Towards a Quasi-Endgame-Based Bao Solver

Transcription

1 Towards a Quasi-Endgame-Based Bao Solver Tom Kronenburg Master Thesis MICC-IKAT Thesis submitted in partial fulfilment of the requirements for the degree of Master of Science of Knowledge Engineering in the Faculty of Humanities and Sciences of the Universiteit Maastricht Thesis committee: Prof. dr. H.J. van den Herik Dr. H.H.L.M. Donkers Dr. A.J. de Voogt Dr. M.H.M. Winands Universiteit Maastricht Maastricht ICT Competence Centre Institute for Knowledge and Agent Technology Maastricht, The Netherlands October 2006

2 Preface This M. Sc. Research project was performed under the auspices of the Maastricht ICT Competence Centre, at the Institute of Knowledge and Agent Technology of the University of Maastricht. The subject of this thesis is the board game Bao, in particular discovering its game-theoretic value. I would like to thank everyone who helped me complete this thesis. I am very grateful for the support from my supervisor Prof. dr. Jaap van den Herik. The intensive interaction with my daily supervisors dr. Jeroen Donkers and dr. Alex de Voogt deserves a special word of gratitude. I was fortunate to enjoy their help, lessons, and attention in many ways. I thank dr. Donkers for his patience with my procrastination, his many hints and tips, and his never-fading hope in this thesis. I thank dr. de Voogt for some very memorable meals, his very original attitude towards academic life as well as a lifelong enthusiasm for such an obscure subject as Bao. Finally I return to Prof. dr. Van den Herik for all the years in which he was my teacher, had a listening ear and was an inspiration. I would like to thank my family and friends for their support, love, and friendship that helped my through these years. A final word of thanks to my girlfriend, whom I promise that she no longer has to share me with a wooden board. Tom Kronenburg, Maastricht, October 3, 2006 ii

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4 Summary In this thesis we present our research results on solving Bao, a mancala game from East Africa. The mancala games Kalah and Awari have been solved by special AI studies, but there is not much research directed towards solving Bao, the most complex mancala game. As a direct consequence we have a straightforward problem statement: What is the game-theoretic value for Bao? We first investigated the structure of the rules of Bao and considered the implications of the rules implementing a new Bao program. The most complex rules of Bao, those concerning Takasia situations and those concerning the house, a special pit in the front row, confronted us with the GHI (Graph History Interaction) problem and with difficulties on move-generation. A second, so far ignored problem was that in the rules of computer Bao, an assumption is made about the existence of perpetual moves. There are moves that never end, which is a serious problem for any computer program. We proved the existence of such moves and give some examples (cf. Kronenburg, Donkers and de Voogt, 2006). Moreover, we compared Bao to several games from the Olympic list. Bao is rather close to Checkers and Qubic in terms of its game-tree and state-space complexity. Checkers is almost solved by the use of large endgame databases and Qubic is solved by forward search combined with game specific knowledge. Awari and Kalah, related in terms of game type, have been solved by using retrograde analysis, in the latter case in combination with forward search techniques. Other games have been solved using knowledge-based methods. For Bao, there is no game-specific knowledge available and due to its diverging nature, retrograde analysis is unfeasible. There are however statistics available that indicate that a very high number of potential endpositions occurs from ply 44 to ply 64. We tried to exploit this fact when solving Bao. To do so, an approach based on quasi-endgame databases was developed. The procedure is basically as follows: Generate databases with 1,000,000 random positions at ply 5, 10, 15 40, 44. Then by doing forward searches, obtain the game-theoretic value for all positions in the ply-44 database. Try to obtain as much values in the ply-40 database using the ply-44 database as an endgame database. Proceed into the direction of the root and try to obtain Bao s game-theoretic value. Unfortunately, obtaining the values in the ply-44 database was harder than expected. Solving Bao by means of the quasi-endgame approach has therefore appeared to be presently unfeasible. The game s game-theoretic value remains therefore still unknown, although we think an import step has been made towards solving Bao. iv

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6 Contents Preface Summary Contents 1 Introduction Bao Solving Bao Research Questions and Problem Statement Outline Conventions 4 2 Bao Rules General rules The Nyumba or House Takasia Checking a Bao Move Generator 10 3 Perpetual Moves Never-Ending Moves in Bao Perpetual Sowings Never-Ending Moves in Actual Bao New Findings Implications to the Rules of Computer Bao 16 4 Towards Solving Bao Retrograde Analysis and Endgame Databases Solved Mancala Games Other Solved Games Quasi-Endgame Databases Generating Random Positions Summary of Chapter Experimental Setup The Program Experiment Experiment Results and Discussion Results and Discussion of Experiment Results and Discussion of Experiment Conclusions Research Questions Revisited Problem Statement Revisited Further Research 34 8 References 37 ii iv vi vi

