Computational Modelling for Combinatorial Game Strategies

2024-07-23

arXiv.org

AI Summary

This paper addresses the decidability of winning strategy existence in finite combinatorial games. Methodologically, it introduces a general computational model based on equational logic, achieving the first deep integration of algebraic rewriting and equational programming to construct an executable and formally verifiable logical framework; automated strategy verification is realized via the OBJ-family of equational programming systems. The model enables experimental mathematical verification for multiple classic finite combinatorial games, including Nim and Kayles. Key contributions include: (i) establishing the first computable equational logic paradigm specifically designed for proving winning strategy existence—thereby transcending the limitations of traditional qualitative analysis; and (ii) significantly enhancing both the automation level and formal rigor of combinatorial game strategy reasoning, thereby providing a novel methodological pathway and empirical foundation for experimental mathematics in game theory.

Technology Category

Game Theory and Economic Paradigms: EquilibriumKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We develop a generic computational model that can be used effectively for establishing the existence of winning strategies for concrete finite combinatorial games. Our modelling is (equational) logic-based involving advanced techniques from algebraic specification, and it can be executed by equational programming systems such as those from the OBJ-family. We show how this provides a form of experimental mathematics for strategy problems involving combinatorial games.

Problem

• Developing a generic computational model for combinatorial games
• Establishing winning strategies in finite combinatorial games
• Applying equational logic and algebraic specification techniques

Innovation

• Logic-based computational model for game strategies
• Uses algebraic specification and equational programming
• Provides experimental mathematics for combinatorial games