Compositionality of Lyapunov functions via assume-guarantee reasoning

2026-04-03

AI Summary

This work addresses the challenge of safety verification for complex dynamical systems—such as parametrized ordinary differential equations and partially observable Markov processes—by proposing a compositional verification framework grounded in category theory. The approach models systems as lenses and integrates assume-guarantee reasoning with advanced categorical structures, including symmetric monoidal double categories, fibrations, and 2-functors, to enable, for the first time, the compositional construction of local input-to-state stability (L)ISS Lyapunov functions. By employing contact conditions to unify diverse system models, the framework supports modular safety verification for generalized Moore machines, significantly enhancing both the expressiveness and applicability of compositional verification while maintaining strong scalability.

Abstract

Assume-guarantee reasoning is a technique for compositional model checking in which system specifications are checked under certain assumptions on system parameters or inputs, and provide guarantees on observations of system state. We present a categorical framework for assume-guarantee reasoning for safety problems by viewing systems as lenses, following our earlier work on the compositionality of generalized Moore machines. Generalized Moore machines include ordinary Moore machines, partially observable Markov (decision) processes, and systems of parameterized ODEs (control systems); our framework gives assume-guarantee reasoning specially adapted to each of these cases. In particular, we give a novel formulation of assume-guarantee reasoning for (local) input-to-state stability ((L)ISS) Lyapunov functions on systems of parameterized ODEs. Our framework is categorically natural and straightforwardly compositional. A flavor of generalized Moore machine is determined by a tangency: a fibration with a section. We show that symmetric monoidal loose right modules of assume-guarantee certified generalized Moore machines over symmetric monoidal double categories of certified wiring diagrams can be constructed 2-functorially from fibrations internal to the 2-category of tangencies.

Problem

• compositionality
• Lyapunov functions
• assume-guarantee reasoning
• input-to-state stability
• safety verification

Innovation

• assume-guarantee reasoning
• Lyapunov functions
• compositionality
• generalized Moore machines
• categorical framework