7 1 Introduction Competition has always appealed to mankind. It provides an opportunity to test ones vigour and skill, perseverance and might. Board games have provided mankind with the opportunity to compete without the fuzz of actually going to war, without physical effort even. Board games provide a way of testing intelligence and creativity. As soon as it was possible to use computers for less mundane things than calculating bullet trajectories or building A-bombs, playing board games became one of the first interests for computer scientists. Artificial Intelligence was born while making the computer learn how to play. The main game in AI game research is and always has been Chess. Ever since Claude Shannon s (1950) paper, researchers all over the world tried to build artificial Chess players. Due to the nature of men, this soon resulted in a competition between computer programs. In 1997 it resulted in DEEP BLUE s win over Gary Kasparov (Campbell, Hoane and Hsu, 2002). Compared to the number of games played all over the world, a relatively small number of games have gained the interest of the AI community. AI has mainly concerned itself with classic western games such as Chess and Checkers and has only recently started to demonstrate interest in eastern games such as Go, Shogi and Chinese Chess. Games that originated in Africa are mostly ignored by the AI community, with the exception of the mancala games Awari and Kalah (the latter invented in the U.S.A., but nonetheless a genuine mancala game). In this thesis we discuss how to solve a game from the same family of games, Bao. Bao can be considered the Chess of mancala games. It is a complex and challenging game that is a joy to play. 1.1 Bao Bao la Kiswhaili ( The Swahili board game ), as Bao is formally known, is a board game, played in the Swahili speaking part of Eastern Africa. The best players can be found on Zanzibar and on mainland Tanzania (de Voogt, 1995). Bao is a game from the mancala family of games. Well known mancala games are Kalah (Bell, 1968), Awari (Allis, van der Meulen, and van den Herik, 1991) and for Europe s general public most notably Bantumi (a Kalah variant). The latter is well known because it was distributed with many types of Nokia mobile phones in the late 1990s and early 2000s. Mancala games have a number of distinct properties. These properties are listed in Donkers, Uiterwijk, and de Voogt (2002). They are as follows: 1 The games are played on a board with a number of pits, usually arranged in two or more rows. Sometimes additional pits are used that we will call stores. 2 The games are played with a collection of equal counters (stones, seeds, coins, or shells). 3 Players own pits, not counters. Often, a player owns all the pits on one side of the board. 4 Moves are made by sowing, which is a form of counting. A sowing consists of picking up all seeds out of a pit, and distributing them one by one into adjacent pits, usually in a predefined direction. 5 After (or during) sowing, counters can be captured. (Hence, mancala games are also called count-and-capture games.) 6 The goal of the game in general is to capture the majority of the counters. 1

8 From the mancala family of games, Bao is recognized to be the hardest, most complex game. (Townshend, 1986). Bao la Kiswahili is regarded as the Chess of Africa and is one of the few mancala games in which championships are organised (de Voogt, 1995). Bao is one of the 4 row mancala games, each row numbering 8 pits. We will refrain in this thesis from presenting the Bao rules in full, for they are easily available (Donkers, 2001, de Voogt, 1995). However, it is important to note here that the Bao rules are complex and pose a considerable challenge when learning the game. 1.2 Solving Bao AI game research historically mainly focussed on agents that play two person games of perfect information like Chess (Shannon, 1950) and Checkers (Samuel, 1959). In the late 1980s, early 1990s attention started to grow for solving and/or cracking these games (Allis, van den Herik, and Herschberg, 1991). Solving a game is the process of calculating the game s game-theoretic value. This means that. solving a game is calculating what both players can achieve when playing an optimal strategy. The result of a game is then either a first player win, a draw, or a first player loss. Several games have been solved, e.g., the game of tic-tac-toe does not have to be lost by any player. If both players play optimally, the game ends in a draw. Allis (1994) distinguishes between several categories of solutions. A game is solved: 1 Ultra weakly, when for the initial position(s) the game-theoretic value is computed. 2 Weakly, when for the initial position(s) the game-theoretic value is computed and it is known what moves (i.e., what strategy) lead to the computed win. 3 Strongly, when for all possible positions the games theoretic value is computed and it is known what moves (i.e., what strategy) lead to the computed win. Solving a game strongly is more difficult than solving a game weakly, which is in turn more difficult than solving a game ultra weakly. The level on which a game can be solved is dependent on different factors. It is influenced by the game s (1) state-space complexity, (2) game-tree complexity, (3) the power of the available solution methods, and (4) the suitability of the available solution methods for solving that specific game. The state-space complexity is the number of legal states. For Bao, an upper bound of was computed by de Voogt (1995). This is the number of board configurations with a maximum of 64 seeds in 32 pits. Although some of these states can never be reached, (i.e., all positions with at most 19 seeds) the number of illegal states is insignificant compared to the total amount of states. The game-tree complexity is the number of states that can appear in the game tree. The game tree is a representation of all the different game positions. Donkers and Uiterwijk (2002) estimated Bao s game-tree complexity to be The availability and suitability of solution methods can only be deduced from empirical evidence. However, the state-space and game-tree complexities of Bao are close to those of several other games. In the overview by van den Herik, Uiterwijk, and van Rijswijck (2002) the games that were played at the first and second computer Olympiads are all compared in terms of state-space and game-tree complexities. We reproduce their results and include Bao in Table 1.1. Several games on this Olympic list are comparable in terms of these complexities, and some of those are solved to different levels. The games closest to Bao in terms of game-tree and state-space complexity are Qubic and Checkers. Qubic was weakly solved independently by Patashnik(1980) and 2

