Concise QBF Encodings for Games on a Grid (extended version)
This work addresses the problem of efficiently and correctly encoding games for QBF solving, which is incremental as it builds on ideas from planning domains like PDDL.
The authors tackled the challenge of encoding 2-player grid games into Quantified Boolean Formulas (QBF) by introducing the Board-game Domain Definition Language (BDDL), which enables concise specifications and uniform encodings, and they demonstrated its feasibility by computing critical depths of winning strategies for various games using QBF solvers.
Encoding 2-player games in QBF correctly and efficiently is challenging and error-prone. To enable concise specifications and uniform encodings of games played on grid boards, like Tic-Tac-Toe, Connect-4, Domineering, Pursuer-Evader and Breakthrough, we introduce Board-game Domain Definition Language (BDDL), inspired by the success of PDDL in the planning domain. We provide an efficient translation from BDDL into QBF, encoding the existence of a winning strategy of bounded depth. Our lifted encoding treats board positions symbolically and allows concise definitions of conditions, effects and winning configurations, relative to symbolic board positions. The size of the encoding grows linearly in the input model and the considered depth. To show the feasibility of such a generic approach, we use QBF solvers to compute the critical depths of winning strategies for instances of several known games. For several games, our work provides the first QBF encoding. Unlike plan validation in SAT-based planning, validating QBF-based winning strategies is difficult. We show how to validate winning strategies using QBF certificates and interactive game play.