Abstract
We study a class of integrity constraints for tree-structured data modelled as data trees, whose nodes have a label from a finite alphabet and store a data value from an infinite data domain. The constraints require each tuple of nodes selected by a conjunctive query (using navigational axes and labels) to satisfy a positive combination of equalities and a positive combination of inequalities over the stored data values. Such constraints are instances of the general framework of XML-to-relational constraints proposed recently by Niewerth and Schwentick. They cover some common classes of constraints, including W3C XML Schema key and unique constraints, as well as domain restrictions and denial constraints, but cannot express inclusion constraints, such as reference keys. Our main result is that consistency of such integrity constraints with respect to a given schema (modelled as a tree automaton) is decidable. An easy extension gives decidability for the entailment problem. Equivalently, we show that validity and containment of unions of conjunctive queries using navigational axes, labels, data equalities and inequalities is decidable, as long as none of the conjunctive queries uses both equalities and inequalities; without this restriction, both problems are known to be undecidable. In the context of XML data exchange, our result can be used to establish decidability for a consistency problem for XML schema mappings. All the decision procedures are doubly exponential, with matching lower bounds. The complexity may be lowered to singly exponential, when conjunctive queries are replaced by tree patterns, and the number of data comparisons is bounded.
Highlights
Static analysis is an area of database theory that focuses on deciding properties of syntactic objects, like queries, integrity constraints, or data dependencies
Results of Björklund, Martens, and Schwentick give 2ExpTime upper bound for containment in unions of conjunctive queries (UCQs) over signav ∪ {∼} and UCQs over signav ∪ { }
They amount to an observation that in counter-examples to containment p ⊆ q, all data values can be set equal or different, except for a bounded number of them needed to witness satisfaction of p; such counter-examples can be encoded as trees over a finite alphabet, and recognized by an automaton evaluating p and q in the usual way
Summary
Static analysis is an area of database theory that focuses on deciding properties of syntactic objects, like queries, integrity constraints, or data dependencies. Non-mixing integrity constraints can be seen as a special case of the general framework of XML-to-relational constraints (X2R constraints) introduced by Niewerth and Schwentick [25] Within this framework they cover a wide subclass of functional dependencies, dubbed XKFDs, which are well suited for tree-structured data and include W3C XML Schema key and unique constraints [14], as well as absolute and relative XML keys by Arenas, Fan, and Libkin [2], and XFDs by Arenas and Libkin [3]. We present the decision procedure for consistency of non-mixing constraints (Section 3), followed by a detailed discussion of the entailment problem, the lower-complexity fragment, the relationships with existing constraint formalisms, and the two alternative interpretations of our results (Section 4). An appendix containing the missing proofs is available at: http://www.mimuw.edu.pl/ ~fmurlak/papers/concon.pdf
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