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 a
₀
b
₀
... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a
₀
b
₀
... a
_n
b
_n
as the vertices and (a
₀
, b
₀
),...,(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 a
₀
to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2
^k+1
- 2 from the vertex a
₀
to the vertex b
_n
. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets.