Parallel Graph Rewriting with Overlapping Rules

We tackle the problem of simultaneous transformations of networks represented as graphs. Roughly speaking, one may distinguish two kinds of simultaneous or parallel rewrite relations over complex structures such as graphs: (i) those which transform disjoint subgraphs in parallel and hence can be simulated by successive mere sequential and local transformations and (ii) those which transform overlapping subgraphs simultaneously. In the latter situations, parallel transformations cannot be simulated in general by means of successive local rewrite steps. We investigate this last problem in the framework of overlapping graph transformation systems. As parallel transformation of a graph does not produce a graph in general, we propose first some sufficient conditions that ensure the closure of graphs by parallel rewrite relations. Then we mainly introduce and discuss two parallel rewrite relations over graphs. One relation is functional and thus deterministic, the other one is not functional for which we propose sufficient conditions which ensure its confluence.