Algebraic topology and concurrency

1. Introduction This article is intended to provide some new insights about concurrency theory using ideas from geometry, and more speciically from algebraic topology. The aim of the paper is twofold: we justify applications of geometrical methods in concurrency through some chosen examples and we give the mathematical foundations needed to understand the geometric phenomenon that we identify. In particular we show that the usual notion of homotopy has to be reened to take into account some partial ordering describing the way time goes. This gives rise to some new interesting mathematical problems as well as give some common grounds to computer-scientiic problems that have not been precisely related otherwise in the past. The organization of the paper is as follows. In Section 2 we explain to which extent we can use some geometrical ideas in computer science: we list a few of the potential or well known areas of application and try to exemplify some of the properties of concurrent (and distributed) systems we are interested in. We rst explain the interest of using some geometric ideas for semantical reasons. Then we take the example of concurrent databases with the problem of nding deadlocks and with some aspects of serializability theory. More general questions about schedules can be asked as well and related to some geometric considerations, even for scheduling micro-instructions (and not only coarse-grained transactions as for databases). The nal example is the one of fault-tolerant protocols for distributed systems, where subtle scheduling properties come into play. In Section 3 we give the rst few deenitions needed for modeling the topological spaces arising from Section 2. Basically, we need to deene a topological space containing all traces of executions of the concurrent systems we want to characterize plus the information about how time ows. This is the main diierence with standard topological reasoning in which there is no information about relation \in time" among points. The central notion here is that of a local po-space, which is a topological space with a local partial-order of time on it. Some examples are given, but we will only see in Section 6 that cubical complexes (or Higher-Dimensional Automata, 28] and 50]) give rise naturally to such spaces, hence most \combinatorial" concurrency models are instances of these local po-spaces. It is worth noting that some models in General Relativity 49] consider timed spaces, and the authors beneeted from some of these …