Geometric and Topological Methods in Concurrency Theory

Inmodern computers (or computer networks), calculations are typically no longer perfomed sequentially (one step at a time) but in cooperation between several processors doing parts of the job in parallel. Through concurrency, one aims to save time and memory. To do so, the proper scheduling of processors is a crucial task. Some of the issues involved are to make sure that the calculation does not stop (no deadlocks), and that the result is correct for all possible schedules. In many instances, it seems to be difficult to prove the correctness of a concurrent program under all possible circumstances, and this is certainly one of the reasons why concurrency is not applied as much as one would hope for, in particular in critical software. One of the difficulties is to obtain feasible mathematical models for concurrent computations. Like for sequential computation, one starts with transition models of a graph-theoretic nature. These have been extended and applied in an area called process algebra, with undeniable success. Still, it is difficult to avoid the so-called state space explosion problem: in most models, the number of states grows exponentially in the number of processors, and it can therefore be hard to handle the complexity of the state space. It has hence been important to be able to reduce the number of states by considering states with the same “potential” as being equivalent. How can one formulate this potential in mathematical terms? Here it seems, that ideas and methods from the geometric/topological world can play a role. A small but growing group of researchers on the interface between topology and theoretical computer science has been working on concurrency models that allow one to formulate interesting notions with the potential to facilitate the future development of algorithms. To say it briefly, one replaces a huge discrete state space by a continuous model, whose main additional feature is the existence of directed paths (dipaths) corresponding to transitions in the original model. It is important that executions (transitions), that are equivalent for structural reasons – not only by chance – can be deformed into each other by a geometric process (oriented homotopy or dihomotopy).