Deciding KAT and Hoare Logic with Derivatives

TL;DR

This paper introduces a new decision procedure for Kleene algebra with tests (KAT) using derivatives, extending the decidability of propositional Hoare logic, with experimental validation.

Contribution

It presents a novel derivative-based decision procedure for KAT equivalence and extends it to propositional Hoare logic, enhancing verification tools.

Findings

New derivative-based decision procedure for KAT equivalence

Extension of procedure to propositional Hoare logic

Experimental results demonstrating effectiveness

Abstract

Kleene algebra with tests (KAT) is an equational system for program verification, which is the combination of Boolean algebra (BA) and Kleene algebra (KA), the algebra of regular expressions. In particular, KAT subsumes the propositional fragment of Hoare logic (PHL) which is a formal system for the specification and verification of programs, and that is currently the base of most tools for checking program correctness. Both the equational theory of KAT and the encoding of PHL in KAT are known to be decidable. In this paper we present a new decision procedure for the equivalence of two KAT expressions based on the notion of partial derivatives. We also introduce the notion of derivative modulo particular sets of equations. With this we extend the previous procedure for deciding PHL. Some experimental results are also presented.