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 lengtha0 b0 ... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a0 b0 ... an bn 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 2^k-1 from the vertex a0 to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2^{k+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.