Pebble-Intervals Automata and FO2 with Two Orders (Extended Version)

TL;DR

This paper introduces pebble-intervals automata (PIA), a new automata model for a decidable fragment of first-order logic involving data values, providing automata-theoretic insights and complexity bounds.

Contribution

The paper presents pebble-intervals automata (PIA), a novel automata model tailored for a specific FO fragment with data values, and proves its relevance for automata-theoretic decision procedures.

Findings

PIA can accept string projections of data word languages definable in the logic.

Automata-theoretic proof of EXPSPACE upper bound for finite satisfiability.

Establishment of closure properties and power of PIA.

Abstract

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.