalry would start throwing their javelins at the enemy. After that, t hey would retreat through the gaps left by the first line. The first line would form a solid line of soldiers by making the posterior century marched to the left then forward. When the first line had engaged in the battle, the second line advanced to be cl osely behind. When the men in the first line needed to be rescued, got exhausted, or felt that they were losing the fight, the men in the second line progressed and took their places. As fresh units were deployed through the gaps relieving those in the fro nt, a continual pressure will be allowed to be placed forward If needed, the third line would advance and rescue the men in first and second lines. In this kind of arrangement, light armed troops and cavalry were sometimes placed on the wings and someti mes in the front of the first line. ! Third Line Second Line First Line Light armed troops and cavalry Enemy Figure 1 : Roman triplex acies deployment
! 9 "#$%&!'()$(*+& ,-./*!0+1#2!3+4456!$(2! 7$%$8+& 7$15 7$#6$+! 9#+:-(.#*4+-; Figure 2 : Legend to figure battles. Light fill: Caesar troops; Dark fill: Adversarial troops.
! 10 CHAPTER VI LG BASED ANALYSIS OF CAESAR'S BATTLES We are going to analyze Caesar's great battles employing LG. Action s of commanders as LG optimization steps are going to be considered. This will imply covering resource allocation, strategy generation and implementation, and deception. Figures and maps are going to be used for this purpose. B efore the B attle of G ergovia After the Gaul defeat in the battle of Avaricum Vercingetorix, the Gauls leader, plan ned to avoid getting in real battle with Caesar's men. Instead he was only harassing them. His goal was to win more tribes to his side. The Battle of Ger gov ia took place in 52 BC  It was one of the Gallic war battles. Marching along the River Allier, to avoid outflanking, Vercingetorix kept pace with the Roman line on the opposite bank. As part of Vercingetorix initial resource allocation to avoid a battle with Caesar, he sent some of his men to destroy all of the brid ges and guard a ll of the spots where Caesar might think to build a new one. We will include the details and challenges faced by Caesar as reported by historians , [ 10 ]. Caesar could not afford crossing the river at this time of the year. The Roman LG res ource allocation will show the same outcome. Gallic LG knows that, and as a way to block all the Roman trajectories they should destroy all of the means that Caesar's army may use to cross the river. However, Roman LG will also know that waiting till the river could be crossed is not an option for Caesar in his drive to conquer the Gaul. Thus, the Roman LG will sugg est a deception. B ased on the built in library of deceptive schemes, it will suggest a combination of static and dynamic schemes. The static schemes will include concealment,
! 11 camouflage and dummies, while the dynamic schemes will i nclude demonstration and decoy ( Stilman et al. 2011) Given the topography of the land, the Roman LG will provide a detailed implementation of those schem es together with the courses of actions that will eventually lead to success. In this deceptive scheme, the Roman LG together with Caesar will gamble on the inferiority of the Gallic LG, i.e., on the inferiority of Vercingetorix tactical thinking. Caesar i mplemented a strategic ruse based on constructing a bridge without the notice of Vercingetorix. When the Roman s camped in wooded country near the destroyed bridge, Caesar attempted to implement his ruse. The day later for this purpose Ca esar and two of his legions hid between the trees and didn't march with the others. To cover the absence of these two legions, the Roman army opened out some of their cohorts. Based on the terrain features, the Roman LG will probably suggest the same concealed way that Ca esar used. The rest of the deception schema implemented by Caesar included camouflage, dummies, demonstration and decoy. The Roman column marched on and in the end of the day they built their camp as usual that ma de the Gauls to suspect nothing. By that t ime, Caesar started to construct the bridge. When it was finished, they crossed the river. At the time Vercingetorix heard about Caesar's crossing, it was too late. Since Vercingetorix didn't want to have a battle with Caesar, he headed away from the river In five days, Caesar reached him at Gergovia. Analogously to the actual historical events, the Roman LG deception scheme with high probability will bid on the inferiority of the Gallic LG. The Roman LG resource allocation procedure will generate a detailed resource allocation including high quality concealed crossing trajectories. I t will also determine required supplies, force division, moving over and remaining, etc. This operation of the Roman LG will be based on
