Binary multirelations form a model of alternating nondeterminism useful for analysing games, interactions of computing systems with their environments or abstract interpretations of probabilistic programs. We investigate this alternating structure with inner or demonic and outer or angelic choices in a relation-algebraic language extended with specific operations on multirelations that relate to the inner layer of alternation.

Abstract We formalize the modal operators from the concurrent dynamic logics of Peleg, Nerode and Wijesekera in a multirelational algebraic language based on relation algebras and power allegories, using relational approximation operators on multirelations developed in a companion article. We relate Nerode and Wijesekera’s box operator with a relational approximation operator for multirelations and two related operators that approximate multirelations by different kinds of deterministic multirelations. We provide an algebraic soundness proof of Goldblatt’s axioms for concurrent dynamic logic and one for a multirelational Hoare logic based on Nerode and Wijesekera’s box as applications.

We formally introduce IsaVODEs (Isabelle verification with Ordinary Differential Equations), an open, compositional and extensible framework for the verification of cyber-physical systems. We extend a previous semantic approach with methods and techniques that increase its expressivity, proof automation, and scalability to the level of state-of-the-art deductive verification tools. Our contributions include a user-friendly specification language, a flexible hybrid store model, including vectors and matrices, and separation-logic-style rules for local reasoning with hybrid stores using a novel form of differentiation called framed Fréchet derivatives. The formalisation of correctness specifications with forward predicate transformers, the certification of flows as unique solutions to systems of ordinary differential equations, and invariant reasoning for such systems also contribute to the scalability and usability of our framework. In combination, these features make our framework flexible and adaptable to several verification workflows. A suite of examples and hybrid systems verification benchmarks validate our framework relative to other state-of-the-art approaches.

Binary multirelations allow modelling alternating nondeterminism, for instance, in games or nondeterministically evolving systems interacting with an environment. Such systems can show partial or total functional behaviour at both levels of alternation, so that nondeterministic behaviour may occur only at one level or both levels, or not at all. We study classes of inner and outer partial and total functional multirelations in a multirelational language based on relation algebra and power allegories. While it is known that general multirelations do not form a category, we show in the multirelational language that the classes of deterministic multirelations mentioned form categories with respect to Peleg composition from concurrent dynamic logic, and sometimes quantaloids. Some of these categories are isomorphic to the category of binary relations. We also introduce determinisation maps that approximate multirelations either by binary relations or by deterministic multirelations. Such maps are useful for defining modal operators on multirelations.

We prove a Kleene theorem for higher-dimensional automata. It states that the languages they recognise are precisely the rational subsumption-closed sets of finite interval pomsets. The rational operations on these languages include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce higher-dimensional automata with interfaces, which are modelled as presheaves over labelled precube categories, and develop tools and techniques inspired by algebraic topology, such as cylinders and (co)fibrations. Higher-dimensional automata form a general model of non-interleaving concurrency, which subsumes many other approaches. Interval orders are used as models for concurrent and distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.

We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders are used as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.

Abstract We present a semantic framework for the deductive verification of hybrid systems with Isabelle/HOL. It supports reasoning about the temporal evolutions of hybrid programs in the style of differential dynamic logic modelled by flows or invariant sets for vector fields. We introduce the semantic foundations of this framework and summarise their Isabelle formalisation as well as the resulting verification components. A series of simple examples shows our approach at work.

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

Abstract We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Abstract We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step, we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

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.

Concurrent Kleene algebras support equational reasoning about computing systems with concurrent behaviours. Their natural semantics is given by series(-parallel) rational pomset languages, a standard true concurrency semantics, which is often associated with processes of Petri nets. We use constructions on Petri nets to provide two decision procedures for such pomset languages motivated by the equational and the refinement theory of concurrent Kleene algebra. The contribution to the first problem lies in a much simpler algorithm and an EXPSPACE complexity bound. Decidability of the second, more interesting problem is new and, in fact, EXPSPACE-complete.

Abstract We present a principled modular approach to the development of construction and verification tools for imperative programs, in which the control flow and the data flow are cleanly separated. Our simplest verification tool uses Kleene algebra with tests for the control flow of while-programs and their standard relational semantics for the data flow. It is expanded to a basic program construction tool by adding an operation for the specification statement and one single axiom. To include recursive procedures, Kleene algebras with tests are expanded further to quantales with tests. In this more expressive setting, iteration and the specification statement can be defined explicitly and stronger program transformation rules can be derived. Programming our approach in the Isabelle/HOL interactive theorem prover yields simple lightweight mathematical components as well as program construction and verification tools that are correct by construction themselves. Verification condition generation and program construction rules are based on equational reasoning and supported by powerful Isabelle tactics and automated theorem proving. A number of examples shows our tools at work.

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 proposed for analysing the trajectories of dynamical or real-time systems. A Hoare logic can be 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.

Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential, and parallel composition, as well as finite and infinite iteration. Starting from a set-theoretic investigation, we propose axiom systems for multirelations in contexts ranging from bi-monoids to bi-quantales.

We reconstruct Peleg’s concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an axiomatization of concurrent dynamic algebras from that basis. In this context, sequential composition is not associative. It interacts with parallel composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg’s axioms are shown to be derivable in these algebras, and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg’s axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra by simple equational axioms for a domain and a codomain operation. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of programs and state transition systems. We develop the basic calculus, present the most interesting models and discuss some related theories. We demonstrate applicability by two examples: algebraic reconstructions of Noethericity and propositional Hoare logic based on equational reasoning.