# THE QUAD HEURISTIC IN LINES OF ACTION. Mark H.M. Winands, Jos W.H.M. Uiterwijk and H. Jaap van den Herik 1. Maastricht, The Netherlands ABSTRACT

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2 4 ICGA Journal March 2001 how the quad heuristic is used there; it turns out that the normal evaluator and the quad evaluator perform equally well. Section 7 contains our conclusions and some suggestions for future research. 2. RULES OF LOA LOA is played on an 8 8 board by two sides, Black and White. Each side has twelve pieces at its disposal. Sackson (1969) describes the original LOA rules of the inventor Claude Soucie in the first edition of A Gamut of Games. Below they are formulated in eight points. 1. The black pieces are placed in two rows along the top and bottom of the board, while the white pieces are placed in two files at the left and right side of the board. The initial position of the game is shown in Diagram The players alternately move, starting with Black. 3. A player to move must move one of its pieces. A move takes place in a straight line, exactly as many squares as there are pieces of either colour anywhere along the line of movement. (These are the Lines of Action.) 4. A player may jump over its own pieces. 5. A player may not jump over the opponent s pieces, but can capture them by landing on them. 6. The goal of a player is to turn all own pieces on the board into one connected unit. The first player to do so is the winner. The connections within the group may be either orthogonal or diagonal. For example, in Diagram 2 Black has won because the black pieces form one connected unit. 7. If one player s pieces are reduced by captures to a single piece, the game is a win for this player. 8. If a move simultaneously creates a single connected unit for both players, the game is drawn 2. Diagram 1: The initial position. Diagram 2: A terminal position. Diagram 3: Movement of pieces. Since some situations are not covered by the original rules, ad hoc rules are introduced. For instance: (1) repetition of positions is considered as a draw, and (2) if a player cannot move, this player loses. 3 In the article we use the standard chess notation. The possible moves of the black piece on d3 in Diagram 3 are indicated by arrows. The piece cannot move to f1 because its path is blocked by an opposing piece. The move to h7 is not allowed because the square is occupied by a black piece. 3. COMPLEXITY OF LOA Below we distinguish two types of complexity, viz. the state-space complexity and the game-tree complexity. Of both complexities we give an estimation of their size. Moreover, we argue that LOA is not crackable with the current state-of-the-art computer techniques. For this goal, new techniques should be developed. 2 In the second edition of A Gamut of Games (1982) simultaneous connection is described as a win for the moving player, but the draw variant is still in force in the main tournaments, such as the Mind Sports Olympiad 2000, and on Richard s PBeM server ( championship). 3 The use of this rule is disputable, some LOA players allow the opponent to pass in such positions.

3 The Quad Heuristic in Lines of Action State-Space Complexity An upper bound for the state-space complexity of LOA can easily be derived (cf. Schaeffer and Lake, 1996). Let B be the number of black pieces and W the number of white pieces. There are at most twelve black and twelve white pieces. Each of the 64 squares on the board can be occupied by a piece. The number of positions with 24 pieces or fewer is derived by the following formula Num( B, W ) Num(1,1) B= 1 W= 1 where 64 Num( B, W ) = B 64 W B In the formula we have corrected the counting formula for all positions with one piece at each side, because they are impossible to reach. One of the players would have won earlier. For the same reason we do not count all positions with pieces of one colour only. In Table 1, we present the number of positions distinguished by the number of pieces on the board. For instance, the number of positions of three pieces equals Num(2,1) + Num(1,2), resulting in 249,984. Totalling the numbers for 3 pieces up to 24 pieces results in different positions. Of course, we then have included some positions, which are not reachable from the start (Dyer, 2000), but we believe that this fact only lowers the actual state-space complexity marginally. If we take the board symmetries into account, the statespace complexity is estimated to be O(10 23 ). 3.2 Game-Tree Complexity # of pieces # of positions 3 249, ,895, ,735, ,648,410, ,273,240, ,124,246,003, ,045,698,101, ,805,625,541, ,521,396,969,611, ,445,574,994,311, ,590,873,672,114, ,632,117,126,476, ,188,514,989,534,830, ,335,094,798,158,420, ,930,216,128,039,066, ,283,294,857,675,368, ,807,935,897,946,939,333, ,158,514,055,288,777,729, ,164,643,498,259,191,458, ,592,373,542,262,268,414, ,757,628,751,471,486,670, ,794,282,450,430,456,394,720 Total: 1,308,338,020,676,454,019,380,152 Table 1: State-space complexity. The game-tree complexity can be approximated by the size of a uniform game tree in which the depth is the average game length and the branching factor the average number of legal moves (Allis, 1994). Since we do not have a large database of LOA games we have approximated the numbers (i.e., game length and branching factor) by experiments in which our program played against itself. The results were: an average game length of 38 ply and an average branching factor of 30 (Winands, 2000). The game-tree complexity thus is about O(10 56 ).

