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

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Refinement algebra for probabilistic programs

We identify a refinement algebra for reasoning about probabilistic program transformations in a total-correctness setting. The algebra is equipped with operators that determine whether a program is enabled or terminates respectively. As well as developing the basic theory of the algebra we demonstrate how it may be used to explain key differences and similarities between standard (i.e. non-probabilistic) and probabilistic programs and verify important transformation theorems for probabilistic action systems.29 page(s

Towards a linear algebra of programming

The Algebra of Programming (AoP) is a discipline for programming from specifications using relation algebra. Specification vagueness and nondeterminism are captured by relations. (Final) implemen- tations are functions. Probabilistic functions are half way between relations and functions: they express the propensity, or like- lihood of ambiguous, multiple outputs. This paper puts forward a basis for a Linear Algebra of Programming (LAoP) extending standard AoP towards probabilistic functions. Because of the quantitative essence of these functions, the allegory of binary relations which supports the AoP has to be extended. We show that, if one restricts to discrete probability spaces, categories of matrices provide adequate support for the extension, while preserving the pointfree reasoning style typical of the AoP.Fundação para a Ciência e a Tecnologia (FCT

On kleene algebras for weighted computation

Kleene algebra with tests (KAT) was introduced as an alge- braic structure to model and reason about classic imperative programs, i.e. sequences of discrete actions guarded by Boolean tests. This paper introduces two generalisations of this structure able to ex- press programs as weighted transitions and tests with outcomes in a not necessary bivalent truth space, namely graded Kleene algebra with tests (GKAT) and Heyting Kleene algebra with tests (HKAT). On these contexts, in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT [10], we discuss the encoding of a graded PHL in HKAT and of its while-free fragment in GKAT.This work is financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundacao para a Ciencia e a Tecnologia, within projects POCI-01-0145-FEDER-016692 and UID/MAT/04106/2013. The second author is also supported by the individual grant SFRH/BPD/103004/2014

Convolution, Separation and Concurrency

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page

Hoare Semigroups

A semigroup-based setting for developing Hoare logics and refinement calculi is introduced together with procedures for translating between verification and refinement proofs. A new Hoare logic for multirelations and two minimalist generic verification and refinement components, implemented in an interactive theorem prover, are presented as applications that benefit from this generalisation

Probabilistic Demonic Refinement Algebra

We propose an abstract algebra for reasoning about probabilistic programs in a total-correctness framework. In contrast to probablisitic Kleene algebra it allows genuine reasoning about total correctness and in addition to Kleene star also has a strong iteration operator. We define operators that determine whether a program is enabled, has certain failure or does not have certain failure, respectively. The algebra is applied to the derivation of refinement rules for probabilistic action systems