! 12 assumption that the Gallic LG has been deceived, i.e., Vercingetorix and his army would not know about the absence of the two Roman legions and they would not have time to return back and intercept the Roman army. The resource allocation will be similar to the historical maneuvers. The se maneuvers were as follows. Vercingetorix destroyed all the bridges and secured all possible spots where the Romans might build a new one. Caesar took the advantage of the wooded land and carefully planned to hide there with two of his legions. After a certain time, the bridge was constructed and the legions crossed the river. Although these actions are not considered fighting they are part of resource allocation, in LG terms. Those are also the factors used to evalua te the quality of the river crossing trajectories up to the level that led them to be chosen by the Roman LG and discard all other trajectories. C onstruction of the LG H ypergames Constructing the LG Hypergames is important in jo ining operations that happened at different speed s and in different operational environments [ 3 ]. The above events that happened before the battle of Gergovia could be represented in two ABGs. These ABGs will have the same space time resolution. One might think that in this case events ha ppen ing in one ABG are not seen by the other until the final result appeared (i.e., the construction of the bridge and the Roman's crossed the river). However, from the LG system point of view, events in both ABGs will happen simultaneously and their effec ts will be reflected immediately in one another. The first ABG will reflect the bridge constructi on by the two legions that will hide with Caesar in the wooded land as well as the bridge crossing after that. The second ABG will reflect the marching of the two armies along the River Allier. Figure 3 shows this possible simulation of the ABGs.
! 13 S iege of A lesia The Roman had retreated from Gergovia after they suffered lots of causalities caused by Vercingetorix and his men ,  Caesar met Labienus a skilled cavalry commander and Caesar's right hand man, on his way and the Roman army get joined up again Cae sar also asked the German for reinforcement. After the reinforcement arrived, the R oman army became stronger and dealt with the attacks on Transalpine Gaul. After that, Caesar marched back to fight Vercingetorix. On their way, the Roman's legions were att acked by the Gaulish cavalry. Although, the Romans were heavily outnumbered, with the help of the Germans, the Roman forced the Gaulish warriors to retreat. Vercingetorix chose to retreat to a hilltop town of the Mandubii called Alesia, which provided the Roman with a clear target. After one day when Vercingetorix and his men camped in Alesia, Caesar with his full army arrived and camped in front of the town. We are going to include the details and challenges of this battle as reported by the histor ians [ 5 ]  Caesar's huge army <-%#+!088-#+ <-%#+!088-#+ Figure 3 : LG Hypergame of the events happened before the battle of Gergovia
! 14 contained 80,000 infantry in addition to his cavalry as he claimed [ 5 ]. As Vercingetorix and his men had the advantage of the ground, a direct assault from Caesar and his men would have result ed in many causalities even if the Roman succeed in the attack. Given the terrain features and positions of the Roman and the Gaul, the Roman LG will indicate the quality of the direct attack trajectories as very poor because of the highest attrition rate fo r a direct attack from Caesar toward Vercingetorix. The Roman LG would not take the risk of this attack. Instead, The Roman LG will identify all possible attack zones, that is, all possible main trajectories coming from the Gaul. Figure 4 illustrates some of these possible attack trajectories. After the Roman LG will process these data and apply the quality function on these attack trajectories blocking Alesia and building a siege around it will be suggested by the Figure 4 : Gauls potential attack zones (main trajectories).
