A Deductive Refinement Calculus for Differential-Algebraic Programs

2026-05-11

AI Summary

This work addresses the challenge of preserving trajectory properties during simplification and transformation of hybrid systems involving differential-algebraic equations (DAEs). To this end, the paper introduces differential-algebraic refinement logic (dARL), a formal framework that builds upon trajectory semantics to support stepwise verification and simplification of DAE-based programs while guaranteeing semantic preservation at each transformation step. The core contribution lies in the first complete and provably correct refinement calculus for index reduction of DAEs, thereby establishing a formal foundation and syntactic assurance for incremental verification of complex DAE systems.

Abstract

This paper presents differential-algebraic refinement logic (dARL) with which one can deductively verify both properties and relations of differential-algebraic programs (DAPs) that extend hybrid dynamical systems with differential-algebraic equations (DAEs). A refinement calculus is introduced that enables the sound comparison of trajectories of differential-algebraic equations, crucially utilizing a novel trace-based semantics. This enables the incremental verification/simplification of complicated DAEs, while ensuring correctness at each step by the soundness of the calculus. The calculus is shown to be complete for certifying index reductions of DAEs, providing trustworthy syntactic proofs of correctness at each step of the reduction.

Problem

• differential-algebraic programs
• refinement calculus
• deductive verification
• differential-algebraic equations
• hybrid dynamical systems

Innovation

• differential-algebraic equations
• refinement calculus
• deductive verification
• trace-based semantics
• index reduction