Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness

Abstract

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and while-loops. It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO). Our main contribution is a completeness theorem for CKAO. Our result resorts on a more general study of CKA "with hypotheses", of which CKAO turns out to be an instance: this analysis is of independent interest, as it can be applied to extensions of CKA other than CKAO. PreliminariesWe recall basic definitions on pomset languages, used in the semantics of CKA, which generalise languages to allow letters in words to be partially ordered. We fix a (possibly infinite) alphabet Σ. When defining sets parametrised by Σ, say S(Σ), if Σ is clear from the context we use S to refer to S(Σ). Pomsets [9,10] are labelled posets, up to isomorphism. Posets and PomsetsDefinition 2.1 (Labellet poset). A labelled poset over Σ is a tuple u = S, ≤, λ , where S is a finite set (the carrier of u), ≤ u is a partial order on S (the order of u), and λ : S → Σ is a function (the labelling of u).

On Tools for Completeness of Kleene Algebra with Hypotheses

“…The following result from [16] (cf. Remark 3.5) gives a sufficient condition for the existence of a reduction.…”

“…As illustrated above, the overall strategy is to reduce completeness of KA H , for a given set of hypotheses H, to completeness of Kleene algebra. The core idea is to provide a map r from expressions to expressions, which incorporates the hypotheses H in the sense that r(e) = cl H ( e ), and such that r(e) is provably equivalent to e under the hypotheses H. This idea leads to the unifying notion of reduction, developed in [25,10,16]. Definition 3.1 (Reduction).…”

“…Remark 3.5. The idea to use two sets of hypotheses in Definition 3.1 is from [16], where reductions are defined slightly differently: the alphabet is fixed (that is, Σ = Γ ), and the last condition is instead defined as cl…”

“…By allowing a change of alphabet, we do not need to distinguish reductions and strong reductions. Lemma 3.3 is in [16,Lemma 4.23], adapted here to the case with two alphabets (this is taken care of in loc. cit.…”

Models of Concurrent Kleene Algebra

“…The work described in this invited talk is based on [1,2,3], and it is joint with a wonderful group of people: Paul Brunet, Simon Docherty, Tobias Kappé, Jurriaan Rot, Jana Wagemaker, and Fabio Zanasi.…”