On Tools for Completeness of Kleene Algebra with Hypotheses

TL;DR

This paper develops modular tools to establish the completeness of Kleene algebra variants with hypotheses, and applies them to prove completeness for several existing and new variants like KAT, KAO, and NetKAT.

Contribution

It introduces a unified, modular approach for proving completeness of Kleene algebra with hypotheses and extends it to new variants.

Findings

Provided modular proofs of completeness for KAT, KAO, and NetKAT.

Proved completeness for new KAT variants with additional constants and operations.

Extended the theoretical framework for Kleene algebra with hypotheses.

Abstract

In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice.