Almost Fair Simulations

2026-05-26

AI Summary

Standard simulation techniques struggle to verify liveness properties, and existing notions of fair simulation are often too complex for interactive verification. This work proposes a family of “approximate fair simulation” relations tailored to transition systems equipped with Büchi fairness conditions. By simplifying the nested inductive–coinductive structures inherent in traditional approaches, our method introduces a stronger and more user-friendly reasoning mechanism. We formalize this framework within a fixed-point logic and develop a corresponding deductive system, which we mechanize and prove correct in the Rocq proof assistant. Case studies demonstrate the effectiveness and practicality of our approach for interactive verification of fairness properties.

Abstract

It is well known that liveness properties cannot be proven using standard simulation arguments. This issue has been mitigated by extending standard notions of simulation for transition systems to fairness-preserving simulations for systems equipped with an additional fairness condition modeling liveness assumptions and/or liveness requirements. In the context of automated verification of finite-state systems, proofs by simulation are an appealing method as there exist efficient algorithms to find a simulation between two systems. However, applications of fair simulation to interactive verification have been much less studied. Perhaps one reason is that the definitions of fair simulation relations typically involve non-trivial nestings of inductive and coinductive relations, making them particularly difficult to use and to reason about. In this paper, we argue that in many cases, stronger notions of fair simulation involving more controlled alternations of fixed points are sufficient. Starting from known fair simulation techniques, we progressively build up a family of almost fair simulation relations for transition systems equipped with a Buechi fairness condition. The simulation relations we present can all be equipped with intuitive reasoning rules, leading to elegant deductive systems to prove fair trace inclusion. We mechanized our simulation relations and their associated deductive systems in the Rocq proof assistant, proved their soundness, and we demonstrate their use through a selection of examples.

Problem

• fair simulation
• liveness properties
• interactive verification
• inductive-coinductive relations
• Buechi fairness

Innovation

• almost fair simulation
• Büchi fairness
• fixed-point alternation
• deductive verification
• proof assistant