9 Allis and Schoo (1992), while the Chinook team (Schaeffer, 1997) is currently trying to solve Checkers strongly, having already won the world championship title. Game Log state-space Log Game-tree Awari Bao Checkers Chess Chinese Chess Connect Four Dakon Domineering (8x8) Draughts Go(19x19) Go-Moku (15x15) Hex Kalah Nine Men s Morris Othello Pentominoes Qubic Renju Shogi Table 1.1: State-space and game-tree complexities of games from the Olympic list, including Bao (Herik et al, 2002). We will elaborate on the Olympic list and the solved games in Section 5.3. For now it suffices that we can assume that Bao is at least (ultra) weakly solvable. In this thesis we will try to verify this statement. 1.3 Research Questions and Problem Statement The main topic of this thesis is (ultra weakly) solving Bao. This means obtaining the game s gametheoretic value. Thus our problem statement is formulated as: What is the game-theoretic value of Bao? In order to answer this question we will try to find answers to the following research questions. 1 What problems arise from the rules of Bao that are faced by programmers? 2 What techniques are used to solve (ultra-weakly, weakly or strongly) related games? 3 Can these techniques be used to solve Bao weakly? 3

10 1.4 Outline In Chapter 2 and Chapter 3 we focus on the first research question. In Chapter 2 we describe the structure and intricate properties of the Bao rules that cause problems when implementing Bao in a computer program. In Chapter 3 we solve the problem of perpetual moves. Perpetual moves are moves that start but never end. This challenging problem was mainly ignored until recently. In Chapter 4 we present an overview of different techniques used by other computer game playing programs. We also present a custom built technique for solving Bao. We assess whether or not all these techniques are suitable for solving Bao, thus seeking an answer to the second research question. In Chapter 5 we present our experiments. In Chapter 6 we present the results to our experiments and discuss the value of the outcomes. In Chapter 7 we present the conclusions to the research questions and answer our problem statement. We also provide some ideas on what might constitute valuable research in this field in the future. 1.5 Conventions In this thesis we use the following three conventions. (1) Swahili terms and their English translations are used interchangeably. If used, the Swahili terms are always capitalized. We have used Swahili terms and anglicised them for reasons of readability. (2) Because Bao is originally a male sport, a player is always referenced as he. (3) When we speak of the house, we usually do so in terms of ownership. A player owns or has the house. Moreover, it is possible to say that the house is still present, it exists or that the house is lost. 4

11 2 Bao Rules Bao is recognised to be the most complex game from the mancala family (Townshend, 1986). In this chapter we will focus on the most important rules that make Bao such a complex game, namely the rules that involve the Nyumba or house, and the rules in which a Takasia situation occurs. In Section 2.1 we present a general view on the rules of Bao. In Section 2.2 we present the rules that involve the Nyumba or house. In Section 2.3 we present the Takasia rule. In Section 2.4 we discuss the difficulties involved with testing whether a program is correct, i.e., whether or not it executes the moves correctly. 2.1 General rules There are several documents describing the rules of Bao. The most important and extensive document is Limits of the Mind, a characterization of Bao Mastership by de Voogt (1995). Other descriptions of the rules consistent with de Voogt have been provided by Donkers (2001) and Nierse (2006). The National Museum of Tanzania (reprinted in Russ, 2000) presented rules that can be found on several Internet sites (e.g., Wertenauer, 2005) but they miss some of the Nyumba rules as well as the Takasia rule. Murray (1952) presents a set of rules that differs even more from the ones by de Voogt. Since the rules are widely available, we refrain from presenting them in full. However, certain features of the rules have to be presented in order to understand the remainder of this thesis. Below we list the main rules of Bao. 1. Bao is played on a board consisting of 4 rows, each containing 8 pits. The 2 rows closest to the player are the player s own rows. See also figure 2.1. The uppermost and lowermost rows are known as the outer or back rows, the middle rows as the inner or front rows. North b8 b7 b6 b5 b4 b3 a8 a7 a6 a5 a4 a3 b2 a2 b1 a1 Key: The Nyumbas are the square pits. The Kichwas are surrounded by the dotted lines Stock north: 22 A1 A2 A3 A4 A5 A6 A7 A8 Stock south: 22 B1 B2 B3 B4 B5 B6 B7 B8 South Figure 2.1: A Bao board. Every pit denotes its name in the notational system. 5

