Witnesses for Fixpoint Games on Lattices

2026-03-12

AI Summary

This work addresses the construction of verifiable witnesses for fixed-point games over lattices to determine whether the least fixed point of a function satisfies a given lower bound. By employing a Galois connection to unify the “logical universe” and the “behavioral universe”—two distinct lattice structures—it establishes, for the first time, a bidirectional correspondence between strategies and witnesses in both primal and dual fixed-point games. The proposed framework extends the applicability of fixed-point verification to novel domains such as certifying lower bounds on termination probabilities in probabilistic systems. It has been successfully applied to verify distinguishing formulas for bisimulation, behavioral metrics, and lower bounds on termination probabilities in Markov chains, thereby demonstrating its generality and effectiveness.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We construct witnesses that can be used to derive strategies in fixpoint games and provide proof that the least fixpoint of a function is either above or not below some given bound. We rely on a lattice-theoretical approach, including a Galois connection that connects a lattice representing the "logic universe", where the witness lives, with another lattice representing the "behaviour universe", over which the function is defined. In fact we consider two types of games -- primal and dual games -- and in both cases show how to derive winning strategies in the game from witnesses and construct witnesses from strategies. The two games differ wrt. their rules and the choice of basis of the lattice. The theory can be instantiated to well-known examples: in particular we compare with the construction of distinguishing formulas in standard bisimilarity and behavioural metrics for probabilistic systems. As a new case study we consider witnesses for certifying lower bounds for the termination probability for Markov chains.

Problem

• fixpoint games
• lattices
• witnesses
• Galois connection
• termination probability

Innovation

• fixpoint games
• lattice theory
• Galois connection
• witnesses
• Markov chains