Decomposition of graph grammar productions and derivations

Abstract

Given a production p* in a graph grammar we consider the problem to find all productions p and p′ and all dependency relations R between p and p′ such that p* is equal to the concurrent production p*_Rp′. In view of the Concurrency Theorem — shown in an earlier paper — this means that there is a bijective correspondence between direct derivations G⇒ X via p* and R-related derivations G⇒ H⇒ X via (p,p′). We are able to give a general procedure for the decomposition of p*=p*_Rp′ which leads to all possible decompositions at least in the case of injective relations R. An important application of this decomposition theorem is the problem to find all possible decompositions of manipulation rules into atomic manipulation rules of a data base system. The theorem is proved within the framework of the algebraic theory of graph grammars using pushout and pullback techniques.