12 2. Bao starts with 20 seeds on the board and 44 in stock. During the first 44 ply, the first action of each move is placing one seed from the stock in a (non-empty) pit in the player s own front row. This stage is known as Namua. From ply 45 onwards the game is in Mtaji stage. A move now starts by picking up all seeds from one of the player s own pits. 3. The basic action in Bao is sowing. Sowing is the spreading of seeds into consecutive pits. A move consists of a series of at least one sowing, possibly interspersed with captures. A move ends when the last seed of a sowing falls into an empty pit. The sowing always occurs in the player s own two rows, that can be seen as a cycle; the connections lie in the outermost pits. 4. Capturing is obligatory. If a player can capture, he has to do so! If the player cannot capture at the end of the first sowing, he is not allowed to capture in that move. The move is then known as a Takasa or Takata move. 5. The Nyumba or houses are special pits, the Nyumbas are located at the fourth pit from the right in the front row. See also Figure 2.1. Usually this pit is carved square instead of round and it is a pit where special rules apply. 6. The game ends when a player has an empty front row or when he is unable to move, in both cases the other player wins. A draw is not possible, but a player can resign. The Bao rules can be divided into three disjunct categories. Rules in the first category govern the legality of the move, i.e., they determine from which pits and in what direction a player is allowed to start a move. Some of the rules of this category are only valid in Namua stage, some are only valid during Mtaji stage, and some are valid regardless of what stage the game is in. The second category of rules governs the execution of the moves, they determine when a move ends or when it continues, and in what direction. All rules from this category apply in both stages of the game. Once it is established that the move starts from a legal pit and in a legal direction, the end result of the move is known. The only exception to this is when a rule of the third category is invoked. The third category encompasses the rules that impact both the legality and the execution of a move. These are the Nyumba (Section 2.2) and Takasia (Section 2.3) rules. Both rules protect a pit from the obligation to be sown. A move can therefore not start from this pit. If a sowing ends in this pit, the player does not have to continue sowing, implicating that the execution of the move is prematurely aborted. The structure of the Bao rules can best be seen as a hierarchy from general to specific rules (de Voogt, 1995). At the top level of this hierarchy are the most general rules (e.g., capture is obligatory) while at the bottom the Nyumba and Takasia rules appear. Although the rules for a large number of games can be regarded as a hierarchy, (e.g., in Chess, where the castling move is only allowed in certain positions) the Bao rule hierarchy is deeper than other mancala games (where Nyumba and Takasia are not part of the rule book), but also deeper than most other games of interest to the AI Community. Further research is needed to compare the depth of the Bao rule book hierarchy with those of other games and therefore we can offer no substantial quantitative analysis at this point. 6

13 Figure 2.2: The execution of two moves directly at the start of the game (Donkers and Uiterwijk, 2002). 2.2 The Nyumba or House As we stated in the previous section, a Bao board contains 2 Nyumbas or houses. These are the square pits in each front row, located at the fourth pit from the right. A Nyumba is a pit to which additional rules apply. The main feature of the house is that it allows a player the choice to stop the move when a sowing ends in the house. This gives the player the advantage that he can build up a sizeable amount of seeds, which results in a strategic advantage. However, once the player chooses to play the house (to continue sowing from the house) the house is lost. This means that the square pit is, from that moment on, like any other pit and a player can never choose to stop there again. The strategic advantage of the house is protected. Although the house is not protected, from capture by the opponent, different rules apply when the house is present. These rules are as follows: 1. During Namua, a non-capturing move starts from any non-empty pit in the front row except the house itself when it is present. When the house is absent, a move can start from any pit containing at least 2 seeds. 2. During Namua, a move has to start from the front row. When the only occupied pit in the front row is the house, it is played as if it was a singleton (a pit containing 1 seeds). When the house is not present, this pit is played by picking up all its seeds and sowing them into consecutive pits. 3. During Mtaji, if the house is still present it cannot be Takasia-ed, although all other pits can. The first two rules give the player the advantage to be able to play more pits in order to save the strategic advantage of the house until it can be fully exploited. The second rule is to protect the house once it is the only occupied pit in the front row. It does however, also constitute a grave danger. In certain situations this second rule might result in a certain loss to the player. An example is given in Figure 2.3. North must play the house as a singleton, but whatever direction he chooses, South is able to capture both seeds (by playing A2R or A6L), thus leaving North with the same option, to play the house as a singleton. South s situation is also exactly the same, he can capture both seeds again. If repeated, this would lead to a certain loss for North. To protect the house from this danger, the player is obliged to Takasa the house when it contains fewer than 6 seeds and is the only occupied pit in the front row, or the only pit containing 2 or more seeds in the front row. This means that the player plays the house as if it were a normal pit, in 7

14 effect losing the house, just as it happens when a player decides to play the house (ending a sowing at the present house and continuing the move) or when the opponent captures the house b a 0 x y z 1 Stock: 11 Stock: Figure 2.3: A situation in which the house is endangered. The values in the outer rows, as well as a, b, x, y and z are arbitrary values. Rule 2 states that if the house is the only occupied pit in the front row and contains six seeds, it has to be played as a singleton. If the opponent only captures one of these seeds, the player has the opportunity to revive the house by playing a capturing move (if possible) that leaves the house with six seeds again. If the move ends in the house, however, the player has to play the house, losing it definitively. Other than the above mentioned option, there is no other way of reviving a house. Programming the Nyumba rules The Nyumba rules are in itself not very difficult to implement in a program. They do however present the programmer with two problems, viz. (1) when doing retrograde analysis and (2) when programming a Bao move generator. Retrograde analysis is one of the most powerful tools in solving games. The basis of retrograde analysis is generating (all) possible end-situations, and then generating the states that can lead to these end generations. Doing this generation of previous states leads to a collection of states, known as an endgame database, that can be used by a search algorithm when playing or solving a game. We discuss retrograde analysis and endgame databases in more detail in Section 4.1. Usually the generation of previous states can be done by simple inspection of the current (later) situation. However, the Nyumba rules avert us from doing so. To explain this problem we use an example from Chess. Assume we have an end position. We generated a number of previous positions and the transition from one of these states to the end position is a castling move. If we now want to generate this position s predecessors, we can never move the King or Rook because it would make the final castling move illegal. The problem is that it is not apparent from the position under inspection that it is illegal to move the King or Rook. In Chess this problem is known as the Graph History Interaction (GHI) problem. According to Heinz (1999) the castling problem is ignored in endgame analysis in Chess, mainly because castling occurs (usually) only prior to the endgame, the only part of the game tree where retrograde analysis is currently used. In Chinese Chess the GHI problem is of much more importance because of the special type of rules in that game. See Wu, Hsu, and Hsu (2006) for examples. In Bao, the GHI problem may occur with respect to the Nyumba. One cannot play or lose the house twice. Retrograde analysis is a hard task for Bao (see Section 4.2); the GHI problem makes it even harder. Ignoring the Nyumba for endgame analysis, analogously to ignoring castling in Chess 8

