The Constructive $μ$-calculus: Game Semantics and Non-Wellfounded Proof Systems

2026-04-25

AI Summary

This work proposes a novel semantic and proof-theoretic framework for the μ-calculus over the constructive modal logic CK. We first introduce a game semantics and establish its equivalence with the standard two-sorted Kripke semantics. Building on this game-theoretic foundation, we develop the first non-wellfounded labeled proof system tailored to the constructive μ-calculus. By integrating game semantics with non-wellfounded proof methods—a combination previously unexplored in this setting—our approach yields a sound and complete deductive system for the logic. This contribution not only provides a robust reasoning framework for constructive μ-calculi but also opens new avenues for proof-theoretic investigations of other non-classical modal logics.

Technology Category

Knowledge Representation and Reasoning: Logic ProgrammingReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

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 semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Data transparency and provenance

Abstract

We study a variant of the modal $μ$-calculus based on the constructive modal logic $\mathsf{CK}$. We define game semantics for the constructive $μ$-calculus and prove its equivalence to the birelational Kripke semantics. We then use the game semantics to prove the soundness and completeness of a fully-labeled non-wellfounded proof system for it. At last, we briefly describe how to adapt the game semantics and proof system to the $μ$-calculus over other non-classical modal logics.

Problem

• constructive μ-calculus
• game semantics
• non-wellfounded proof systems
• modal logic
• Kripke semantics

Innovation

• constructive modal logic
• game semantics
• non-wellfounded proof system
• μ-calculus
• birelational Kripke semantics