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 ... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a₀b₀ ... a_nb_n as the vertices and (a₀, b₀),...,(a_n, b_n) 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 b_n. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets.