Homomorphisms and inverse homomorphisms on graph-walking automata

Abstract

Graph-walking automata analyze an input graph by moving between its nodes, following the edges. This paper investigates the effect of node-replacement graph homomorphisms and inverse homomorphisms on recognizability by these automata. For deterministic graph-walking automata, it is shown that the family of graph languages they recognize is closed under inverse homomorphisms: for an n-state automaton, the inverse homomorphic images of the graphs it accepts can be recognized by an automaton with at most kn+1 states, where k is the number of labels of edge end-points in the pre-image graphs. At the same time, it is proved that in the worst case these inverse homomorphic images require a deterministic graph-walking automaton with at least kn states. The upper bound kn+1 also holds for nondeterministic graph-walking automata. The second result is that already for tree-walking automata, both deterministic and nondeterministic, the families they recognize are not closed under injective homomorphisms. Here the proof is based on a new homomorphic characterization of regular tree languages: every regular tree language is representable as h−1(g(all trees)), for some injective node-replacement homomorphisms g and h.