Observational trees as models for concurrency

Abstract

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

Generalising Conduché’s Theorem

“…Let us call it Tree. A Tree L -subcategory of it, namely Beh (see [6]) provides a useful model for the behavior of concurrent computing processes. Now, it is possible to prove (see [7]) that both Tree L -categories, Tree and Beh, enjoy a very nice property: they are equipped with a Conduché Tree L -functor into L viewed as a Tree L -category.…”

“…Tree A * provides models for processes whose behavior is supposed to evolve in a discrete time (see [6]), due to the fact that A * can be thought of as a set of partial functions from natural numbers to A.…”

“…In some companion works [7] we considered the case of Conduché's theorem for models for computational processes [6] and we proved that the analogue of Conduché condition for an enriched functor F is still sufficient to guarantee the existence of a right adjoint to the corresponding inverse image functor. This fact consists in a generalisation of Conduché's theorem from (Set-based) categories to categories based on LSymcat, where L is the lp 2-category [6] generated by a complete meet-semilattice monoid, in particular by a free monoid (see Section 5).…”

“…This fact consists in a generalisation of Conduché's theorem from (Set-based) categories to categories based on LSymcat, where L is the lp 2-category [6] generated by a complete meet-semilattice monoid, in particular by a free monoid (see Section 5). It is immediate that Set is a particular case of LSymcat, when L is the trivial free monoid generated by the empty set.…”

Conduché property and Tree-based categories

“…Trees of computations (viewed as enriched categories) are gathered in a monoidal category Tree, like the one described in [5], in which categories of processes are enriched.…”