Supermartingale Certificates for Quantitative Omega-regular Verification and Control

2025-05-24

AI Summary

Existing martingale-based verification methods for discrete-time infinite-state stochastic systems support only quantitative reachability/safety or qualitative (probability-one) ω-regular properties, fundamentally limiting their applicability to general quantitative ω-regular specifications. Method: We propose the first verifiable martingale certificate framework for quantitative ω-regular properties, based on the product space of the system and a limit-deterministic Büchi automaton (LDBA). We introduce a novel *LDBA-based supermartingale (LDBSM)* certificate and develop a template-based automated synthesis algorithm grounded in polynomial inequality solving. Contribution/Results: Our approach enables fully automated verification and controller synthesis for general quantitative ω-regular properties—e.g., “visit a target set infinitely often with probability ≥ 0.95”—extending supermartingale techniques beyond prior capabilities. We demonstrate its effectiveness on polynomial dynamical systems, successfully solving multiple benchmark quantitative ω-regular problems previously intractable via supermartingale methods.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We present the first supermartingale certificate for quantitative $omega$-regular properties of discrete-time infinite-state stochastic systems. Our certificate is defined on the product of the stochastic system and a limit-deterministic B""uchi automaton that specifies the property of interest; hence we call it a limit-deterministic B""uchi supermartingale (LDBSM). Previously known supermartingale certificates applied only to quantitative reachability, safety, or reach-avoid properties, and to qualitative (i.e., probability 1) $omega$-regular properties. We also present fully automated algorithms for the template-based synthesis of LDBSMs, for the case when the stochastic system dynamics and the controller can be represented in terms of polynomial inequalities. Our experiments demonstrate the ability of our method to solve verification and control tasks for stochastic systems that were beyond the reach of previous supermartingale-based approaches.

Problem

• Verify quantitative ω-regular properties in stochastic systems
• Extend supermartingale certificates beyond reachability and safety
• Automate synthesis for polynomial-inequality-based systems

Innovation

• First supermartingale certificate for ω-regular properties
• Limit-deterministic Büchi supermartingale (LDBSM) approach
• Automated template-based synthesis for polynomial systems