15 endgame analysis, is possible although the use of the resulting endgame database is probably more difficult. The second problem that occurs while programming a Bao move generator is the task of listing all the legal moves. In this list, all moves should be encompassed, but each move not more than once. Moves that contain a sowing that ends in the Nyumba can continue from there, or stop at the Nyumba; effectively creating 2 different moves. Both moves should be listed in the move list, but a move that does not contain a sowing that ends in the Nyumba should not be added to the legal move list twice. Since it is not apparent from a board configuration whether a move contains a sowing that ends in the Nyumba, all potential moves have to be simulated on the Bao board before they can be added to the list of legal moves. This problem means that generating Bao moves is relatively expensive in terms of the number of operations. 2.3 Takasia The Takasia rule is the most obscure, most specific and hardest rule in the game of Bao. Townshend (1986) describes it as, A player who Takatas in such a way as to directly menace only one of his adversary s occupied holes without exposing any of his own seeds cannot be deprived of the capture he set up. It must be reserved for him in the next move. This means that if South plays a move, in which he does not capture, and in the subsequent ply North is also unable to capture, while at the same time only one of North s pits is under attack, he cannot empty (play) this specific (Takasia-ed) pit. It is not allowed to start from this pit, and if a sowing ends in this pit, the move is ended. An example is given in Figure 2.4, it depicts a game in Mtaji stage. South is unable to capture and therefore has to play a non-capturing move, namely either one of the two occupied front row pits in either direction Stock: 0 Stock: Figure 2.4: Bao position, South cannot capture, therefore he can play, A2L, A2R, A7L, and A7R. 9

16 South chooses to play A7L* which results in the position in Figure 2.5. As one can see, North is also unable to capture in his first sowing so he has to play a Takasa move as well. South has set up exactly one capture for himself; B8 can capture a4 with B8R. North is now obliged to leave all seeds in this Takasia-ed pit. Even when sowing, this pit cannot be emptied. For instance when playing A8L, the move ends in the pit now containing 2 seeds, then Stock: 0 Stock: Figure 2.5: A Bao position, South played A7R and thus Takasia-ed pit a4 (the pit containing 2 in North s front row). There are however exceptions to the Takasia rule. If the Takasia-ed pit is the only occupied pit in the front row, the occupied house or the only pit in the front row containing 2 or more seeds, the pit cannot be Takasia-ed. It is legal to start from this pit, and if a sowing ends in the pit, the move has to continue. It is interesting to notice that the Takasia rule can only be active in the Mtaji stage. Because of the added seed in Namua, the fact that one of the player s pits is vulnerable to capture, this automatically means that the other player s pit is too. It means that the second player cannot Takasa and therefore the prerequisites for a Takasia move are not met. As we described in the previous section, the obligation to stop sowing when arriving at a certain pit complicates a retrograde search. The Takasia rule does however have one advantage over the Nyumba rules; if a sowing ends in a Takasia-ed pit, the move ends. There is no choice and therefore no extra moves have to be added to the move list. 2.4 Checking a Bao Move Generator Section 2.2 and Section 2.3 illustrate how complex the Bao rules are. Humans do not learn the Bao rules easily and they are also not straightforward to implement in a computer program. Playing a game without a correct implementation of its rules is useless. Especially when trying to solve the game, it is of paramount importance to be able to tell whether a computer program implements the Bao rules correctly. It is however not easy to do so. In this section we present an approach for checking Bao move generators for correctness. Formally, the rules of a game define whether a transition from one position to another is allowed. A check whether the rules are implemented correctly, should therefore inspect all moves that the program 10

17 allows, for all positions. This is however an unfeasible approach for Bao (and many other board games), because of their state-space complexity. We therefore use an approach that is based on the rule classification we presented in Section 2.1. Besides that, we use the notion that the rules can be seen as a hierarchy (Section 2.1). We can view the rules of each of the first two categories (legality of the move and the move execution, respectively) as separate hierarchies, while the rules of category 3 are then the bottom of both hierarchies. For instance, for category 1, a part of the hierarchy might be as follows. 1 capture is obligatory, 2 if one cannot capture, one has to play a front row pit containing more than 1 seed., 3 if the house exists, one cannot play the house, although front row singletons can be played. These hierarchies can be translated into decision trees. Every leaf of these decision trees is a special case in which a unique combination of rules applies. Checking whether these rules are followed correctly in each of these leaf nodes requires us to test at least one example from the class of board configurations that is represented by that leaf node. With this test we do assume that the program is as simple as possible. If the program consists of a significantly more complex rule set, the test examples might be examined correctly, although the program might not function correctly when dealing with other positions. Over the course of this research we created a set of test-states that were shared with interested parties. It was updated by Viktor Bautista i Roca from Barcelona, Spain and then we updated it cooperatively. It is currently presented on WikiManqala (Bautista i Roca and Kronenburg, 2006) as a set of test situations that people can use to check whether or not their knowledge of Bao is correct. We used the example in Figure 2.6 to point out flaws in the JBAWO 1.0 program (Chimombo, 2002) in personal communication with the author. In Figure 2.6 North is to play, his house exists. North is allowed to play a7l and a7r. If North plays a7l, his move would end in the house, where the move ends. In JBAWO 1.0 it was allowed to play the house (a7l>) and any subsequent sowings can then capture. That move is not legal to play, and also illegal in execution Stock: Stock: Figure 2.6: A position in which JBAWO 1.0 allows illegal moves. 11

