Uniform Substitution for Differential Refinement Logic

TL;DR

This paper develops a uniform substitution calculus for differential refinement logic dRL, enhancing reasoning about hybrid systems and simplifying proofs by enabling direct axiom instantiation and decidability in certain fragments.

Contribution

It introduces a uniform substitution rule for dRL, allowing more streamlined and sound proof procedures for hybrid system properties and refinements.

Findings

Uniform substitution enables direct use of axioms without schemata.

Decidability of refinement relations in a fragment of hybrid programs.

Enhanced proof simplicity and soundness in differential refinement logic.

Abstract

This paper introduces a uniform substitution calculus for differential refinement logic dRL. The logic dRL extends the differential dynamic logic dL such that one can simultaneously reason about properties of and relations between hybrid systems. Refinements are useful e.g. for simplifying proofs by relating a concrete hybrid system to an abstract one from which the property can be proved more easily. Uniform substitution is the key to parsimonious prover microkernels. It enables the verbatim use of single axiom formulas instead of axiom schemata with soundness-critical side conditions scattered across the proof calculus. The uniform substitution rule can then be used to instantiate all axioms soundly. Access to differential variables in dRL enables more control over the notion of refinement, which is shown to be decidable on a fragment of hybrid programs.