A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

TL;DR

An algebraic axiomatization of Kleene Algebra is presented and two completeness results are proved for a large fragment of GKAT consisting of skip-free programs.

Abstract

Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs. In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa's axiomatization of Kleene Algebra. In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT consisting of skip-free programs.