Stefano Kasangian

Bicategories of spans and relations

Abstract A new kind of bicategorical limit is used to characterize bicategories of the form Spant( E ) and Rel( E ) where in the former case E is a category with pullbacks and in the latter E is a regular category. The characterization of Rel( E ) differs from those in the literature which require involutions on the bicategories.

Enriched categorial semantics for distributed calculi

Abstract Algebraic models for distributed computations, synchronization and concurrency describe computations distributed among agents performing ‘local’ tasks and their interactions as well, which allow sharing and exchanging information for the solution of a common problem. Many calculi have been introduced for this purpose, like for instance Milners CCS and SCCS. In our approach, we describe composition of computations as a tensor product in a suitable monoidal category, generalizing the free monoid generated by the alphabet A of elementary actions. The interactive aspect, namely the synchronization operation, is given by a monoidal functor ∗, which by its very definition exhibits the interchange law between the two operations. Several examples of this kind of monoidal category are introduced. The typical one, treated in full detail, is the topos BT of budding trees , constructed out of the topos of labelled A-trees T / F ( A ) and the (base) monoidal closed category of nice trees NT , which provides the semantics of computations , and several process semantics as categories enriched over it. Good semantics are characterized in terms of a notion of hereditary fullness and computations are synchronized considering the ‘effect’ of the monoidal functor ∗. It is then possible to describe operators such as relabelling , restriction and insertion-deletion of an idle move , showing that our set-up covers the classical semantics for synchronization calculi, like for instance Milners CCS and Bergstra and Klops synchronization algebras. Further, we introduce a distinction between those structures (and properties) which make sense at the level of computations and those which occur at the level of process semantics. Although the stress is generally on the enriched-categorical structure, nice topos-theoretical aspects arise, connected with Benabous work on the topos of trees.

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.

Applications of the Calculus of Trees to Process Description Languages

Benabous notion of motor is extended to cover labelled finite trees. Operations on them are defined that permit to easily define the semantics for a finite concurrent calculus. Then, suitable motors that constructively define canonical representatives for strong and observational congruence based on the notion of bisimulation are introduced in a clean and straightforward way. This enables us to provide the calculus with a fully abstract semantics up to the above congruences.

Enriched Categories for Local and Interaction Calculi

The construction of models for distributed computations plays a very important role in designing and developing parallel computing systems. Various algebraic approaches have been proposed in the past as, for instance, the communicating computing agents of [Mil80], [BeK85], and [BHR84].

On Continuous Time Agents

Continuous time agents are studied in an enriched categorical framework that allows for a comprehensive treatment of both the interleaving and the true concurrent paradigms in parallelism. The starting point is a paper by Cardelli, where actions have a duration in a (dense) time domain. More recent works are also briefly considered and some possible directions towards timed “true concurrent” processes are indicated.

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.

Liver system. II - A categorical coarse model for its autoisodiasostasis

SummaryIn the context of the system theoretic theory of the liver, a new categorical approach is introduced, by which autoisodiasostasis is characterized in terms of the splitting of certain idempotents. Equifinal behaviors are then described in this setting.