A Coalgebraic Approach to Kleene Algebra with Tests

TL;DR

This paper develops a coalgebraic framework for Kleene algebra with tests, introducing new automata-theoretic interpretations and a coinductive proof principle to reason about program equivalences.

Contribution

It presents a novel coalgebraic approach to Kleene algebra with tests, including a new automata interpretation and a coinductive proof method.

Findings

Established a coalgebraic theory for Kleene algebra with tests

Introduced a new automata-theoretic interpretation

Developed a coinductive proof principle for expression equivalence

Abstract

Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene algebra with tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene algebra with tests expressions and a corresponding automata-theoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene algebra with tests expressions.