Algebraic Principles for Rely-Guarantee Style Concurrency Verification Tools

We provide simple equational principles for deriving rely-guarantee-style inference rules and refinement laws based on idempotent semirings. We link the algebraic layer with concrete models of programs based on languages and execution traces. We have implemented the approach in Isabelle/HOL as a lightweight concurrency verification tool that supports reasoning about the control and data flow of concurrent programs with shared variables at different levels of abstraction. This is illustrated on two simple verification examples

Higher Catoids, Higher Quantales and their Correspondences

We establish modal correspondences between omega-catoids and convolution omega-quantales. These are related to J\'onsson-Tarski style-dualities between relational structures and lattices with operators. We introduce omega-catoids as generalisations of (strict) omega-categories and in particular of the higher path categories generated by polygraphs (or computads) in higher rewriting. Convolution omega-quantales generalise the powerset omega-Kleene algebras recently proposed for algebraic coherence proofs in higher rewriting to weighted variants. We extend these correspondences to ({\omega},p)-catoids and convolution ({\omega},p)-quantales suitable for modelling homotopies in higher rewriting. We also specialise them to finitely decomposable ({\omega}, p)-catoids, an appropriate setting for defining ({\omega}, p)-semirings and ({\omega}, p)-Kleene algebras. These constructions support the systematic development and justification of higher quantale axioms relative to a previous ad hoc approach.Comment: 46 pages, 8 figure

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

Semiring-based Specification Approaches for Quantitative Security

Our goal is to provide different semiring-based formal tools for the specification of security requirements: we quantitatively enhance the open-system approach, according to which a system is partially specified. Therefore, we suppose the existence of an unknown and possibly malicious agent that interacts in parallel with the system. Two specification frameworks are designed along two different (but still related) lines. First, by comparing the behaviour of a system with the expected one, or by checking if such system satisfies some security requirements: we investigate a novel approximate behavioural-equivalence for comparing processes behaviour, thus extending the Generalised Non Deducibility on Composition (GNDC) approach with scores. As a second result, we equip a modal logic with semiring values with the purpose to have a weight related to the satisfaction of a formula that specifies some requested property. Finally, we generalise the classical partial model-checking function, and we name it as quantitative partial model-checking in such a way to point out the necessary and sufficient conditions that a system has to satisfy in order to be considered as secure, with respect to a fixed security/functionality threshold-value