Compressed Game Solving
Jeffrey Considine
TL;DR
This work addresses the challenge of solving complex board games by replacing traditional position-by-position retrograde analysis with compressed set representations of positions. It introduces a set-oriented framework for both retrograde and meet-in-the-middle analyses and demonstrates a practical instantiation using deterministic finite automata (DFAs) to store and manipulate position sets. The approach achieves sublinear empirical space usage $O(n^c)$ (with $c$ around $0.5$–$0.7$) and constant-time membership tests, enabling strong-solving performance on Breakthrough and confirming efficiency on Nim. Breakthrough results include solving larger board sizes on a commodity laptop and achieving impressive compression ratios, suggesting the method’s potential for broader games, while noting limited leverage for Amazons and Chess and pointing to future work in integrating knowledge and symmetries. Overall, the paper contributes a novel, scalable representation for move generation and set-based game solving with practical implications for sublinear-time exploration of large state spaces.
Abstract
We recast move generators for solving board games as operations on compressed sets of strings. We aim for compressed representations with space sublinear in the number of game positions for interesting sets of positions, move generation in time roughly linear in the compressed size and membership tests in constant time. To the extent that we achieve these tradeoffs empirically, we can strongly solve board games in time sublinear in the state space. We demonstrate this concept with the game Breakthrough where we empirically realize compressed representations taking roughly $n^{0.5}$ to $n^{0.7}$ space to store relevant sets of $n$ positions.