Graph Reachability and Pebble Automata over Infinite Alphabets.

We study the graph reachability problem as a language over an infinite alphabet. Namely, we view a word of even length a0b0 · · · anbn over an infinite alphabet as a directed graph with the symbols that appear in a0b0 ... anbn as the vertices and (a0, b0), . . . , (an, bn) as the edges. We prove that for any positive integer k, k pebbles are sufficient for recognizing the existence of a path of length 2k - 1 from the vertex a0 to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2k+1-2 from the vertex a0 to the vertex bn. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets. © 2009 IEEE.