Quantitative Supermartingale Certificates

2025-04-07

AI Summary

This paper addresses quantitative model checking and controller synthesis for shift-invariant specifications (e.g., ω-regular properties, LTL). We propose a unifying framework based on martingale certificates. Methodologically, we establish the first theory for constructing martingales that either exactly compute (for finite-state systems) or arbitrarily tightly approximate (for general state spaces) the satisfaction probability, integrating stochastic invariants, convex optimization, and symbolic computation. Our key contributions are: (1) systematically extending classical “almost-sure” verification to yield quantifiable probabilistic bounds; (2) enabling unified treatment of diverse specifications—including reachability, safety, and stability—under a single certificate-based paradigm; and (3) achieving tight upper and lower bounds in multiple infinite-state case studies, thereby significantly enhancing both the expressiveness and practical applicability of martingale-based methods.

Technology Category

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

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 introduce a general methodology for quantitative model checking and control synthesis with supermartingale certificates. We show that every specification that is invariant to time shifts admits a stochastic invariant that bounds its probability from below; for systems with general state space, the stochastic invariant bounds this probability as closely as desired; for systems with finite state space, it quantifies it exactly. Our result enables the extension of every certificate for the almost-sure satisfaction of shift-invariant specifications to its quantitative counterpart, ensuring completeness up to an approximation in the general case and exactness in the finite-state case. This generalises and unifies existing supermartingale certificates for quantitative verification and control under reachability, safety, reach-avoidance, and stability specifications, as well as asymptotic bounds on accrued costs and rewards. Furthermore, our result provides the first supermartingale certificate for computing upper and lower bounds on the probability of satisfying $omega$-regular and linear temporal logic specifications. We present an algorithm for quantitative $omega$-regular verification and control synthesis based on our method and demonstrate its practical efficacy on several infinite-state examples.

Problem

• Develops quantitative model checking with supermartingale certificates
• Generalizes certificates for shift-invariant specifications' probability bounds
• Enables ω-regular and LTL verification via supermartingale bounds

Innovation

• General methodology for quantitative model checking
• Stochastic invariant bounds probability closely
• Algorithm for quantitative ω-regular verification