A case for po-manifolds — in chase after a good topological model for concurrency
Stefan Sokołowski · Electronic Notes in Theoretical Computer Science · 2003
It has repeatedly been argued that the semi-cubical complexes3 , which derive directly from the higher-dimensional automata, are as general a model for concurrency as one may ever need. In fact, most (but not all) classical examples of interacting concurrent processes have an adequate semi-cubical description. I believe this is only due to the fact that most classical examples are built around the notion of discrete event, or action, which makes the transition from the example's idea to higher-dimensional automata, to semi-cubical complexes, very natural. But it is not difficult to provide realistic examples of concurrency not based on such discrete events, which leads to local po-spaces with no “cubification”. Although the concept of local po-space is very general and admits many pathologies, the examples of concurrency I have in mind are well-behaved and not a pathology at all, so the generality of the local po-spaces is for them a large overkill. Still, they do not fit into the semi-cubical strait jacket. This report puts forward the notion of po-manifold, a subcategory of the local po-spaces slightly bigger than the semi-cubical complexes. A po-manifold is locally homeomorphic to a block, which is an extremely nice global po-space. The local homeomorphisms satisfy a simple consistency condition. This means the po-manifolds are “locally nice” even though they may be too smooth to admit a cubification.