Homomorphisms on graph-walking automata

TL;DR

This paper explores how graph homomorphisms affect the recognizability of graph languages by graph-walking automata, revealing state complexity bounds and limitations in closure properties under certain homomorphisms.

Contribution

It establishes bounds on state increase for inverse homomorphic images in GWA and shows non-closure under injective homomorphisms for tree-walking automata.

Findings

Inverse homomorphic images require kn+O(1) states in the worst case.

GWA family is closed under inverse homomorphisms.

Tree-walking automata are not closed under injective homomorphisms.

Abstract

Graph-walking automata (GWA) are a model for graph traversal using finite-state control: these automata move between the nodes of an input graph, following its edges. This paper investigates the effect of node-replacement graph homomorphisms on recognizability by these automata. It is not difficult to see that the family of graph languages recognized by GWA is closed under inverse homomorphisms. The main result of this paper is that, for $n$-state automata operating on graphs with $k$ labels of edge end-points, the inverse homomorphic images require GWA with $kn+O(1)$ states in the worst case. The second result is that already for tree-walking automata, the family they recognize is not closed under injective homomorphisms. Here the proof is based on an easy homomorphic characterization of regular tree languages.