Pebble-Intervals Automata and FO2 with Two Orders.

We introduce a novel automata model, which we call 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 , and obtain as a corollary an automata-theoretic proof of the $$\textsc {ExpSpace}$$ upper bound for finite satisfiability due to Schwentick and Zeume.