有限自动机方法在组合博弈中的应用
A Finite Automaton Approach to Combinatorial Games
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中文总结 AI 辅助
本研究将有限自动机用于组合博弈自动求解，提出两阶段算法、广义 misère 商，解决 Kotzig's nim 等博弈的关键猜想，为组合博弈论提供新工具与范式。
中文摘要 AI 辅助
本研究将有限自动机应用于多种组合博弈的自动求解。对于局面和移动可表示为正则语言及其运算的博弈，我们设计了两阶段自动求解算法：首先构建候选有限自动机以确定$\boldsymbol{\textit{P}}$位置（必败位置）和$\boldsymbol{\textit{N}}$位置（必胜位置），随后对该自动机进行严格形式化验证；验证通过后即可得到该博弈的完整解。对于 partisan 八进制博弈（ partisan games，即 partisan 博弈，指双方可走移动规则不同的博弈），我们引入了广义 misère 商（misère play，即 misère 博弈，指最后一步走棋者输的博弈），克服了传统理论仅适用于 impartial 博弈（双方规则相同的博弈）的局限。利用上述算法，我们成功求解了绝大多数两位 partisan 八进制博弈，并基于这些结果提出了 Guy 猜想的 partisan 版本。我们还成功求解了相当数量处于 misère 玩法下的 partisan 八进制博弈，并对其中呈现“代数周期性”的博弈结构提出了猜想。对于 Kotzig's nim（Kotzig 尼姆博弈），我们解决了最重要的相关猜想：证明其在正常玩法和 misère 玩法（包括对应的 partisan 版本）下的结果和 SG 值（Sprague-Grundy 值，组合博弈论中的核心指标）具有周期性。我们的算法成功求解了多个小规模案例，包括 misère 玩法和 partisan 版本。本研究为组合博弈的自动求解开创了新的理论工具和算法范式，在组合博弈论领域具有广泛的拓展和应用前景。
英文摘要
This study applies finite automata to the automatic solving of a variety of combinatorial games. For games whose positions and moves can be represented as regular languages and their operations, we design a two-stage automatic solving algorithm: first, construct a candidate finite automaton to determine the $\mathcal{P}$- and $\mathcal{N}$-positions, and then perform rigorous formal verification on this automaton; once verified, a complete solution of the game is obtained. For partizan octal games, we introduce a generalized misère quotient, overcoming the limitation that traditional theory applies only to impartial games. Using the above algorithm, we successfully solve the majority of two-digit partizan octal games, and based on these results, we propose a partizan version of Guy's conjecture. We also successfully solve a considerable number of partizan octal games under misère play, and give a conjecture on the structure of those games exhibiting ``algebraic periodicity'' among them. For Kotzig's nim, we resolve the most important related conjecture: we prove that the outcomes and SG values are periodic under both normal and misère play (including their partizan versions). Our algorithm successfully solves several small-scale cases, including misère play and partizan versions. This study pioneers a new theoretical tool and algorithmic paradigm for the automatic solving of combinatorial games, and has broad prospects for further extension and application in the field of combinatorial game theory.
