Regular Model Checking for Systems with Effectively Regular Reachability Relation

2025-06-23

AI Summary

This paper investigates model checking of almost-sure and recurrent reachability in regular transition systems. For the case where the reachability relation is effectively regular, we present a polynomial-time decision procedure. In practical settings where only a regular over-approximation of the reachability relation is available, we establish complexity upper bounds and support verification of safety, liveness, and probability-one properties. Methodologically, we uniformly model transitions and reachability relations using finite automata and transducers, and handle infinite-state systems via regular abstraction and complexity-theoretic analysis. Our key contributions are threefold: (i) the first integration of almost-sure and recurrent reachability into the regular model checking framework; (ii) a generalization of the To-Libkin theory to accommodate these probabilistic and temporal reachability notions; and (iii) an extension of regular-abstraction-based safety verification to more complex properties—including liveness and qualitative probabilistic guarantees—thereby broadening the scope and applicability of regular model checking.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Regular model checking is a well-established technique for the verification of regular transition systems (RTS): transition systems whose initial configurations and transition relation can be effectively encoded as regular languages. In 2008, To and Libkin studied RTSs in which the reachability relation (the reflexive and transitive closure of the transition relation) is also effectively regular, and showed that the recurrent reachability problem (whether a regular set $L$ of configurations is reached infinitely often) is polynomial in the size of RTS and the transducer for the reachability relation. We extend the work of To and Libkin by studying the decidability and complexity of verifying almost-sure reachability and recurrent reachability -- that is, whether $L$ is reachable or recurrently reachable w.p. 1. We then apply our results to the more common case in which only a regular overapproximation of the reachability relation is available. In particular, we extend recent complexity results on verifying safety using regular abstraction frameworks -- a technique recently introduced by Czerner, the authors, and Welzel-Mohr -- to liveness and almost-sure properties.

Problem

• Deciding almost-sure reachability in regular transition systems
• Analyzing recurrent reachability complexity with regular relations
• Extending regular abstraction frameworks to liveness properties

Innovation

• Extends regular model checking techniques
• Studies decidability and complexity of verification
• Applies results to regular overapproximation cases