Automatic Synthesis of Generalized Winning Strategies of Impartial Combinatorial Games Using SMT Solvers

Strategy representation and reasoning has recently received much attention in artificial intelligence. Impartial combinatorial games (ICGs) are a type of elementary and fundamental games in game theory. One of the challenging problems of ICGs is to construct winning strategies, particularly, generalized winning strategies for possibly infinitely many instances of ICGs. In this paper, we investigate synthesizing generalized winning strategies for ICGs. To this end, we first propose a logical framework to formalize ICGs based on the linear integer arithmetic fragment of numeric part of PDDL. We then propose an approach to generating the winning formula that exactly captures the states in which the player can force to win. Furthermore, we compute winning strategies for ICGs based on the winning formula. Experimental results on several games demonstrate the effectiveness of our approach.

Queens in Exile: Non-attacking Queens on Infinite Chess Boards

Number the cells of a (possibly infinite) chessboard in some way with the numbers $0, 1, 2, \ldots$. Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size $\mathbb{Z} \times \mathbb{Z}$ numbered along a square spiral, and an infinite single-quadrant chessboard (of size $\mathbb{N} \times \mathbb{N}$) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.