Detecting Deadlocks in Concurrent Systems

A local po-space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, "spatial" and "temporal", leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local po-spaces.We introduce here a particular kind of local po-space, the "globular CW-complexes", for which we formalize these notions of deformations and which are sufficient to formalize higher dimensional automata. The existence of the category of globular CW-complexes was already conjectured in [11].After localizing the category of globular CW-complexes by spatial and temporal deformations, we get a category (the category of dihomotopy types) whose objects up to isomorphism represent exactly the higher dimensional automata up to deformation. Thus globular CW-complexes provide a rigorous mathematical foundation to study from an algebraic topology point of view higher dimensional automata and concurrent computations.

Section: Proposition 38 Let M Be a Hda The Following Diagram Is Comentioning

confidence: 99%

“…are compact Hausdorff spaces with a closed (global) partial order. More general topological models are needed in general, in which the partial order is only defined locally, and have been introduced and motivated in [7], [5] and [6]. The precise definitions and properties are given in Section 3.1.…”

Section: Introductionmentioning

confidence: 99%

Topological deformation of higher dimensional automata

Gaucher

Goubault

2003

“…1. For so-called semaphore programs (explained below), these progress graphs have been exploited for an algorithmic determination of deadlocs and unreachable states [23,5,9]. A systematic framework for studying schedules of actions of distributed computations by means of geometric properties was proposed by V. Pratt [25] and subsequently R. van Glabbeek [30].…”

Section: Background and Historymentioning

confidence: 99%

“…Even if the po-space X does not have any deadlock point x (i.e., π 1 (X)(x, X 1 ) = ∅ for all x ∈ X, cf. [23,5,9]), the mapping spaces very often have lots of them. If X is the po-space from the left part of Fig.…”

Section: Higher Homotopy Categoriesmentioning

confidence: 99%

State spaces and dipaths up to dihomotopy

Raussen

2003

Self Cite

Geometric models have been used by several authors to describe the behaviour of concurrent sytems in computer science. A concurrent computation corresponds to an oriented path (dipath) in a (locally) partially ordered state space, and di homotopic dipaths correspond to equivalent computations. This paper studies several invariants of the state space in the spirit of those of algebraic topology, but taking partial orders into account as an important part of the structure. We use several categories of fractions of the fundamental category of the state space and define and investigate the related quotient categories of "components". For concurrency applications, the resulting categories can be interpreted as a dramatic reduction of the size of the state space to be considered.

“…The "topological" formalization that follows is one of the most recent ones, and essentially dates back to [14] and [15], but is based on much older results [10].…”

Section: • S Is a Set Of States With Initial State I • L Is A Set Of mentioning

confidence: 99%

Some geometric perspectives in concurrency theory

Goubault

2003

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of "geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the "direction" of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: "directed" or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ω-categorical and topological techniques.