A Geometric Approach to the Problem of Unique Decomposition of Processes

This paper proposes a geometric solution to the problem of prime decomposability of concurrent processes first explored by R. Milner and F. Moller in [MM93]. Concurrent programs are given a geometric semantics using cubical areas, for which a unique factorization theorem is proved. An effective factorization method which is correct and complete with respect to the geometric semantics is derived from the factorization theorem. This algorithm is implemented in the static analyzer ALCOOL.Comment: 15 page

The Boolean Algebra of Cubical Areas as a Tensor Product in the Category of Semilattices with Zero

In this paper we describe a model of concurrency together with an algebraic structure reflecting the parallel composition. For the sake of simplicity we restrict to linear concurrent programs i.e. the ones with no loops nor branching. Such programs are given a semantics using cubical areas. Such a semantics is said to be geometric. The collection of all these cubical areas enjoys a structure of tensor product in the category of semi-lattice with zero. These results naturally extend to fully fledged concurrent programs up to some technical tricks.Comment: In Proceedings ICE 2014, arXiv:1410.701

Some Invariants of Directed Topology towards a Theoretical Base for a Static Analyzer Dealing with Fine-Grain Concurrency

We define the geometric models of conservative programs. Those models belong to a class of objects, the isothetic regions, that is contained in most of the categories introduced as framework for directed topology. We describe some invariants of directed topology and prove that they are well-behaved for isothetic regions. In particular, the class of isothetic regions satisfy a unique decomposition property that is related to parallelization of programs.On décrit les modèles géométriques des programmes concurrents dits conservatifs puis on montre que tous ces modèles appartiennent à une classe d'objets commune à plusieurs catégories ayant été introduites comme cadre de la topologie dirigée. On introduit divers invariants et on montre que pour tous les objets de cette classe, ces derniers ont un bon comportement. En particulier on montre que cette classe d'objets admet une propriété de décomposition unique et que l'on peut paralléliser un programme à partir de la décomposition de son modèle

Topologie Algébrique Dirigée et Concurrence

In order to analyze concurrency by means of algebraic topology, we study the properties of the category of partially ordered spaces. The "fundamental category" functor sends any such space to a small loop-free category whose size of the set of objects is way too large in respect with the information it contains. Thus we are led to define the notion of category of components of a small loop-free category and prove a theorem which establishes the interest of this definition as well as a "van Kampen like" theorem which leads to effective calculations. Then we are able to modelize programs written in PV langage (the original Dijkstra's one) : several examples are detailed.Afin d'étudier la concurrence au moyen de techniques issues de la topologie algébrique, on étudie les propriétés de la catégorie des espaces ordonnés. Le foncteur "catégorie fondamentale" associe à chaque tel espace une petite catégorie sans boucle, dont la taille de l'ensemble des objets est trop grand par rapport à l'information qu'elle contient. On définit alors la catégorie des composantes d'une petite catégorie sans boucle et l'on prouve un théorème qui justifie le bien fondé de cette définition ainsi qu'un théorème "à la van Kampen" qui ouvre la voie vers des calculs effectifs. On représente ainsi les programmes écrits en langage PV (on entend ici la version originale de Dijkstra) : plusieurs exemple sont traîtés

Géométrie des programmes conservatifs

The programs we consider are written in a restricted form of the language introduced by Dijkstra (1968). A program is said to be conservative when each of its loops restores all the resources it consumes. We define the geometric model of such a program and prove that the collection of directed paths on it is a reasonable overapproximation of its set of execution traces. In particular, two directed paths that are close enough with respect to the uniform distance result in the same action on the memory states of the system. The same holds for weakly dihomotopic directed paths. As a by-product, we obtain a notion of independence which is favourably compared to more common ones. The geometric models actually belong to a handy class of local pospaces whose elements are called isothetic regions. The local pospaces we use differ from the original ones, we carefully explain why the alternative notion should be preferred. The title intentionally echoes the article by Carson and Reynolds Jr. (1987).Les programmes auxquels on s'intéresse font partie d'un fragment du langage introduit par Dijkstra dans «Cooperating Sequential Processes» (1968). Un programme est dit conservatif lorsque chacune de ses boucles restaure les ressources qu'elle a utilisées. On construit le modèle géométrique d'un tel programme et on montre que la collection de ses chemins dirigés est une sur-approximation raisonnable de l'ensemble des traces d'exécution du programme. En particulier, deux chemins dirigés proches au regard de la norme uniforme produisent le même effet sur l'état mémoire du système. Le résultat est en valable pour deux chemins dirigés faiblement dihomotopes. Au passage, on introduit une notion d'indépendance qui se compare favorablement à deux autres, plus communes. Les modèles géométriques appartiennent a une classe d'espaces localement ordonnés particulièrement pratique, dont les éléments sont appelés régions isothétiques. Les espaces localement ordonnés auxquels nous avons recours diffèrent des originaux, nous expliquons avec soin pourquoi nous préférons la version alternative. Le titre fait intentionnellement écho à l'article «The Geometry of Semaphore Programs» de Carson et Raynolds (1987)

The Boolean Algebra of Cubical Areas as a Tensor Product in the Category of Semilattices with Zero

In this paper we describe a model of concurrency together with an algebraic structure reflecting the parallel composition. For the sake of simplicity we restrict to linear concurrent programs i.e. the ones with no loops nor branchings. Such programs are given a semantics using cubital areas that we call geometric. The collection of all these cubical areas enjoys a structure of tensor product in the category of semi-lattice with zero. These results naturally extend to fully fledged concurrent programs up to some technical tricks

CATEGORIES OF COMPONENTS AND LOOP-FREE CATEGORIES

Abstract. Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F (Definition 1.11) from G to its set of connected components (seen as a discrete category). From the observation that E and F differ unless G[x, x] = {idx} for every object x of G, we prove there is a fibered equivalence (Definition 1.12) from C[Σ-1] (Proposition 1.1) to C/Σ (Proposition 1.8) when Σ is

Colimits of local orders

International audienc

Unique decomposition of homogeneous languages and application to isothetic regions

International audienceA language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group Sn on the collection of homogeneous languages of length n ∈ N. One recovers the isothetic regions from (Haucourt (2017)) by considering the alphabet of connected subsets of the space |G|, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by Rn|G|) are co-unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections Rn|G| satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections Rn|G| to the geometric properties of the space |G|. On the way, the algebraic structure Rn|G| is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R1|G|