Parallel Coherent Graph Transformations

Abstract : Cellular automata as well as simultaneous assignments in Python can be understood as the parallel application of local rules to a grid or an environment that can be easily represented as an attributed graph. Since the result of such transformations cannot generally be obtained by a sequential application of the involved rules, this situation infringes the standard notion of parallel independence. An algebraic approach with production rules of the form L←K←I→R is adopted and a condition of parallel coherence more general than parallel independence is formulated, that enable the definition of the Parallel Coherent Transformation (PCT). This transformation supports a generalisation of the Parallelism Theorem in the theory of adhesive HLR categories, showing that the PCT yields the expected result of sequential rewriting steps when parallel independence holds. Categories of finitely attributed structures are proposed, in which PCTs are guaranteed to exist. These notions are introduced and illustrated on several detailed examples.