Feasible Automata for Two-Variable Logic with Successor on Data Words

TL;DR

This paper introduces weak data automata, a restricted automata model for data words, which precisely captures a fragment of logic with successor and data equality, and analyzes their nonemptiness problem complexity.

Contribution

It defines weak data automata, establishes their logical correspondence, and proves the nonemptiness problem is decidable in 2-NEXPTIME, extending to omega-strings.

Findings

Weak data automata are less expressive than data automata.

Nonemptiness for weak data automata is decidable in 2-NEXPTIME.

Extension to omega-strings maintains the same complexity bound.

Abstract

We introduce an automata model for data words, that is words that carry at each position a symbol from a finite alphabet and a value from an unbounded data domain. The model is (semantically) a restriction of data automata, introduced by Bojanczyk, et. al. in 2006, therefore it is called weak data automata. It is strictly less expressive than data automata and the expressive power is incomparable with register automata. The expressive power of weak data automata corresponds exactly to existential monadic second order logic with successor +1 and data value equality \sim, EMSO2(+1,\sim). It follows from previous work, David, et. al. in 2010, that the nonemptiness problem for weak data automata can be decided in 2-NEXPTIME. Furthermore, we study weak B\"uchi automata on data omega-strings. They can be characterized by the extension of EMSO2(+1,\sim) with existential quantifiers for infinite sets. Finally, the same complexity bound for its nonemptiness problem is established by a nondeterministic polynomial time reduction to the nonemptiness problem of weak data automata.