! 15 Roman LG resource allocation procedure as the optimal solution to block or weaken these trajectories and to turn the whole Gaul army into a vulnerable target (Figure 5) By proceedin g with this plan, the disadvantage of the position of the Roman will be turned into an advantage. Caesar plan was similar to the Roman LG suggestion. Maneuvers Were As Follows The Roman LG will enhance the quality of this siege by building rampart, twenty three fortlets, and camps. The rampart was 11 miles long. This is the major resource allocation suggestion to be made by Roman LG (Figure 6). The Gallic LG will try to preven t the Roman from blocking them so it will issue a direct attack plan to intercept the Roman warriors. An attack zone will be created Figure 5 : Building inward siege
! 16 by the Gaul's cavalry who will be sent to attack the Romans while they are working. As a response to the coming attack, the Roman LG will initiate a counter attack led by the Roman's auxiliary and allied cavalry together with Caesar's reserve of the German horsemen who will join later and succeed in driving the Gaul's cavalry back (Figure 6 A). Figure 7 shows the network language interpretation of the above maneuvers. P 0 is the piece related to the main trajectory trying to attack the target q 1 (i.e., the Gaul's cavalry attacking the Roman warriors who are building the siege), and q 0 is the piece related to the first negation trajectory trying to intercept the main piece P 0 at location 3 before it destroys the target (i.e., the Roman auxiliary, allied cavalry, and Caesar's reserve of the German horsemen). Figure 6 : Skirmishes while building the siege
! 17 After the Gaul attack will be repulsed and the Roman Siege is about to finish, the Gallic LG will realize the level of danger and the huge pressure that the Gaul was under Being blocked will tie the Gaul moves and without any help they will be defeated, so the Gallic LG will suggest sending the cavalry away to ask for a relief army before Caesar's army completes the blockade (Figure 6, B). In this battle, the Roman LG will suggest a numbe r of resource allocations and reallocations A detailed implementation of these resou rce allocation procedures and calculation of the courses of actions that will lead to a Roman victory are going to be suggested by the Roman LG. In addition to surrounding the hill and maintaining a line of circumvallation, the Roman LG will suggest a number of obstacles to be added. The purpose of these obstacles will be to worsen the Gaul attacking trajectories and to warn the Romans of a coming attack and to delay it as well. One of these obstacles will be a twenty feet width of a straight sided ditch running from one stream to another to be dug Figure 7 : Network language interpretation for the attack zone generated for the Gaul's cavalry.
! 18 to the west. The Roman LG will continue the resource re allocation process and will suggest digging another ditch of 400 paces le ngth (c. 130 yards) behind the first ditch, which will be considered as the Roman main defense line. Also, a twelve foot high rampart together with high towers within eighty foot intervals will be built behind the double ditch. To protect the Romans from a night attack and to slow down any attack attempt from the Gaul, the Roman LG will suggest maintaining a series of obstacles and macabre traps in front of the double d itch. Maintaining such strong lines of defenses will allow a small number of Caesar's men to withstand different kinds of the Gaul attacks except for the heaviest ones, where more men would be needed. With those defenses Alesia will be completely blocked and enclosed. All these obstacles will result from the extensive analysis done by the Roman LG in order to protect Romans from surprising attacks, to help them in defending against coming attacks by weaken ing the enemy's attack trajectories. Knowing that the relief army is on its way to support the G aul, the Roman LG will expect s ome of the relief army attack zones and will start to prepare for these attacks. Figure 8 shows some of the expected attack trajectories to be initiated by the relief army on the eye of the Roman LG.
! 19 The Roman LG on its last stage of resource re allocation will suggest building another longer line of contravallation facing outwards as the best way to defend against the relief army attack zones The Roman LG will be able to generate courses of action leading to victory even with this double encircl ement. As a result, the Roman LG will decide that this double encirclement is not so dangerous for Romans At this point, the i nwards and outwards lines of defenses w ill be suggested by the Roman LG (Figure 9 ). Figure 8 : Some of the expected attackin g trajectories from the Relief a rmy.
