Interpolation-Inspired Closure Certificates

2025-12-09

IEEE Conference on Decision and Control

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Learning to Search

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• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Barrier certificates, a form of state invariants, provide an automated approach for safety verification of dynamical systems. Similar to barrier certificates, recent works explore the notion of closure certificates, a form of transition invariants, to verify dynamical systems against ω-regular properties including safety. A closure certificate, defined over state pairs of a dynamical system, is a real-valued function whose zero superlevel set characterizes an inductive transition invariant of the system. The search for such a certificate can be effectively automated by assuming it to be within a specific template class, e.g. a polynomial of a fixed degree, and then using optimization techniques such as sum-of-squares programming (SOS) or satisfiability-modulo-theory solvers (SMT) to find it. Unfortunately, one may not be able to find such a certificate for a fixed template. In such a case, one must change the template, e.g. increase the degree of the polynomial. In this paper we consider a notion of multiple closure certificates dubbed interpolation-inspired closure certificates. An interpolation-inspired closure certificate consists of a set of functions which jointly over-approximate a transition invariant by first considering one-step transitions, then two, and repeat until a transition invariant is obtained. The advantage of interpolation-inspired closure certificates is that they allow us to prove properties even when a single function for a fixed template cannot be found using standard approaches. We present an SOS programming approach to search for these sets of functions and demonstrate the effectiveness of our proposed method for verifying safety and persistence (or refuting recurrence) over some case studies.

Problem

• closure certificates
• dynamical systems
• ω-regular properties
• barrier certificates
• transition invariants

Innovation

• interpolation-inspired closure certificates
• transition invariants
• sum-of-squares programming
• omega-regular properties
• barrier certificates