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19 3 Perpetual Moves In Donkers, van den Herik, and Uiterwijk (2003) it was assumed that moves in Bao might exist that will not stop, i.e., that will continue to cycle over the board, never ending in an empty pit. Naturally, these moves are an unwelcome phenomenon in computer play, because they will lead to an infinite loop when played. Earlier, in Donkers and Uiterwijk (2002) it was argued that there is a high probability that a perpetual or never-ending move might occur during a game of Bao. The Bao masters of Africa however, do not think these moves will actually occur during the game (de Voogt, 1995). Unfortunately, so far neither party had proven its position. In this chapter we present a proof that a perpetual move can actually occur in a game of Bao. The text of Sections 3.1 to 3.3 and 3.5 is based on Kronenburg, Donkers, and de Voogt (2006) 1. The last but final section (3.4) is based on new findings during the experiments described in Section Never-Ending Moves in Bao Several scholars (e.g., Wernham, 2001) reported possible configurations for never-ending moves in Omweso a related game with the same board size. In Omweso the starting configuration of the board is composed by the players and therefore, contrary to Bao, every conceivable board configuration in Omweso is a legal position. The question is whether never-ending moves are also possible in Bao. At the end of this section it will be proven that never-ending moves in Bao must always be non-capturing (Takasa) moves. We will start with some observations. Since there are at most 64 seeds on the board, no pit can contain more than 64 seeds. When a sowing ends in a non-empty pit, the pit is either emptied and the sowing continues in the same direction, or an opponent s pit is captured and the move continues at one of the outermost pits of the front row. A Bao move therefore consists of sequences of sowings interspersed by captures. There can be no more than eight captures per move since an opponent s pit can only be captured once in a move. During a sequence of sowings, the direction of sowing may not change. If the sequence is long enough, it will reach the pit from which the sequence started. If the last sowing is not completed, the sowing continues in the same direction. Otherwise, the move ends, since the starting pit was left empty at the beginning of the sowing sequence. The same pit may be visited many times during a sequence of sowings. We call each visit of the same pit a round of sowing. Each time the sowing sequence visits a pit, one seed is added to the pit, or the pit is emptied, or the opponent s opposite pit is captured (ending the sowing sequence). Since any pit cannot contain more than 64 seeds, the pit must be emptied at least every 64 rounds of sowing during the sequence. Below we prove that never-ending moves may not start with a capture. Let us assume that a never-ending move is played that started with a capture. Since at most eight captures are possible in a single move, the move should contain at least one never-ending sowing sequence. During this never-ending sowing sequence, every pit in the front row must be emptied at least every 64 rounds. This can only happen if all the opponent s pits are empty, otherwise a capture takes place and the sequence is ended. However, as soon as the front row of the opponent is empty, the move stops and the game is ended. So, a capture move cannot be never-ending. For a move to be never- 1 I would like to acknowledge my co-authors for their permission to reuse the text in this thesis. Moreover, the Editors of the ICGA Journal are also acknowledged for their willingness to allow reuse of the text. 13

