A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational model

Innocent strategies as presheaves and interactive equivalences for CCS

Seeking a general framework for reasoning about and comparing programming languages, we derive a new view of Milner's CCS. We construct a category E of plays, and a subcategory V of views. We argue that presheaves on V adequately represent innocent strategies, in the sense of game semantics. We then equip innocent strategies with a simple notion of interaction. This results in an interpretation of CCS. Based on this, we propose a notion of interactive equivalence for innocent strategies, which is close in spirit to Beffara's interpretation of testing equivalences in concurrency theory. In this framework we prove that the analogues of fair and must testing equivalences coincide, while they differ in the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014

Observational trees as models for concurrency

Given an automaton, its behaviour can be modelled as the sets of strings over an alphabet A that can be accepted from any of its states. When considering concurrent systems, we can see a concurrent agent as an automaton, where non-determinism derives from the fact that its states can offer a different behaviour at different moments in time. Non-deterministic computations between a pair of states can then no longer be described as a ‘set’ of strings in a free monoid. Consequently, between two states we will have a labelled structured set of computations, where the structure describes the possibility of two computations parting from each other while maintaining the same observable steps. In this paper, we shall consider different kinds of observation domains and related structured sets of computations. Structured sets of computations will be organised as a category of generalised trees built over a meet-semilattice monoid formalizing the observation domain. Theorems allowing us to introduce the usual concurrency operators in the models and relating different models will then be obtained by first considering ordinary functors (on and between the observation domains), and then lifting them to the categories of structured sets of computations

Una proprietà del comportamento per gli automi completi

Si dimostra che, considerando gli automi come categorie arricchite, il comportamento è una opfibrazione e la nota aggiunzione con la realizzazione vale anche in questo contesto più generale.It is shown that, in the categorical approach by which automata are enriched categories, behaviour turns out to be an opfibration and its adjointness to realization still holds in this enriched framework

Tree automata and enriched category theor

Si dimostra che un singolo automa ad albero può essere considerato come una categoria basata su un'opportuna bicategoria costruita a partire dagli alberi di input. In questo contesto si estende il teorema di aggiunzione locale fra realizzazione e comportamento.It is shown that tree automata can be described as cate¬gories enriched on a suitable base bicategory built up with input trees. In this setting the known theorem relating realization and behaviour by a local adjunction still holds true

A quasi-universal realization of automata

Si considera un approccio categoriale alla teoria degli automi nel quale la realizzazione di automi non deterministici risulta essere universale in un senso 2-categoriale (lax).We consider a categorical approach to the theory of automata by which the realization of non deterministic automata is universal in a 2-categorical sense (lax)

The Topos of Labelled Trees: A Categorical Semantics for SCCS

In this paper a we give a semantics for SCCS using the constructions of the topos of labelled trees. The semantics accounts for all aspects of the original formulation of SCCS, including unbounded non-determinism. Then, a partial solution to the problem of characterizing bisimulation in terms of a class of morphisms is proposed. We define a class of morphisms of the topos of trees, called conict preserving, such that two trees T and U are bisimilar iff there is a pair of conflict preserving morphisms f : T # U and g : U # T such that f g f = f and g f g = g. It is the first characterization which does not require the existence of a third quotient object. The results can be easily extended to more general transition systems

Generalising Conduché's theorem

In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009) we proved a form of Conduché's theorem for LSymcat-categories, where L was a meet-semilattice monoid. The original theorem was proved in Conduché (CR Acad Sci Paris 275:A891-A894, 1972) for ordinary categories. We showed also that the "lifting factorisation condition" used to prove the theorem is strictly related to the notion of state for processes whose semantics is modeled by LSymcat-categories. In this note we resume the content of Kasangian and Labella (J Pure Appl Algebra, 2009) in order to generalise the theorem to other situations, mainly arising from computer science. We will consider PSymcat-categories, where P is slightly more general than a meet-semilattice monoid, in which the lifting factorisation condition for a PSymcat-functor still implies the existence of a right adjoint to its corresponding inverse image functor. © 2009 Springer Science+Business Media B.V

Split extensions, semidirect product and holomorph of categorical groups

Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences