Barrier Certificates for Uncertain Temporal Specifications

2026-05-10

AI Summary

This work addresses the verification challenge of stochastic dynamical systems subject to temporal logic specifications whose predicates evolve randomly over time. The authors propose a novel framework based on barrier certificates, constructing an augmented state space to transform stochastic predicate temporal specifications into deterministic ones. This approach extends, for the first time, the barrier certificate methodology to settings involving stochastically evolving predicates. By replacing conventional dynamic programming with convex optimization, the method derives analytical upper bounds on violation probabilities for linear systems under safety-type specifications. Numerical experiments demonstrate that the proposed technique achieves a favorable trade-off between computational efficiency and conservativeness.

Abstract

This paper studies satisfying temporal logic specifications on stochastic dynamical systems, where the predicates evolve randomly over time. Such randomness may arise from uncertain environment models or external stochastic processes causing the sets associated with predicate satisfaction to vary in a non-deterministic manner. As a result, verifying whether a stochastic dynamical system satisfies a temporal specification depends also on the uncertainty in the predicates. We develop a certificate-based framework to bound the probability of satisfying temporal logic specifications with randomly evolving predicates. We first show that temporal logic specifications with stochastic predicates can be transformed to specifications with deterministic predicates on an augmented space which is extended to include the stochastic space of predicate's uncertainty. We then utilize barrier certificates on an augmented space to provide tractable optimization-based conditions and to avoid the computational burden of dynamic programming. Focusing on linear dynamics and safety-type specifications, we derive analytical conditions under which barrier certificates guarantee bounds on the probability of violating the stochastic safety predicates. The approach is demonstrated on numerical case studies.

Problem

• stochastic dynamical systems
• temporal logic specifications
• randomly evolving predicates
• uncertain environments
• safety verification

Innovation

• Barrier Certificates
• Stochastic Temporal Logic
• Augmented State Space
• Safety Verification
• Uncertain Predicates