! 20 !!!! !!!!!!!!!! ! The Roman LG will insist on increasing the blockade pressure. To do so, it will allow no one from the Gaul to pass through its lines of circumvallation. As the Roman LG will predict, the food and supply will beco me low and there w ill no t be sufficient amount to feed all the people. Indeed, in reality, Vercingetorix wanted to move the women, children and elderly out of Alesia but Caesar didn't allow any one to pass through his lines. The Roman LG will interpret the information of the Gaul running out of food and supplies as a sign of the enemy weakness. As a result it will increase the pressure on them and will suggest admitting no one th rough the lines of circumvallation Analogously to the actual historical events, given the great number of Caesar's men and the terrain features, the Roman LG resource allocation procedure will show the same outcome. Figure 9 : Siege of Alesia with inwards and outwards lines of defenses
! 21 When the relief army arrived, they camped on the south west on a high ground as was predicted by the Roman LG resource allocation. The day after, they showed both the Roman and their friends their huge army. As a response from Vercingetorix, he ordered hi s men to move out of the town. The warriors waited behind the wide trench that the Roman had dug. Feeding the Gallic LG with the information of the relief army arrival will result in suggesting a concurrent attack from both directions to distract Caesar's men and increase the attrition rate turning the Romans into a vulnerable target due to the attacks coming from the front and back (Figure 10). While observing that the Gaul and the relief army stood in lines prepared to the attack from both direction s the Roman LG will suggest deploying the warriors on both of the siege lines. To show the enemy the Roman confidence to attack the relief army from the rear and from the wing s (dooming Figure 10 : Concurrent attacks from both directions.
! 22 them under a high attrition rate ) the Roman cavalry will be sent to go out of the lines to fight the relief army's horsemen (Figure 10,A). Figure 11 illustrates the network language interpretation for the attack zone generated for the Roman cavalry who went out of the circumvallation lines. The Roman LG w ill provide the same outcome to what happen ed in reality. Caesar deployed his men on both of the siege lines. To show the enemy his confidence and to increase the pressure on the relief army Caesar ordered his cavalry to go out of the lines to fight the r elief army horsemen. The fight lasted through the afternoon and it was for a long time in favor of the Gaul because of their huge numbers Reflecting on those events in the battle field and how the Romans to be loosing, the Roman LG will suggest providing t he warriors with any resources that should strength en their situation. I n this case the German cavalry will be ready to rescue the Roman warriors. Figure 11 : Network language interpretation for the attacking zone issued by the Roman cava lry out of the circumvallation l ines.
! 23 After the Roman won that day, the Gallic LG knew that as long as these obstacles are there, it is hard for t he Gaul to win. It suggested to attack one of them. The Gaul warriors spent the day in preparation. To be able to climb the rampart that the Roman built, the Gal l warriors collected ropes and built ladders. They also collected sticks to fill the ditches th at the Roman had dug. At midnight, the Gallic LG will suggest an attack to be started by the relief army. 0.$-(=!*/#!>$88-:!,>!?(2#+#6*-1$*#2!*/#!54@#+!4)!*/#!<41$(!,>A! As there were no means of communication between the relief army and Vercingeto rix and his men, the relief army raised a great cheer as a signal to let their friends know about the ongoing attack. The Gallic LG will model spreading information that the relief army is initiating an attack. These kinds of information are implemented as communication delays between the independent components of LG, which are in this case the relief army and the Ga u l warriors who are blocked in Ale sia. Even with the great efforts of the relief army, the warriors in Alesia were too late to join their comra des in the attack as they took a very long time to prepare and organize themselves and the Roman had enough time to repulse the relief army attack. Both attacks were repulsed by Mark Antony and Trebon ius, who were responsible of the s e section s of the Roman lines. Because of the strong resource allocation made by the Roman, the Roman LG was able to repulse both attacks. On a hill slope on the north west was the Roman camp that was not included in the Roman's lines because of the huge effort that will be needed to include the hill within the defense lines. The camp was occupied with only two legions. T he relief army found the Roman camp as a vulnerable spot that could be the key to defeat the Roman. The Gallic LG will suggest a deception using the buil t in library of deceptive schemes. A
! 24 combination of static and dynamic schemes will be suggested. Implementation of these schemes and calculations of the courses of actions that will lead to a victory will be provided by the Gallic LG [ 1 ]. As Vercingetorix and his men were o n a hill top they noticed some of the relief army preparations. However, they didn't know anything about the plan as there were no means of communication between them. Feeding the Gallic LG with this information will result in suggesting an all out attack on the inner lines to be launched by Vercingetorix's men as a way to support the deception plan and to insure its success. The Gallic LG will gamble on the inferiority of the Roman LG and will confirm the deception plan. The Gallic LG wi ll suggest to move Vercassivellaunus's men, under the cover of the night, to the reverse slope of the hill where the Roman camp is and hide there ( F igure 12 ). To distract the Roman LG from the real attack, the diversionary attacks would be suggested in different places. In reality, even though Vercingetorix had no idea about the plan that Vercassivellaunus was pursuing he knew that t he relief army is up to something as he noticed some of the preparations. As a way to help Vercassivellaunus and the relief army, he issued an all out attack from the inner lines. Figure 12 : The relief army trajectories to attack the Roman camp.
