A Hierarchy of Supermartingales for $ω$-Regular Verification

2025-11-28

AI Summary

This work addresses the almost-sure verification of ω-regular properties—specifically Streett conditions—over discrete-time Markov chains. To this end, we introduce three novel supermartingale certificates: Generalized Streett Supermartingales (GSSMs), Distribution-Valued Streett Supermartingales (DVSSMs), and Lexicographic Progress Measures Supermartingales (LexPMSMs). Compared to classical Streett supermartingales, these certificates are strictly more expressive and more powerful for verification; DVSSMs are proven to achieve theoretical optimality in verification strength. Methodologically, our approach integrates fixed-point characterizations of positive/null recurrence, probabilistic extensions of parity progress measures, lexicographic composition, and distribution-valued function techniques. We implement a prototype verifier based on LexPMSMs, which successfully verifies several benchmark instances where existing methods fail, thereby substantially improving verification coverage and practical applicability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

We propose new supermartingale-based certificates for verifying almost sure satisfaction of $ω$-regular properties: (1) generalised Streett supermartingales (GSSMs) and their lexicographic extension (LexGSSMs), (2) distribution-valued Streett supermartingales (DVSSMs), and (3) progress-measure supermartingales (PMSMs) and their lexicographic extension (LexPMSMs). GSSMs, LexGSSMs, and DVSSMs are derived from least-fixed point characterisations of positive recurrence and null recurrence of Markov chains with respect to given Streett conditions; and PMSMs and LexPMSMs are probabilistic extensions of parity progress measures. We study the hierarchy among these certificates and existing certificates, namely Streett supermartingales, by comparing the classes of problems that can be verified by each type of certificates. Notably, we show that our certificates are strictly more powerful than Streett supermartingales. We also prove completeness of GSSMs for positive recurrence and of DVSSMs for null recurrence: DVSSMs are, in theory, the most powerful certificates in the sense that for any Markov chain that almost surely satisfies a given $ω$-regular property, there exists a DVSSM certifying it. We provide a sound and relatively complete algorithm for synthesising LexPMSMs, the second most powerful certificates in the hierarchy. We have implemented a prototype tool based on this algorithm, and our experiments show that our tool can successfully synthesise certificates for various examples including those that cannot be certified by existing supermartingales.

Problem

• Develops new supermartingale certificates for verifying ω-regular properties.
• Establishes a hierarchy comparing verification power of different certificate types.
• Provides algorithms and tools for synthesizing these certificates effectively.

Innovation

• Generalized Streett supermartingales for Markov chain verification
• Distribution-valued Streett supermartingales for null recurrence
• Lexicographic progress-measure supermartingales with synthesis algorithm