Some geometric perspectives in concurrency theory

Abstract

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.

Homological properties of non-deterministic branchings of mergings in higher dimensional automata

“…For other examples of similar investigations with different algebraic topological models of concurrency, cf. for example [9] [2] [8].…”

Modeling Slicer and Control Algorithm for Distributed Computation in Metrized Weaker Topological Spaces

“…Researchers have proposed to employ homotopy theory to model the complexity of mutual exclusion involving concurrently executing programs [6,16]. Interestingly, the stability properties of distributed concurrent processes can be analyzed by applying homotopy theory [15]. It is illustrated that, the semaphore objects (for mutual exclusions) can be formed in topological spaces having partial ordering and, it can be extended to analyze deadlock and serializability properties of concurrent processes [14].…”

“…The hybrid geodistributed computation is essentially asynchronous in nature [27,29]. The formal models of asynchronous distributed systems are constructed by employing algebraic topology and homotopy theory [5,6,10,15]. The algebraic topological models impose a set of very rigid and fixed geometric structures, whereas a more flexible structure would benefit the modeling of asynchronous distributed systems exposing finer states of control.…”

Events, Causality and Symmetry

“…The mathematical connections move us closer to topology and geometry and their use in reasoning about processes [31,47,48,61].…”