! 25 At midday, the Gallic LG will issue an attack by Vercassivellaunus and his men to the Roman camp at the same time where other attacks were launched creating multiple zones with different main trajectories ( F igure 13 ). The Roman LG had expected these concurrent attacks long time ago and that's why it added all of these ob stacles in its resource allocation stage to help in defending against these attacks The Roman LG will respond to theses multiple attacks and will suggest spreading the army thinly to defend against them. The Gallic LG had put the Roman LG under huge pre s sure. The Roman LG will request an immediate reinforcement to the north w est side were the Roman camp is as soon as it realizes the coming attack from Vercassivellaunus and his men there (Figure 13 A). Since protecting the north west side was very importa nt as it was the weakest part of the Roman, trajectories heading to the north west side to reinforce the legions there had the highest priority and considered as the main trajectories to those zones. Similarly, to reinforce the two legions holding the Roma n camp, Caesar had sent Labienus with six cohorts. Also, given each attack zone features and obstacles provided by the Roman, the Roman LG would suggest how to spread the warriors to deal with the attacks. The Galli c LG will suggest many attack trajectorie s with different start and end points. The Gaul warriors will start moving along tho se trajectories unt il they succeed. The first attack trajectories were directed through the weakest sections of the circumvallation lines but they were repulsed by the Roma n warriors. This means that the Gallic LG was mistaken in those choices Next, they started pursuing the attack trajectories going through the sp ots that were thinly guarded by the Romans but highly protected by th e slopes. The Gaul warriors succeeded at one point by getting over the rampart and pulling down one of the Roman towers.
! 26 When the Gaul warriors will succeed in getting over the rampart, the Roman LG will consider that attack zone to have immediate reinforcements to return the Gaul's warriors back and block them again, adding a new Roman attack zone. After the arrival of reinforcements to this attack zone, th e Gallic LG will issue a retreat order. As all attack zones have been either controlled or disappeared except for the one in the north west side, the Roman LG will suggest safe trajectories heading to there and considering outflanking the enemy turning the Gaul into the vulnerable target (Figure 13 B). This counter attack will change the battle situation to the Roman's favor and the relief army will fail # Figure 13 : The Gallic LG deception plan.
! 27 In reality, Caesar tried to rescue the situation by sending Decimus Brutus with some of the tr oops. More reinforcement were needed to drive the Gaul back, so Caesar sent Caius Fabius with some cohorts. In the end, the Roman succeeded in ending this attack and plugging the gap in the line. After that, Caesar took four of his cohorts and went to supp ort Labienus at the fort on the hillside. Caesar divided the army's cavalry into two groups planning to attack Vercassivellaunus and his men from the flank. So he sent one of the groups outside the lines of contravallation for this purpose and kept the oth er one with him. This reinforcement changed the battle situation to the Roman's favor. C onclusion The Gallic LG did not succeed in both attacks that it had launched by its relief army. The Gallish warriors were totally trapped and have no way to run. o This means that the Gallic LG was inferior. The Roman LG great effort and time spent on allocating and reallocating its resources before the actual battle have proven its superiority LG H ypergame C onstruction Siege of Alesia is one of the battles that has lots of details and concurrent m ultiple attack zones. In order to simplify the complexity of the problem, decomposition is needed. Complex problems can be seen as multiple games that fall under one LG hypergame  Each ga me is represented by its ABG. These ABGs have different space time resolutions, each represents a different set of operations. Within an ABG, each piece is assigned a list of possible actions it can perform A change in the state of one ABG within the LG h ypergame, may change the state of the other ABGs. Interlinking
! 28 mappings between ABGs are used to spread the information of a changed state to all other connected ABGs. The events of Siege of Alesia could be represented in different levels of resolution. In order for the LG system to have the full view and to suggest good courses of actions, ABGs with higher levels of abstraction are needed in addition to the ABGs with lower levels of abstractions. For example, figure 1 4 illustrates a possible representation of the LG hypergame hierarchy with two levels of resolution for one of the concurrent attacks that happened during the battle. However, higher levels of resolution could be obtained as needed for example, a view could be obtained in terms of a cohort le vel. This will add another level of resolution to the hypergame. Getting deeper, even a soldier level could be obtained. This higher level of resolution is very important for the LG system to organize the army and suggest courses of actions that will lead to a victory. Similar decomposition is utilized by human commanders in actual battles including the Siege of Alesia.
