Solving First-Order Fixed-Point Logics via a Least-to-Greatest Transformation Based on Game Semantics

2026-07-12

AI Summary

This study addresses the validity problem for first-order fixed-point logic by proposing a novel μ-to-ν transformation method grounded in game semantics. The approach leverages parity conditions to convert least fixed points into greatest fixed points, thereby reducing the verification task to the search for winning strategies in parity games. This work establishes, for the first time, a unified game-semantic framework for μ-to-ν transformations, demonstrating the equivalence between Tsukada et al.’s model and standard winning conditions. A solver implementing this transformation is developed, and experimental results show that the proposed optimizations significantly enhance solving efficiency, outperforming existing tools.

Abstract

Fixed-point logics provide an expressive intermediate framework for reasoning about temporal properties of programs. One of the key approaches to solving their validity checking problem is via transformations from least fixed points to greatest fixed points ($μ$-to-$ν$ transformations), which generalizes a reduction from termination verification to safety verification studied in binary reachability analysis. In this paper, we introduce game-semantic interpretations of $μ$-to-$ν$ transformations. We first introduce a new $μ$-to-$ν$ transformation based on parity relations. We show that solving $μ$-to-$ν$-transformed fixed-point equation systems corresponds to finding winning strategies in the game semantics of the original fixed-point equation systems. We apply the same game-semantic framework to interpret two existing $μ$-to-$ν$ transformations, one by Kobayashi et al.\ and the other by Unno et al, and show that they admit analogous game-semantic interpretations. Furthermore, we show that the game introduced by Tsukada et al.\ corresponds to an alternative characterization of the winning condition. On the implementation side, we propose optimization techniques for efficiently solving our new $μ$-to-$ν$ transformation. We implement these techniques in a fixed-point logic solver, compare our approach with existing solvers, and demonstrate the effectiveness of the proposed optimizations through experiments.

Problem

• fixed-point logics
• validity checking
• mu-to-nu transformation
• temporal properties
• program verification

Innovation

• game semantics
• mu-to-nu transformation
• fixed-point logic
• parity relations
• winning strategies