# Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time

## TL;DR

This paper introduces Guarded Kleene Algebra with Tests (GKAT), a simplified algebraic framework that enables nearly linear time verification of certain programs, contrasting with the more complex KAT.

## Contribution

It develops the algebraic theory of GKAT, proves its near-linear time decidability, and establishes a Kleene theorem and completeness results.

## Key findings

- GKAT's equational theory is almost linear time decidable.
- A full Kleene theorem for GKAT is established.
- Completeness of GKAT's axiomatization is proved.

## Abstract

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union ($+$) and iteration ($*$) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa's axiomatization of Kleene Algebra.