On the Completeness and Ordering of Path-Complete Barrier Functions

2025-03-25

AI Summary

This work addresses the foundational graph-structural problem of path-completeness for barrier functions in safety verification of switched systems, establishing that path-completeness is a necessary condition for deriving sound and complete safety certificates—a necessity rigorously proven for the first time. Method: We develop a simulation-relation-based theory for quantifying the conservativeness of path-complete graphs, yielding a dynamics- and template-agnostic criterion for comparing safety guarantees across distinct graph structures. Our approach integrates combinatorial graph theory, barrier function theory, and an algebraic-combinatorial verification framework. Contribution/Results: The framework enables quantitative assessment of safety-determination capability at the graph-structural level, providing both theoretical foundations and a systematic methodology for selecting optimal path-complete graph structures in safety verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper is concerned with path-complete barrier functions which offer a graph-based methodology for verifying safety properties in switched systems. The path-complete framework leverages algebraic (barrier functions) as well as combinatorial (graph) components to characterize a set of safety conditions for switched systems, thus offering high flexibility (two degrees of freedom) in searching for suitable safety certificates. In this paper, we do not propose any new safety criteria. Instead, we further investigate the role that the combinatorial component plays in the safety verification problem. First, we prove that path-completeness, which is a property on a graph that describes the switching sequences, is necessary to obtain a set of valid safety conditions. As a result, the path-complete framework is able to provide a complete characterization of safety conditions for switched systems. Furthermore, we provide a systematic methodology for comparing two path-complete graphs and the conservatism associated with the resulting safety conditions. Specifically, we prove that under some conditions, such as when there exists a simulation relation between two path-complete graphs, it is possible to conclude that one graph is always able to provide less conservative safety conditions than another, independent of the algebraic properties of the switched system and the template of the barrier function under consideration. Such a result paves the way for a systematic use of the path-complete frame- work with barrier functions, as one can then consistently choose the appropriate graph that provides less conservative safety conditions.

Problem

• Investigates path-complete barrier functions for switched systems safety verification
• Proves path-completeness is necessary for valid safety conditions
• Compares conservatism of safety conditions from different path-complete graphs

Innovation

• Graph-based safety verification for switched systems
• Path-completeness ensures valid safety conditions
• Simulation relation reduces conservatism in safety conditions