7 The Quad Heuristic in Lines of Action 9 Diagram 5: A position with a threat. Diagram 6: A solid formation. Blocking Because a piece is not allowed to jump over the opponent s pieces, it can happen that the piece is blocked, i.e., cannot move. Blocking a piece far away from the other pieces is an effective way of preventing the opponent to win. For instance, in Diagram 7 White must first spend many moves to free the blocked piece, i.e., capturing pieces around the blocked piece on a8, before it may attempt to reach the connection goal. During that time, Black may set up a winning strategy. Even partial blocking can be quite effective, especially if it forces a player to find a way around the opponent s pieces. Handscomb (2000a) describes that it is a common tactic to create a wall on the b- or g-file, or the 2 nd or 7 th rank in the opening phase of the game, which partially blocks pieces of the opposing side. In MIA, blocking is a separate component. Diagram 7: Blocking a piece. Diagram 8: LOA tournament, 1999, Roessner vs. Handscomb, after 10. e1-c3. Centralisation Pieces in the centre or controlling the centre are assumed to be more important than pieces without any centre relation. However, Handscomb (2000b) argues that the benefits of actual centralisation in LOA are sometimes exaggerated. Because of the nature of the game, pieces huddled together in the middle of the board are incapable of capturing each other. Even with only two pieces in a given line of action a distance of two squares is needed for a capture, and if there are three pieces a considerable distance between the pieces is required. In Diagram 8 the white pieces are scattered around the edges of the board. Nevertheless, the white piece on h4 can destroy Black s formation in the middle by capturing the black piece on d4. So, the white pieces on the edges of the board are acting as cruise missiles. Of course, this is not possible when all the pieces are in the centre. This example shows that controlling the centre might be better than occupying it. Hence, actual centralisation has a limited importance, but successfully controlling the centre is an important component in the evaluation function. Through an effective control of the centre, the two starting groups of the opponent remain separated, whereas the own groups can be brought together. The notion of centralisation seems to be more important in LOA than in chess. As a direct consequence we have found that the dynamic principle of favouring pieces to move towards their centre of mass is more profitable than the static principle of favouring pieces to move towards the centre of the board.

10 12 ICGA Journal March 2001 on the left and right became physically separated by a wall of white pieces. The wall demonstrated the drawback of ignoring the position of the opponent s pieces (Billings, 2001). Diagram 11: The fifth Computer Olympiad, 2000 MIA vs. MONA, fifth game, after 6. g8-g4. Diagram 12: The fifth Computer Olympiad, 2000 MIA vs. MONA, fifth game, after 14. e5-f4. 6. QUIESCENCE SEARCH When the search reaches the depth limit, a static evaluation function should be applied in the leaf node reached. This approach can have disastrous consequences because of the approximate nature of the evaluation function (Kaindl, 1982). Obviously, a more sophisticated depth limit is needed. The evaluation function should only be applied to positions that are quiescent (cf. Shannon, 1950). 6.1 Quiescence Search in LOA Tactical capture moves are mostly responsible for swings in the evaluation of a LOA position. However, it is hard to define what a tactical capture move is, because this heavily depends on the board position. In general capture moves, which destroy connections or form connections, are considered tactical. Based on this definition of tactical, we use a quiescence search that involves only this kind of capture moves. For instance, if Black plays d7 d4 in Diagram 13, Black will both connect its pieces in the centre and destroy White s connection. Using the quad heuristic, capture moves destroying or forming connections can be detected easily. In Section 4 we have seen that the use of the Euler number to detect the number of isolated groups is not always correct. Therefore, we have to adjust the result in some particular cases (E 1). Yet, in general using the quad heuristic leads to a fast and efficient quiescence search. We have implemented in our program an extended version of quiescence search (cf. Schrüfer, 1989). This kind of quiescence search limits the set of moves considered and uses the evaluations of interior nodes as lower / upper bounds of the resulting search value. Diagram 13: A tactical capture move. Diagram 14: The fifth Computer Olympiad, 2000, MIA vs. MONA, seventh game, after 12. c7 e5.

