Brzozowski Goes Concurrent - A Kleene Theorem for Pomset Languages

Journal of Logical and Algebraic Methods in Programming

Brunet

Luttik

et al. 2019

Self Cite

Concurrent Kleene Algebra (CKA) is a formalism to study concurrent programs. Like previous Kleene Algebra extensions, developing a correspondence between denotational and operational perspectives is important, for both foundations and applications. This paper takes an important step towards such a correspondence, by precisely relating bi-Kleene Algebra (BKA), a fragment of CKA, to a novel type of automata, pomset automata (PAs).We show that PAs can implement the BKA semantics of series-parallel rational expressions, and that a class of PAs can be translated back to these expressions. We also characterise the behavior of general PAs in terms of context-free pomset grammars; consequently, universality, equivalence and series-parallel rationality of general PAs are undecidable. ✩ This paper is an extended version of a paper published at CONCUR'17 [1]. (Tobias Kappé) c 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http: // creativecommons. org/ licenses/ by-nc-nd/ 4. 0/ Definition 2.2Without loss of generality, we can assume that C U and C V are disjoint.The sequential composition of U and V , denoted U · V , is the pomsetThe parallel composition of U and V , denoted U V , is the pomsetAs a convention, sequential composition takes precedence over parallel composition, i.e., U · V W is read as (U · V ) W .Sequential composition forces the events in the left pomset to be ordered before those in the right pomset. We note that these operators are welldefined modulo isomorphism of labelled posets, and that the empty pomset 1 is the unit for both sequential and parallel composition.Definition 2.3. The set of series-parallel pomsets [12,13], Pom sp , is the smallest subset of Pom that includes the empty and primitive pomsets and is closed under sequential and parallel composition.In this paper we concern ourselves with series-parallel pomsets. For inductive reasoning, it is useful to recall part of [12, Theorem 3.1].Lemma 2.4. Let U ∈ Pom sp . If U is non-empty, then exactly one of the following is true: (i) U = a for some a ∈ Σ, or (ii) U = V · W for non-empty V, W ∈ Pom sp , strictly smaller than U, or (iii) U = V W for non-empty V, W ∈ Pom sp , strictly smaller than U.We can quantify the degree of nesting of parallel and sequential composition of a series-parallel pomset as follows.

Section: Introductionsupporting

confidence: 65%

Section: Pomset Automatamentioning

confidence: 99%

On series-parallel pomset languages: Rationality, context-freeness and automata

Journal of Logical and Algebraic Methods in Programming

Brunet

Luttik

et al. 2019

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“…The extension to the existing result for BKA provides a clear understanding of the difficulties introduced by the presence of the exchange axiom and shows how to separate concerns between CKA and BKA, a technique also useful elsewhere. For one, our construction also provides an extension of (half of) Kleene's theorem for BKA [14] to CKA, establishing pomset automata as an operational model for CKA and opening the door to decidability procedures similar to those previously studied for KA. Furthermore, it reduces deciding the equational theory of CKA to deciding the equational theory of BKA.…”

Section: Introductionmentioning

confidence: 69%

“…We furthermore note that the algorithm to compute downward closure can be used to extend half of the result from [14] to a Kleene theorem that relates the CKA-semantics of expressions to the pomset automata proposed there: if e ∈ T , we can construct a pomset automaton A with a state q such that L A (q) = e CKA .…”

Section: Discussion and Further Workmentioning

confidence: 99%

See 1 more Smart Citation

Concurrent Kleene Algebra: Free Model and Completeness

Lecture Notes in Computer Science

Brunet

Silva

et al. 2018

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the axioms for CKA with bounded parallelism are complete for the semantics proposed in the original paper; consequently, these semantics are the free model for this fragment. This result settles a conjecture of Hoare and collaborators. Moreover, the technique developed to this end allows us to establish a Kleene Theorem for CKA, extending an earlier Kleene Theorem for a fragment of CKA. ments.The notion of N-freeness for pomsets is useful for proving the lemmas to come.Definition A.1. Let U = [u] be a pomset. We say that U is N-free if there are no u 0 , u 1 , u 2 , u 3 ∈ S u such that u 0 ≤ u u 1 , u 2 ≤ u u 3 and u 0 ≤ u u 3 and no other relation between them, i.e., the graph of these elements has the shape of an N.Note that N-freeness is well-defined for pomsets, for the presence of an Nshape does not depend on the particular representative u. It is not hard to see that all series-parallel pomsets are N-free. Perhaps surprisingly, this N-freeness provides a complete characterisation of series-parallel pomsets [6].It is also useful to restrict a labelled poset to a part of its carrier, as follows.Definition A.2. Let u be a labelled poset, and let S ⊆ S u . We write u ↾ S for the restriction of u to S, i.e., labelled poset given by S u↾S = S, ≤ u↾S = ≤ u ∩ S × S, and λ u↾S (z) = λ u (z).A.1 Subsumption of empty or primitive pomsets Lemma A.2. Let u be a labelled poset such that u ⊑ 1 or 1 ⊑ u. Then u = 1.Proof. We treat the case where u ⊑ 1; the case where 1 ⊑ u is similar. Let h : 1 → u witness that u ⊑ 1. Then h is a bijection from S 1 = ∅ to S u ; accordingly, S u = ∅. But then u = 1, because the labelled poset with empty carrier is unique.Proof. First, suppose that U = 1. We then have that U = [1] and V = [v] such that u ⊑ 1 or 1 ⊑ u. By Lemma A.2, we find that v = 1 and thus V = [1] = 1.Second, suppose that U = a for some a ∈ Σ. Then U = [u] for some pomset with singleton carrier S u , with λ u (u) = a for all u ∈ S u . Since V = [v] and u ⊑ v or v ⊑ u, we find that u ≃ v by Lemma A.3. This establishes that U = V . A.2 The factorisation lemmaLemma 3.3 (Factorisation). Let U , V 0 , and V 1 be pomsets such that U is subsumed by V 0 · V 1 . Then there exist pomsets U 0 and U 1 such that:Also, if U 0 , U 1 and V are pomsets such that U 0 U 1 ⊑ V , then there exist pomsets V 0 and V 1 such that:Proof. We start with the first claim. Let U , V 0 and V 1 be as in the premise, and write U = [u], V 0 = [v 0 ] and V 1 = [v 1 ]. Without loss of generality, we can assume that v 0 and v 1 are disjoint, that S v0 ∪ S v1 = S u , and that the identity function S v0 ∪ S v1 → S u is the subsumption witnessing that u ⊑ v 0 · v 1 .We then choose u i = u ↾ vi for i ∈ 2, and claim that u 0 · u 1 = u.-For the carrier, we already know that

Learning Pomset Automata

Heerdt

Lecture Notes in Computer Science

Rot

et al. 2021

Self Cite

We extend the $$\mathtt {L}^{\!\star }$$ L ⋆ algorithm to learn bimonoids recognising pomset languages. We then identify a class of pomset automata that accepts precisely the class of pomset languages recognised by bimonoids and show how to convert between bimonoids and automata.