Graph Rewriting Systems and their Application in Biology

Abstract

In sequential rewriting systems on strings, which are called Chomsky-grammars, derivation is defined as the replacement of a substring by another one according to a rewriting rule. The rest of the host string remains unchanged. Since their introduction in 1959, a broad theory has been developped (see e.g. [34]). Starting with [28] and [35], in the last years several authors have generalized this concept to get more or less complex rewriting systems, named graph or web grammars[1], [2], [10], [15], [21], [22], [23], [27], [29], [31], [36], and[38]. In keeping up the idea of sequential rewriting, in one derivation step on graphs only one subgraph is replaced by another one while the rest of the host graph remains unchanged (cf. Fig. 1). In each derivation step exactly one rewriting rule is applied. This rule has to specify which subgraph is to be replaced (left hand side), which subgraph has to be inserted Open image in new window (right hand side) and, furthermore, how the embedding (incoming and outgoing edges) of the left hand side is transformed if the right hand side is substituted for it. Thus a graph production for sequential rewriting is a triple $$ p = \left( {{d_i},{d_r},E} \right) $$ being graphs, namely the left and right hand side of p respectively, and E being any algorithmic specification for the transformation of the embedding. All approaches mentioned above differ mainly in the way the embedding transformation E is defined and which embedding manipulations the definition allows.