Combining combinatorial game theory with an a-ß solver for CRAM

Combinatorial games are a special category of games sharing the property that the winner is by definition the last player able to move. To solve such games two main methods are being applied. The first is a general a-ß search with many possible enhancements. This technique is applicable to every game, mainly limited to the size of the game due to the exponential explosion of the solution tree. The second way is to use techniques from Combinatorial Game Theory (CGT) to calculate exact values of combinatorial games. This method is only applicable to relatively small (sub)games. In previous research we have shown that the methods can be combined in a fruitful way by using endgame databases filled with CGT values. For two partisan combinatorial games (where both players have different possible moves), namely Domineering [2] and the all-small partisan game Clobber [29], we have shown that incorporating CGT endgame databases in a-ß solvers gives large search reductions, both in number of nodes investigated and time needed. In this paper we investigate if it is possible to use CGT to improve an a-ß solver for an impartial game (where the players have the same possible moves). Impartial games are notoriously difficult to solve, since it is hard to use perfect knowledge rules that can prune the search tree. We have implemented an a-ß solver for the impartial game of Cram. This program exploits endgame databases with CGT values to solve subgames of various sizes. We also implemented efficient ways to identify the relevant subgames that can be solved by using these databases. Our test suite consists of 37 boards, namely the 2 × n boards with 2 = n = 21, the 3 × n boards with 3 = n = 12, the 4 × n boards with 4 = n = 8, and the 5×n boards with 5 = n = 6. Using the CGT endgame databases, complex games are solved with reductions of around 50% of the number of nodes needed without databases, going up to 80-90% for the smaller non-trivial boards, and 100% for the trivial ones. Moreover, many Cram boards have been solved that were effectively unsolvable without the databases. We also invented a move-ordering rule based on early splitting of the board into subgames, giving a further modest gain. All in all the results show that incorporating knowledge from CGT into an a-ß solver for Cram gives a large improvement. By this contribution we have obtained more evidence that using CGT for solving games is beneficial for the whole spectrum of combinatorial games, from partisan games (Domineering) to all-small partisan games that in many ways behave impartial-like (Clobber) to impartial games (Cram).