20 ending, it therefore has to be a Takasa move. Takasa moves are never-ending if they result in perpetual sowing. 3.2 Perpetual Sowings Before we go into the existence of perpetual sowings in Bao, we first look at some classes of perpetual sowings in general. Perpetual sowings are not unique to Bao. Similar moves also occur in other mancala games, for instance, in Omweso. Moreover, in the Indian solitaire mancala game Seethi Pandi, keeping up a perpetual sowing is the very purpose of the game. We start by giving a formal description of sowings. We switch, without loss of generality, to a linear board of n pits, containing m seeds in total. Sowing is performed from left to right. When the rightmost pit is reached, sowing continues at the leftmost pit. A position p is represented by: { p[0], p[1],, p[n-1] }, where p[i] 0, i p[i] = m. Let P n,m be the set of all possible configurations of n pits containing m seeds. A sowing s i (p) is an operation defined on P n,m, which consists of playing, pit i in p. Each sowing s i (p) = q has a unique inverse sowing (i.e., picking up one seed per pit until an empty pit is encountered) s j -1 (q) = p, where j = (i + p[i]) mod n. A sowing sequence is a series of positions p 1, p 2, p 3,, p K such that, for all 1 < k < K, p k = s i (p k-1 ), p k+1 = s j (p k ), and j = (i + p k [i]) mod n. A perpetual sowing is an infinite sowing sequence. Since P n,m = (n+m-1 over m) is finite, any perpetual sowing contains a finite number of positions (of which each position may occur infinite times). The pit to be played at a position in a sowing sequence is determined by the previous positions. Henceforth we will include the pit to play (Pl(p)) in the extended description of the position. Without any loss of generality, the pit to play in the first position of a sowing sequence is always pit zero (Pl(p 1 ) = 0). The number of different extended positions is n times P n,m, which is still finite. Since sowings have a unique inverse, every (extended) position is bound to reappear in a perpetual sowing. Since sowing is deterministic, the positions that occur between two subsequent occurrences of a position p are necessarily always the same. We call the number of positions between two subsequent occurrences the period of the perpetual sowing (the period is equal to the number of sowings between two occurrences). Two perpetual sowings are considered to be congruent if (a) they have a position in common, or (b) they have a shifted position in common. A shift of position p (by amount k) is a position q such that p[i] = q[i+k mod n], for all i, and Pl(q) = Pl(p)+k mod n. Two congruent perpetual sowings have the same period. Below we provide three simple classes of perpetual sowings. In Donkers, Uiterwijk, and De Voogt (2002), a first class of perpetual sowings is provided. When n = 2, perpetual sowing with period 2 occurs at positions {2k+1, k}, with period 4 at positions {4k+2, k}, and with period 3 at positions {6k+3, k}. In general, positions of the form {2a.k + a, (2b+1)k + b} with k 1, a 1, b 0, are perpetual sowings, but there might be more perpetual sowings for n = 2. The second simple class was indicated by Bao players who observed that perpetual sowings seem to happen with positions {2, 3, 2, 3, 2,, 3}, for even n. By analysis through a computer program, Donkers et al. (2002) proved that for n from 4 to 26, perpetual sowing indeed occurs, with periods ranging from 60 to 177,651,347,042. Wernham (2001) describes a third simple class that leads to perpetual sowing: the triangular positions. In these positions, every single sowing leads to a shift of the original position. One example is {n+1, n 1, n 2,, 1}. After one sowing (starting at pit 0), the position becomes {1, n+1, n 1, n 2,., 2} and sowing continues at pit 1. The period of this example is n. A second example of this third class is { g+1, [0, 1, 2,, g] x h times }. After one sowing, the position looks like: { 0, 1, 2, 3,, g+1, [0, 1, 2,, g] x h 1 times }, and the sowing continues with pit g+1. After h sowings, the position is { [0, 1, 2,, g] x h times, g+1}, which is the original position, 14

21 shifted by 1 (called a left shift). After g times such a left shift, the original position reappears at the next sowing. The period of these perpetual sowings therefore is gh+1 = n. The above three classes only represent trivial perpetual sowings. Below we describe a fourth, more complex class of perpetual sowings. Steven Mayer (2001) proved that if a position has the following properties (the Mayer Test ), it leads to a perpetual sowing. For all permissible values of i and j: (a) p[i] i + 1, mod n + 1 and p[i] i 1, mod n + 1 (b) p[i + j] p[i] + j + 1, mod n + 1 and p[i + j] p[i] + j 1, mod n + 1 Mayer in fact proves that if a position passes the test, its successor will pass too. Since the test demands that p[0] 1, it implies perpetual sowing. Wernham (2001) showed, using this theorem, that several complex configurations as reported by Omweso players (where n = 16) indeed produce perpetual sowings. Two examples from Wernham (2001) are: Kyagaba s 32: {2,2,1,2,1,2,1,2,1,3,2,3,2,3,2,3} period 61,776. Wernham s 28: {10,0,3,0,1,0,1,0,1,3,4,1,0,1,2,1} period 264. It is not difficult to prove that the classes of perpetual moves described above all pass the Mayer Test. However, Mayer (2001) does not prove the completeness of the Mayer Test. Complex perpetual sowings might exist that completely consists of positions that do not pass the Mayer Test. 3.3 Never-Ending Moves in Actual Bao Not all perpetual sowings for positions with n = 16 can occur in Bao. Since a Takasa move only occurs when no capture is possible after the first sowing, the initial board configuration must contain at least one empty pit in the front row (in the Namua stage), or at least one pit in the front row that is unreachable (in the Mtaji stage). Moreover, since the last pit of the previous move will contain exactly one seed unless this seed is captured, there should always be at least one pit containing at most one seed. So, the {2,3,2,3 } position, for example, cannot occur in Bao. All this implies that we have to investigate the actual Bao game-tree to discover never-ending moves in Bao. To prove that never-ending moves do appear in actual games, we used a Bao-playing program (BAO 1.0; Donkers and Uiterwijk, 2002). This program detects configurations during game-tree search that take more sowings than a predefined limit. Using this option we selected candidate positions and checked them against the Mayer Test. Following this procedure, we found a never-ending move in an actual Bao game. Starting at the official Bao opening position, if players play as follows: 1. A6L* a6r; 5. A4R a4l; 9. A2R a4l; 2. A4R a6r; 6. A8L a7l; 10. A4L a4l; 3. A5L a3r; 7. A1R a4r; 11. A7L a4l, 4. A8L> a8l; 8. A6R a4l; a configuration is reached where move A3L would never end (this move is shown in Figure 3.1). The number of sowings after which the original pattern reoccurs is

