A cyclic proof system for Guarded Kleene Algebra with Tests (full version)

TL;DR

This paper introduces a cyclic proof system for Guarded Kleene Algebra with Tests (GKAT), enabling efficient reasoning and decision procedures that are more space-efficient than those for Kleene Algebra.

Contribution

It develops a non-well-founded sequent system for GKAT, providing completeness and a decision procedure in NLOGSPACE, improving over Kleene Algebra's hypersequent approach.

Findings

Proof search runs in NLOGSPACE.

Set of regular proofs is complete for the guarded language model.

Compared to Kleene Algebra, decision procedures are more space-efficient.

Abstract

Guarded Kleene Algebra with Tests (GKAT for short) is an efficient fragment of Kleene Algebra with Tests, suitable for reasoning about simple imperative while-programs. Following earlier work by Das and Pous on Kleene Algebra, we study GKAT from a proof-theoretical perspective. The deterministic nature of GKAT allows for a non-well-founded sequent system whose set of regular proofs is complete with respect to the guarded language model. This is unlike the situation with Kleene Algebra, where hypersequents are required. Moreover, the decision procedure induced by proof search runs in NLOGSPACE, whereas that of Kleene Algebra is in PSPACE.