12 14 ICGA Journal March 2001 this selection. It has to be investigated whether quiescence search in MIA will still be effective because it might be compensating for inadequacies in other components of the program and its effectiveness might be less with an enhanced evaluation function. ACKNOWLEDGEMENTS The authors thank the anonymous referees for their constructive comments and suggestions for improvements. We also would like to thank Dave Dyer for his additional information on the quad heuristic. Moreover, we thank Kerry Handscomb for making articles of Abstract Games available. Next, we gratefully acknowledge the possibility to use additional LOA knowledge, made available to us by the former LOA World Champion Fred Kok. Finally, we recognise the support of members of the Alberta GAMES Group, Darse Billings, Yngvi Björnsson and Jonathan Schaeffer. 8. REFERENCES Akl, S.G. and Newborn, M.M. (1977). The Principal Continuation and the Killer Heuristic ACM Annual Conference Proceedings, pp ACM, Seattle. Allis, L.V., Herik, H.J. van den and Herschberg, I.S. (1991). Which Games Will Survive? Heuristic Programming in Artificial Intelligence 2: the second computer olympiad (eds. D.N.L. Levy and D.F. Beal), pp Ellis Horwood, Chichester. ISBN Allis, L.V. (1994). Searching for Solutions in Games and Artificial Intelligence. Ph.D. thesis Rijksuniversiteit Limburg, Maastricht, The Netherlands. ISBN Billings, D. and Björnsson, Y. (2000). Mona and YL s Lines of Action Page. Billings, D. (2001). Personal communication. Björnsson, Y. (2000). YL wins Lines of Action Tournament. ICGA Journal, Vol. 23, No. 3, pp Blixen, C. von (2000). Lines-of-Action. Breuker, D.M., Uiterwijk, J.W.H.M. and Herik, H.J. van den (1996). Replacement Schemes and Two-Level Tables. ICCA Journal, Vol. 19, No. 3, pp Dyer, D. (2000). Lines of Action Homepage. Dyer, D. (2001). Personal communication. Gray, S.B. (1971). Local Properties of Binary Images in Two Dimensions. IEEE Transactions on Computers, Vol. C-20, No. 5, pp Handscomb, K. (2000a). Lines of Action Strategic Ideas Part 1. Abstract Games, Vol. 1, No. 1, pp Handscomb, K. (2000b). Lines of Action Strategic Ideas Part 2. Abstract Games, Vol. 1, No. 2, pp Handscomb, K. (2000c). Lines of Action Strategic Ideas Part 3. Abstract Games, Vol. 1, No. 3, pp Kaindl, H. (1982). Quiescence Search in Computer Chess. SIGART Newsletter, 80, pp Kocsis, L., Uiterwijk, J.W.H.M., and Herik, H.J. van den (2000). Learning Time Allocation using Neural Networks. Working Notes of the Second International Conference on Computers and Games CG'2000, pp

13 The Quad Heuristic in Lines of Action 15 Minsky, M. L. and Papert, S.A. (1988). Perceptrons, An Introduction to Computational Geometry. Expanded edition. MIT Press, Cambridge, MA, USA. ISBN (First edition, Massachusetts Institute of Technology, 1969). Sackson, S. (1969). A Gamut of Games. Random House, New York, NY, USA. A second edition (1982) has been republished in 1992 by Dover Publications, New York, NY, USA. ISBN Schaeffer, J. (1983). The History Heuristic. ICCA Journal, Vol. 6, No. 3, pp Schaeffer, J. (1984). The Relative Importance of Knowledge. ICCA Journal, Vol. 7, No. 3, pp Schaeffer, J. and Lake, R. (1996). Solving the Game of Checkers. Games of No Chance (ed. J. Nowakowski). MSRI Publications, Vol. 29, pp Cambridge University Press, Cambridge, UK. ISBN Schrüfer, G. (1989). A Strategic Quiescence Search. ICCA Journal, Vol. 12, No. 1, pp Shannon, C.E. (1950). Programming a Computer for Playing Chess. Philosophical Magazine, Vol. 41, pp Winands, M.H.M. (2000). Analysis and Implementation of Lines of Action. M.Sc. thesis. Universiteit Maastricht, Maastricht, The Netherlands. 9. APPENDIX: EXPERIMENTAL SET-UP All experiments have been performed in the framework of MIA. MIA has been written in Java and runs on every well-known operating system. It can be played at the website: MIA performs an alpha-beta depth-first iterative deepening search. The program uses a entry, two-deep transposition table (Breuker, Uiterwijk, and van den Herik, 1996), the history heuristic (Schaeffer, 1983) and killer moves (Akl and Newborn, 1977). For all experiments described in this article, the null move is not used because of possible negative effects of zugzwang in LOA (Winands, 2000). Below the positions are given, which are used in the experiment of Section 4. These positions are taken from matches MIA played at the fifth Computer Olympiad. TP 1: Black to move TP 2: White to move TP 3: Black to move TP 4: Black to move TP 5: White to move TP 6 Black to move TP 7: White to move TP 8: Black to move TP 9: Black to move TP 10: Black to move TP 11: White to move TP 12: White to move TP 13: Black to move TP: 14 Black to move TP 15: Black to move

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