# Concise QBF Encodings for Games on a Grid (extended version)

arXiv: https://arxiv.org/abs/2303.16949 · HTML: https://mlsanity.com/paper/2303.16949/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 28 references that are in the map.
- First posted on arXiv 29 Mar 2023.

## Builds on (most cited first, 4 of 4)

- [DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL](https://mlsanity.com/paper/1702.08256/) (2017)
- [Classical Planning as QBF without Grounding (extended version)](https://mlsanity.com/paper/2106.10138/) (2021)
- [Solving QBF by Abstraction](https://mlsanity.com/paper/1604.06752/) (2016)
- [Implicit State and Goals in QBF Encodings for Positional Games (extended version)](https://mlsanity.com/paper/2301.07345/) (2023)

## Abstract (arXiv)

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.