Two-variable logic on data trees and XML reasoning

Abstract

Motivated by reasoning tasks in the context of XML languages, the satisfiability problem of logics on data trees is investigated. The nodes of a data tree have a label from a finite set and a data value from a possibly infinite set. It is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result decidability of satisfiability and containment for a dataaware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

Alternation-free modal mu-calculus for data trees

“…In this section, we show that there are two-way translations between first-order logics with 2 variables over data trees which were considered in [4] and certain fragments of CTL with freeze quantification and one register. The main result in [4] is that finitary satisfiability for FO 2 (+1, ∼) is in 3NEXPTIME.…”

“…The main result in [4] is that finitary satisfiability for FO 2 (+1, ∼) is in 3NEXPTIME. By Theorem 17(b) below, it follows that finitary satisfiability for the fragment SCTL ↓,1 (+1) of µ ↓,1 ( , , , ) also is in 3NEXPTIME.…”

“…We show that FO 2 (+1, ∼), the first-order logic for which finitary satisfiability was proved decidable in [4], is as expressive as a certain fragment of the alternation-free modal µ-calculus with one register, which is incomparable with the forward fragment. Indeed, forward modal operators can be iterated by means of fixed points, whereas the "descendant" and "younger sibling" predicates are not available in FO 2 (+1, ∼).…”

“…The variables range over tree nodes, +1 stands for two predicates "child" and "next sibling", and < stands for two predicates "descendant" and "younger sibling". In [4], complexity of satisfiability over finite data trees was studied. For FO 2 (+1, ∼), it was shown to be in 3NEXPTIME, but for FO 2 (+1, <, ∼), to be at least as hard as nonemptiness of vector addition tree automata.…”

“…The only decidability result in [2,17] that allows negation and data does not allow axes which are recursive (such as "self or descendant") or between siblings. By representing XML documents as data trees and translating from XPath to FO 2 (+1, ∼), a decidable fragment with negation, data and all nonrecursive axes was obtained in [4]. Another fragment of XPath was considered in [18], but it lacks concatenation, recursive axes and sibling axes.…”