Deciding KAT and Hoare Logic with Derivatives

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

Deciding Kleene algebra terms equivalence in Coq

“…In the case of the development of the decision procedure for KAT terms equivalence, we are mainly interested in extending the current development in the following ways: first, we are interested in improving the performance of the KAT equivalence procedure along the ideas of Broda et al [63,65] in which a more efficient way of dealing with the propositional layer is proposed; next, we wish to improve the performance of the decision procedure by following along the ideas introduced by Almeida et al [58,61] and whose method proposed dispenses the creation of the KAT terms r and u that are required to automate the proof of partial correctness of imperative programs encoded as KAT terms; the other way we wish to improve in our development is to add support for SKAT [59,60], which we believe that it will approximate the usage of KAT to a more realistic notion of program verification, since at the level of SKAT we have access to first order constructions in programs.…”

“…Almeida et al [58,61] presented a new development of a decision procedure for KAT equivalence. The implementation was made using the OCaml programming language, is not mechanically certified, but includes a new method for proving the partial correctness of programs that dispenses the burden of constructing the terms r and u introduced in the previous section.…”

Constrained Expressions and their Derivatives

“…Derivation and partial derivation have already been used in order to perform the membership test over extensions of regular expressions [6,7,8,10], expressions denoting non-necessarily regular languages [9], guarded strings [1] or even context-free grammars [17]. In the rest of this paper, we extend regular expressions by introducing new operators based on boolean formulae in order to increase the expressive power of expressions.…”

Deciding Synchronous Kleene Algebra with Derivatives

“…There, derivatives are considered with respect to symbols vσ where σ is an action symbol but v corresponds to a valuation of the Boolean tests. This induces an exponential blow-up on the number of states or transitions of the automata and an accentuated exponential complexity when testing the equivalence of two KAT expressions (as noted in [23,3]). A. Silva [28] introduced a class of automata over guarded strings that avoids that blow-up.…”