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

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

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

Conduché property and Tree-based categories

AbstractThis paper focuses on a property of enriched functors reflecting the factorisation of morphisms, used in concurrency semantics. According to Lawvere [F.W. Lawvere, State categories and response functors, 1986, Unpublished manuscript], a functor strictly reflecting morphism factorisation induces a notion of state on its domain, when it is considered as a control functor. This intuition works both in case of physical and computing processes [M. Bunge, M.P. Fiore, Unique factorisation lifting functors and categories of linearly-controlled processes, Math. Structures Comput. Sci. 10 (2) 2000 137–163; M.P. Fiore, Fibered models of processes: Discrete, continuous and hybrid systems, in: Proc. of IFIP TCS 2000, in: LNCS, vol. 1872, 2000, pp. 457–473]. In this note we investigate a more general property in the family of models we proposed elsewhere for communicating processes, and we assess their bisimulation relations [S. Kasangian, A. Labella, Observational trees as models for concurrency, Math. Structures. Comput. Sci. 9 (1999) 687–718; R. De Nicola, D. Gorla, A. Labella, Tree-Functors, determinacy and bisimulations, Technical Report, 02/2006, Dip. di Informatica, Univ. di Roma “La Sapienza” (Italy), 2008 (submitted for publication), http://www.dsi.uniroma1.it/%7Egorla/papers/DGL-TR0206.pdf]. Hence, we adapt the notion of “Conduché condition” [F. Conduché, Au sujet de l’existence d’adjoints à droîte aux foncteurs image reciproque dans la catégorie des catégories, C. R. Acad. Sci. Paris 275 (1972) A891–894] to the context of enriched category theory. This notion, weaker than the original “Moebius condition” used by Lawvere, seems to be more suitable for the description of the concurrency models parametrised w.r.t. a base category via the mechanism of change of base, actually. The base category is a monoidal 2-category; a category of generalised trees, Tree, is obtained from it. We consider Conduché Tree-based categories, where enrichment reflects factorisation of objects in the base category. We prove that a form of Conduché’s theorem holds for Conduché Tree-functors. We also show how the Conduché condition plays a crucial role in modelling concurrent processes and bisimulations between them. The notions of “state preservation” and “determinacy” [R. Milner, Communication and Concurrency, Prentice Hall International, 1989] are formally characterised

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)

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

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