! 29 ! ! Figure 1 4 : LG hypergame hierarchy with two levels of resolution.
! 30 CHAPTER VII CONCLUSION LG based analysis ha s been made for Alexander the G reat [ 1 ], Hannibal [ 2 ] and Julius Caesar battles ( this work ) All of them demonstrated that no mysterious actions occurred in those battles Every decision made by the commanders, every resource allocated and every deception plan have its rational. LG grounds this rational well. As we saw in the analysis the Roman LG and Gallic LG produce d the same courses of actions as those reported by the historians. This is achieved by feeding the LG system with proper ini tial data and ongoing updates. However, i t is important to indicate that the only way to really test the LG hypothesis is to perform the LG software experiments .
! 31 REFERENCES [ 1 ] B. Stilman et al. Revisiting History with Linguistic Geometry ," ISAST Trans. on Computers and Intelligent Systems v ol. 2, n o. 2, pp. 22 38, Oct. 2010. BCD B. Stilman et al. The Primary Language of Ancient Battles, Int. J. of Machine Learning and Cybernetics, Springer v ol. 2, n o. 3, pp. 157 176, 2011. [ 3 ] Stil man, B et al. Adversarial Reasoning and Resource Allocation: the LG Approach in Proc. of the S PIE Conference Enabling Technol ogies for Simulation Science IX Orlando, FL, March 28 April 1, 2005, pp. 177 188. ! BED FA! Stilman, B et al ., G*+$*#.# 6 -(!,$+.# H G:$8#!I+4J8#16 =K!-(! !"#$%&'%(')* +$'&,-(-./*0,1234'4(,-')*!22%,'56$&*4,*+$'"(-.*46$*722,-$-48&*9(-" :* 0A!L4**! $(2!MANA!N:O(#$(#&!O2 ;* F4 :$!<$*4(P!7<7!I+#66=!CQQR=!:/AS=! S= 55A!CTU H CVS A !  A. Goldsworthy, "Proconsul, 58 50 BC," in CAESAR Life of a Colossus New Haven: Yale University Press, 2006 pp. 336 342 [ 6 ] J. Caesar and A.T. Walker, Caesar's Gallic War, Books I IV, with Introduction, Vocabulary and Grammatical Appendix Rev. ed. Chicago: Scott, Foresman, 1970, pp.9 10. [ 7 ] K. Gilliver, Esse ntial Histories Caesar's Gallic Wars 58 50 BC Routledge: Routledge, 2003. [ 8 ] Caesar's legion (n.d.) [Online]. Available: http://fallout.wikia.com/wiki/Caesar's_Legion#cite_note 1 [ 9 ] The Roman art of war in Caesar's time (n.d.) [Online]. Available: h ttp://www.uah.edu/student_life/organizations/SAL/texts/misc/romanwarmach.html [ 10 ] A. Goldsworthy, "Proconsul, 58 50 BC," in CAESAR Life of a Colossus New Haven: Yale University Press, 2006 pp. 328 330