Encoding graphs by derivations and implications for the theory of graph grammars

Abstract

A typical (notion of a sequential) graph grammar G consists of a f i n i t e set of labels z, a set of terminal labels A, (A ~ z) , a f i n i t e set of productions of form YI ~ Y2' where YI and are graphs (with labels from z) , and a s ta r t graph (or a f i n i t e set of s ta r t graphs). A der ivat ion step in G is performed as fol lows. Given a graph X and a production Y1 ~ from G, one locates a subgraph of X isomorphic to YI and replaces i t by a subgraph Y1⁄2 isomorphic to Y2 The crucial part of replacement is to establ ish connections between Y' and remainder of ×. , 2 The way that connections are established is specif ied by so-called embeddingmechanism which may be unique for whole grammar or i n t r i n s i c to each of productions. This embedding mechanism is rea l l y the heart of G. Often also appl icat ion conditions are added to productions in G roughly speaking, they specify which subgraphs of × that are isomorphic to Y1 may be replaced. The language generated by G is set of a l l graphs labeled by terminal labels only which can be derived from a s ta r t graph in one or more steps. (See Rosenfeld & Milgram, 19~2; Della Vigna & Ghezzi, 1978; Nagl, 1979; Ehrig, 1979; or Janssens & Rozenberg, 1980, 1982, for examples of d i f fe ren t types of graph grammars and embedding mechanisms.) We give here a somewhat informal presentation of a very simple idea which is well applicable to (almost) every graph grammar concept independently of embedding mechanism used. Given a graph X in a graph language generated by a graph grammar G, we encode this graph by encoding i t s der ivat ion. In general, such an endoding w i l l be more cOmplex than standard representation of X by i t s nodes, edges, and labels. However, i f der ivat ion of graph is reasonably short , then th is encoding outperforms standard representation. This simple observation has a number of implications for normal forms of graph grammars. In par t icu lar , we show that a graph grammar which generates a l l graphs ( la beled by some arb i t ra ry but f ixed set of labels) cannot be essent ia l l y growing.