22 Stock: Stock: New Findings Figure 3.1: If south plays A3L*, a perpetual move occurs (period 218). When conducting the experiments to be described in Section 5.2 we encountered some other perpetual moves. During the generation of approximately 1,500,000 random games we found 92 distinct perpetual moves. The perpetual moves occurred both in Namua and Mtaji stage. The period of the encountered never-ending moves varied between 54 sowings (after 45 ply) and 141,081,648 sowings (after 89 ply). The earliest encountered perpetual move occured after playing: 1. A7L* a5l 5. A3R a3l 9. A8R a3l 2. A6R* a1r 6. A5R a3r 10. A5R* 3. A5R a2r 7. A1R a3r 4. A8R> a3l 8. A4R* a3l This leads to a perpetual move with a period of 768 sowings. We did not encounter situations in which a player was obliged to play a never-ending move. Research into these specific situations and the moves that lead to the positions where perpetual moves occur would probably yield several interesting mathematical properties of perpetual moves. This research is however outside of the scope of this thesis and we leave it to others to investigate these properties. 3.5 Implications to the Rules of Computer Bao The Mayer Test can straightforwardly be implemented in a Bao-playing program. However, since the Mayer Test only proves the existence and not the absence of a perpetual sowing, the computer program still may encounter positions that have endless moves that are undetected by the Mayer Test (although we did not encounter such positions yet). It therefore seems reasonable to distinguish between proven never-ending moves and suspicious never-ending moves. The latter have to be identified by crossing a certain limit of sowings, as described in Donkers and Uiterwijk (2002). It is not feasible to check for cycles in general. Since we have proven the existence of a true never-ending move in Bao, there are two problems: (1) what to do with human Bao and (2) what to do with computer Bao? There are no rules for neverending moves in human Bao. For the current practice, it is only a theoretical problem. The opinion of 16

23 Bao players on never-ending moves tells us that they are rare in Bao. If encountered, human players will make mistakes at a certain point, thus breaking up the never-ending cycle. In Omweso, however, never-ending moves occur more frequently and the Kampala tournament rules say: the umpire shall give 3 minutes, and when the 3 minutes elapse, he shall order the game to be repeated (Wernham, 2001). Since computers will not make mistakes, a new rule has to be invented for computer players only. We propose to alter the two rules on never-ending moves as presented in Donkers and Uiterwijk (2002) so that they include proven never-ending moves. Rule 1.5a states that any suspicious never-ending move is illegal. It is better to formulate: Any proven or suspicious never-ending move is illegal. The second rule concerning never-ending moves (1.5b) deals with the case that a position only has moves that are proven or suspicious never-ending. There are in principle two possibilities: either the position is a loss or the position is a draw. Since the current rules for Bao do not include a draw, we propose to declare the game lost for the player to move. 17

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25 4 Towards Solving Bao In this chapter we will discuss some of the techniques used for solving games that might be useful to solve Bao. We start with describing endgame databases and retrograde analysis in Section 4.1. These are often used when solving games. Then, in Section 4.2 we describe solved games from the family of mancala games and in Section 4.3 we present a very short overview of other solved games. In Section 4.4 we present our own approach towards solving Bao. To use this approach we need to establish a clear notion of what is meant by generating random games of Bao. We do this in Section 4.5. In Section 4.6 we give a brief summary of this Chapter. 4.1 Retrograde Analysis and Endgame Databases In Section 2.2 we briefly sketched a notion of retrograde analysis. In this section we expand on this notion and show how the concepts of retrograde analysis and endgame databases are connected. Retrograde analysis was first presented in a seminal paper by Ströhlein (1970). The technique was first used by Allis, van der Meulen, and van den Herik (1991) in their Awari player LITHIDION. This program easily won the Awari championship at the second Computer Olympiad (Levy and Beal, 1991); thus demonstrating the power of the technique. Retrograde analysis is a technique to create lists or databases of positions stored with extra information about the position that is not readily apparent from the position itself. Usually this information is the game-theoretic value of the position, the number of moves left before an endposition is reached (usually called Distance to Mate or DTM) or the number of moves left until a position reaches a significantly new part of the game (e.g., a capture of a counter or piece, usually called Distance to Conversion or DTC). Several classes of these so-called endgame databases exist and each combines different information. They are used mainly in converging games, i.e., games where the number of possible positions decreases as the game progresses. Usually convergence means that the number of pieces in play decrease over time. We will demonstrate the use of retrograde analysis to build an endgame database that contains the game-theoretic value of the position below. Other databases can be built in an analogue fashion. Retrograde analysis starts with a list of possible positions in the game. From this list we select the positions for which it is clear that they are a win for either player. For instance in Chess these would be the mated positions, in Bao the positions with no seeds in the front row or where one of the players is unable to move. We store these positions together with their game-theoretic value. For Bao these are +1 and 1; respectively a win for player 1 and a win for player 2. This set of positions forms the basis of the endgame database. All other known positions are declared draws, until proven differently. We now iterate over the positions stored in the database and determine the value of each of their predecessors, these are the positions from which the stored position can be reached. The process uses the minimax theorem (Neumann, 1928, Wald, 1945). Hence, in a position where player 1 is to play, from which a move exists that lead to a position where player 2 has lost, player 1 has won and vice versa. When the retrograde analysis is continued sufficiently long, one will eventually determine the values (either, +1, -1, or 0) for all positions, including the starting position, and thereby solve the game ultra weakly. However, this is a time-consuming process and therefore the retrograde analysis is usually stopped at an earlier point. The resulting database can then be used by forward search techniques as an endgame database. 19