Automata for two-variable logic over trees with ordered data values

TL;DR

This paper introduces ordered-data tree automata (ODTA) for processing data trees with ordered data values, extending existing models to include order comparisons and analyzing their logical and computational properties.

Contribution

It defines ODTA and weak ODTA models, provides logical characterisations, and analyzes their decidability and complexity, extending automata theory to ordered data domains.

Findings

ODTA's non-emptiness is decidable in 3-NEXPTIME.

Weak ODTA's non-emptiness is NP-complete.

Many existing formalisms are captured by weak ODTA.

Abstract

Data trees are trees in which each node, besides carrying a label from a finite alphabet, also carries a data value from an infinite domain. They have been used as an abstraction model for reasoning tasks on {XML} and verification. However, most existing approaches consider the case where only equality test can be performed on the data values. In this paper we study data trees in which the data values come from a linearly ordered domain, and in addition to equality test, we can test whether the data value in a node is greater than the one in another node. We introduce an automata model for them which we call ordered-data tree automata (ODTA), provide its logical characterisation, and prove that its non-emptiness problem is decidable in 3-NEXPTIME. We also show that the two-variable logic on unranked trees, studied by Bojanczyk, Muscholl, Schwentick and Segoufin in 2009, corresponds precisely to a special subclass of this automata model. Then we define a slightly weaker version of ODTA, which we call weak ODTA, and provide its logical characterisation. The complexity of the non-emptiness problem drops to NP. However, a number of existing formalisms and models studied in the literature can be captured already by weak ODTA. We also show that the definition of ODTA can be easily modified, to the case where the data values come from a tree-like partially ordered domain, such as strings.