  We introduce a novel automata model, called pebble-intervals automata (PIA),
and study its power and closure properties. PIAs are tailored for a decidable
fragment of FO that is important for reasoning about structures that use data
values from infinite domains: the two-variable fragment with one total preorder
and its induced successor relation, one linear order, and an arbitrary number
of unary relations. We prove that the string projection of every language of
data words definable in the logic is accepted by a pebble-intervals automaton
A, and obtain as a corollary an automata-theoretic proof of the EXPSPACE upper
bound for finite satisfiability due to Schwentick and Zeume.
