Probabilistic Rely-guarantee Calculus

Jones' rely-guarantee calculus for shared variable concurrency is extended to include probabilistic behaviours. We use an algebraic approach which combines and adapts probabilistic Kleene algebras with concurrent Kleene algebra. Soundness of the algebra is shown relative to a general probabilistic event structure semantics. The main contribution of this paper is a collection of rely-guarantee rules built on top of that semantics. In particular, we show how to obtain bounds on probabilities by deriving rely-guarantee rules within the true-concurrent denotational semantics. The use of these rules is illustrated by a detailed verification of a simple probabilistic concurrent program: a faulty Eratosthenes sieve.Comment: Preprint submitted to TCS-QAP

On Kleene Algebra vs. Process Algebra

We try to clarify the relationship between Kleene algebra and process algebra, based on the very recent work on Kleene algebra and process algebra. Both for concurrent Kleene algebra (CKA) with communications and truly concurrent process algebra APTC with Kleene star and parallel star, the extended Milner's expansion law $a\parallel b=a\cdot b+b\cdot a+a\parallel b +a\mid b$ holds, with $a,b$ being primitives (atomic actions), $\parallel$ being the parallel composition, $+$ being the alternative composition, $\cdot$ being the sequential composition and the communication merge $\mid$ with the background of computation. CKA and APTC are all the truly concurrent computation models, can have the same syntax (primitives and operators), maybe have the same or different semantics

A note on commutative Kleene algebra

In this paper we present a detailed proof of an important result of algebraic logic: namely that the free commutative Kleene algebra is the space of semilinear sets. The first proof of this result was proposed by Redko in 1964, and simplified and corrected by Pilling in his 1970 thesis. However, we feel that a new account of this proof is needed now. This result has acquired a particular importance in recent years, since it is a key component in the completeness proofs of several algebraic models of concurrent computations (bi-Kleene algebra, concurrent Kleene algebra...). To that effect, we present a new proof of this result

Concurrent Kleene Algebra with Tests and Branching Automata

We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings , or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law (x‖y)(z‖w)≤(xz)‖(yw) of CKA, we make use of the subsumption order of Gischer on the gsp-strings. We also define deterministic branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests

Concurrent Kleene Algebra with Observations: from Hypotheses to Completeness

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 $\mathsf{while}$-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

On Series-Parallel Pomset Languages: Rationality, Context-Freeness and Automata

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.Comment: Accepted manuscrip

Concurrent Kleene Algebra: Free Model and Completeness

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 techniques developed along the way are reusable; in particular, they allow us to establish pomset automata as an operational model for CKA.Comment: Version 2 includes an overview section that outlines the completeness proof, as well as some extra discussion of the interpolation lemma. It also includes better typography and a number of minor fixes. Version 3 incorporates the changes by comments from the anonymous referees at ESOP. Among other things, these include a worked example of computing the syntactic closure by han

Algebraic coherent confluence and higher-dimensional globular Kleene algebras

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent proofs by confluence. To this end, we introduce the structure of modal higher-dimensional globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We give a calculation of a coherent Church-Rosser theorem and Newman's lemma in higher-dimensional Kleene algebras. We interpret these results in the context of higher-dimensional rewriting systems described by polygraphs.Comment: Pre-print (second version