Graph Reachability and Pebble Automata over Infinite Alphabets

Let D denote an infinite alphabet -- a set that consists of infinitely many symbols. A word w = a_0 b_0 a_1 b_1 ... a_n b_n of even length over D can be viewed as a directed graph G_w whose vertices are the symbols that appear in w, and the edges are (a_0,b_0),(a_1,b_1),...,(a_n,b_n). For a positive integer m, define a language R_m such that a word w = a_0 b_0 ... a_n b_n is in R_m if and only if there is a path in the graph G_w of length <= m from the vertex a_0 to the vertex b_n. We establish the following hierarchy theorem for pebble automata over infinite alphabet. For every positive integer k, (i) there exists a k-pebble automaton that accepts the language R_{2^k-1}; (ii) there is no k-pebble automaton that accepts the language R_{2^{k+1} - 2}. Based on this result, we establish a number of previously unknown relations among some classes of languages over infinite alphabets.