QBF Encoding of Generalized Tic-Tac-Toe

Harary’s generalized Tic-Tac-Toe is an achievement game for polyominoes, where two players alternately put a stone on a grid board, and the player who first achieves a given polyomino wins the game. It is known whether the first player has a winning strategy in the generalized Tic-Tac-Toe for almost all polyominoes except the one called Snaky. GTTT(p, q) is an extension of the generalized Tic-Tac-Toe, where the first player places q stones in the first move and then the players place q stones in each turn. In this paper, in order to attack GTTT(p, q) by QBF solvers, we propose a QBF encoding for GTTT(p, q). Our encoding is based on Gent and Rowley’s encoding for Connect-4. We modify three parts of the encoding: initial condition, move rule and winning condition of the game. The experimental results show that some QBF solvers can be used to solve GTTT(p, q) on 4× 4 or smaller boards.