On Tools for Completeness of 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 reproving completeness of KAT and KAO, and prove completeness of a new variant of KAT where the collection of tests only forms a distributive lattice.

Words-to-Letters Valuations for Language Kleene Algebras with Variable Complements

“…Interestingly, the equational theory of KAs w.r.t. languages coincides with the language equivalence under the standard language valuation (see also, e.g., [1,11]): for all KA terms (i.e., regular expressions) t, s, we have…”

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

“…Other instances of reductions include [10,4,12,49,21,23,32,36,28]. Recent work has started to study reductions and their compositionality properties [12,22,35].…”

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

“…cit. Other instances of reductions include [9,4,11,47,19,21,30,34,26]. Recent work has started to study reductions and their compositionality properties [11,20,33].…”