On the Regular Structure of Prefix Rewriting

We consider a pushdown automaton as a word-rewriting system with labelled rules applied only in a prefix way. The notion of pushdown transition graph is then extended to the notion of prefix transition graph generated by a word-rewriting system and accessible from a given axiom. Such accessible prefix transition graphs are context-free graphs in the sense of Muller and Schupp (1985), and we show that they are also the rooted pattern graphs of finite degree, where a pattern graph is a graph produced from a finite graph by iterating the addition of a finite family of finite graphs (the patterns). Furthermore, this characterization is effective in the following sense: any finite family of patterns generating a rooted graph G of finite degree, is mapped effectively into a word-rewriting system R such that the accessible prefix transition graph of R is isomorphic to G, and the reverse transformation is effective.