The Algebra of Parity Games

2025-01-30

AI Summary

This paper addresses the algebraic modeling and automated solving of parity games by introducing the first sound and complete equational axiomatization. Methodologically, it models parity games as string diagrams, endows them with a symmetric monoidal category structure, and establishes a bidirectional, semantics-preserving translation—via functorial semantics—between this diagrammatic representation and the symbolic calculus with least fixed points (μ-calculus style). The main contributions are: (1) the first purely algebraic characterization of parity games, enabling direct computation of winning regions via equational reasoning alone; (2) a fully combinatorial, formally verifiable winning-region algorithm that preserves the underlying graph structure; and (3) a semantic bridge between graphical representations and symbolic computation, providing a novel algebraic foundation for game logics, model checking, and higher-order program verification.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint 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 graphs
• Semantics 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In recent work, Watanabe, Eberhart, Asada, and Hasuo have shown that parity games can be seen as string diagrams, that is, as the morphisms of a symmetric monoidal category, an algebraic structure with two different operations of composition. Furthermore, they have shown that the winning regions associated to a given game can be computed functorially, i.e. compositionally. Building on their results, this paper focuses on the equational properties of parity games, giving them a sound and complete axiomatisation. The payoff is that any parity game can be solved using equational reasoning directly at the level of the string diagram that represents it. Finally, we translate the diagrammatic language of parity games to an equally expressive symbolic calculus with fixpoints, and equip it with its own equational theory.

Problem

• Mathematical Properties
• Parity Games
• Graphical Language to Symbolic Computation

Innovation

• Algebraic Framework
• String Diagrams
• Calculational Format with Cycles