Structural Operational Semantics for Heterogeneously Typed Coalgebras

Concurrently interacting components of a modular software architecture are heterogeneously structured behavioural models. We consider them as coalgebras based on different endofunctors. We formalize the composition of these coalgebras as specially tailored segments of distributive laws of the bialgebraic approach of Turi and Plotkin. The resulting categorical rules for structural operational semantics involve many-sorted algebraic specifications, which leads to a description of the components together with the composed system as a single holistic behavioural system. We evaluate our approach by showing that observational equivalence is a congruence with respect to the algebraic composition operation

Structural Operational Semantics for Heterogeneously Typed Coalgebras

Concurrently interacting components of a modular software architecture are heterogeneously structured behavioural models. We consider them as coalgebras based on different endofunctors. We formalize the composition of these coalgebras as specially tailored segments of distributive laws of the bialgebraic approach of Turi and Plotkin. The resulting categorical rules for structural operational semantics involve many-sorted algebraic specifications, which leads to a description of the components together with the composed system as a single holistic behavioural system. We evaluate our approach by showing that observational equivalence is a congruence with respect to the algebraic composition operation.publishedVersio

Coalgebra for the working software engineer

Often referred to as ‘the mathematics of dynamical, state-based systems’, Coalgebra claims to provide a compositional and uniform framework to spec ify, analyse and reason about state and behaviour in computing. This paper addresses this claim by discussing why Coalgebra matters for the design of models and logics for computational phenomena. To a great extent, in this domain one is interested in properties that are preserved along the system’s evolution, the so-called ‘business rules’ or system’s invariants, as well as in liveness requirements, stating that e.g. some desirable outcome will be eventually produced. Both classes are examples of modal assertions, i.e. properties that are to be interpreted across a transition system capturing the system’s dynamics. The relevance of modal reasoning in computing is witnessed by the fact that most university syllabi in the area include some incursion into modal logic, in particular in its temporal variants. The novelty is that, as it happens with the notions of transition, behaviour, or observational equivalence, modalities in Coalgebra acquire a shape . That is, they become parametric on whatever type of behaviour, and corresponding coinduction scheme, seems appropriate for addressing the problem at hand. In this context, the paper revisits Coalgebra from a computational perspective, focussing on three topics central to software design: how systems are modelled, how models are composed, and finally, how properties of their behaviours can be expressed and verified.Fuzziness, as a way to express imprecision, or uncertainty, in computation is an important feature in a number of current application scenarios: from hybrid systems interfacing with sensor networks with error boundaries, to knowledge bases collecting data from often non-coincident human experts. Their abstraction in e.g. fuzzy transition systems led to a number of mathematical structures to model this sort of systems and reason about them. This paper adds two more elements to this family: two modal logics, framed as institutions, to reason about fuzzy transition systems and the corresponding processes. This paves the way to the development, in the second part of the paper, of an associated theory of structured specification for fuzzy computational systems

Involutive Categories and Monoids, with a GNS-correspondence

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive categories

A logic for complex computing systems: Properties preservation along integration and abstraction

International audienceIn a previous paper, we defined both a unified formal framework based on L.-S. Barbosa's components for modeling complex software systems, and a generic formalization of integration rules to combine their behavior. In the present paper, we propose to continue this work by proposing a variant of first-order fixed point modal logic to express both components and systems requirements. We establish the important property for this logic to be adequate with respect to bisimulation. We then study the conditions to be imposed to our logic (characterization of sub-families of formulas) to preserve properties along integration operators, and finally show correctness by construction results. The complexity of computing systems results in the definition of formal means to manage their size. To deal with this issue, we propose an abstraction (resp. simulation) of components by components. This enables us to build systems and check their correctness in an incremental way

A formal abstract framework for modelling and testing complex software systems

International audienceThe contribution of this paper is twofold: first, it defines a unified framework for modeling abstract components, as well as a formalization of integration rules to combine their behaviour. This is based on a coalgebraic definition of components, which is a categorical representation allowing the unification of a large family of formalisms for specifying state-based systems. Second, it studies compositional conformance testing i.e. checking whether an implementation made of correct interacting components combined with integration operators conforms to its specification

Bohrification

New foundations for quantum logic and quantum spaces are constructed by merging algebraic quantum theory and topos theory. Interpreting Bohr's "doctrine of classical concepts" mathematically, given a quantum theory described by a noncommutative C*-algebra A, we construct a topos T(A), which contains the "Bohrification" B of A as an internal commutative C*-algebra. Then B has a spectrum, a locale internal to T(A), the external description S(A) of which we interpret as the "Bohrified" phase space of the physical system. As in classical physics, the open subsets of S(A) correspond to (atomic) propositions, so that the "Bohrified" quantum logic of A is given by the Heyting algebra structure of S(A). The key difference between this logic and its classical counterpart is that the former does not satisfy the law of the excluded middle, and hence is intuitionistic. When A contains sufficiently many projections (e.g. when A is a von Neumann algebra, or, more generally, a Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be compared with the traditional quantum logic, i.e. the orthomodular lattice of projections in A. This time, the main difference is that the former is distributive (even when A is noncommutative), while the latter is not. This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in "Deep Beauty" (ed. H. Halvorson

Coalgebraic Components in a Many-Sorted Microcosm

The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming

Coalgebraic Components in a Many-Sorted Microcosm

Contains fulltext : 75330.pdf (author's version ) (Open Access)Algebra and Coalgebra in Computer Science : Third International Conference, CALCO 2009, 07 september 201