Filtering unsatisfiable XPath queries

Abstract

The satisfiability test checks, whether or not the evaluation of a query returns the empty set for any input document, and can be used in query optimization for avoiding the submission and the computation of unsatisfiable queries. Thus, applying the satisfiability test before executing a query can save processing time and query costs. We focus on the satisfiability problem for queries formulated in the XML query language XPath, and propose a schema-based approach to the satisfiability test of XPath queries, which checks whether or not an XPath query conforms to the constraints in a given schema. If an XPath query does not conform to the constraints given in the schema, the evaluation of the query will return an empty result for any valid XML document. Thus, the XPath query is unsatisfiable. We present a complexity analysis of our approach, which proves that our approach is efficient for typical cases. We present an experimental analysis of our developed prototype, which shows the optimization potential of avoiding the evaluation of unsatisfiable queries.

Introduction

XPath (see [29], [30]) is either a standalone XML query language or is embedded in other XML languages (e.g. XSLT, XQuery, XLink and XPointer) for specifying node sets in XML documents. An important issue in XPath evaluation is the satisfiability problem of XPath queries. An XPath query Q is unsatisfiable, if the evaluation of Q on any XML document returns every time an empty result. Therefore, the satisfiability test of XPath queries plays a critical role in query optimization. The application of the satisfiability test can avoid the submission and the unnecessary evaluation of unsatisfiable queries, and thus saving processing time and query cost. As well as for query optimization, the XPath satisfiablity test is also important for consistency problems, e.g. XML access control [6] and type-checking of transformations [19]. Therefore, many research efforts focus on the satisfiability test of XPath queries with or without respect to schemas, e.g. [2], [10], [11], [14], [17], [18].

In the absence of schemas, the satisfiability test can detect that the structure properties of XPath queries are inconsistent with the XML data model (e.g. [14]). For example, the XPath query Q1 = /parent::a is unsatisfiable, because the document node has no parent node according to the XML data model. The query Q2 = //regions/america is tested as a satisfiable XPath query without respect to a schema. However, according to a given schema, e.g. the schema given in [8], the element regions can have children, which are called namerica and samerica, but does not have children with name america. Therefore, Q2 is unsatisfiable with respect to the given schema. Thus, we can detect more errors in XPath queries if we additionally consider schema information. Therefore, we focus on the satisfiability test of XPath in the presence of schemas.

The most widely used schema languages are XML Schema (see [31], [32]) and DTD (see [28]). In this paper, we focus on XML Schema for the definition of schemas. As well as imposing the constraints of the structure and semantics on XML documents as DTDs do, the XML Schema language provides powerful capabilities for specifying data types on elements and attributes, most of which are not expressible in DTDs. The XML Schema language provides a large number of built in simple types and allows deriving new types for the values of elements and attributes, which are only specified to be character data in DTDs. Thus, if the types of values of elements or attributes in an XPath query do not conform to constraints specified in the XML Schema definition, the XPath query selects an empty set of nodes for any valid XML document. For example, the query meeting[@date = ˈ01-05-06ˈ] does not retrieve anything if the type of the attribute date is declared to have the format DD-MM-YYYY. Therefore, the powerful data-typing facilities supported by XML Schema provide another dimension for the satisfiability test of XPath queries. Since XML Schema can express more restrictions than a DTD, a DTD can be easily transformed into an XML Schema representation, but in general, an XML Schema definition cannot be transformed into a DTD without loosing information. To the best of our knowledge, existing work only deals with DTDs except our previous contributions (see [10], [11]).

Our schema-based approach checks whether or not an XPath query Q conforms to the structure, semantics, data type and occurrence constraints in a given XML schema definition S by evaluating Q on S rather than the instance documents of S. If Q does not conform to the constraints of S, Q cannot be evaluated completely on S, and thus Q is unsatisfiable. For schemas, our approach supports recursive as well as non-recursive schemas, considers a significant part of the XML Schema language and allows arbitrary nesting and references of model groups. For XPath, our approach allows all XPath axes and negation operations in predicates. The satisfiability test for the XPath subset supported by our approach in the presence of the schemas supported by our approach is undecidable (see [2]). Therefore, we present an incomplete, but fast satisfiability tester, i.e. if our tester returns unsatisfiable, then we are sure that the XPath query is unsatisfiable, but if our tester returns maybe satisfiable, then the XPath query may be satisfiable or may be unsatisfiable. Note that we do not loose correctness in the proposed application scenarios of our satisfiability tester when using an incomplete tester.

This paper is an extended version of [10], [11]. We extend the contributions of [10], [11] by significantly extending the supported subset (see Section 2.2) of the XML Schema language, allowing various content models of elements and arbitrary nesting of model groups; by supporting the type-checking of values of elements and attributes (see Section 4.5) and the checking of occurrence constraints (see Section 4.6); by integrating all new contributions into the prototype of [11] and by additional experiments (see Section 6).

The rest of the paper is organized as follows: Section 2 describes the supported subsets of XPath and XML Schema. Section 3 develops a data model for XML Schema. This data model for XML Schema is the basis for our XPath–XSchema evaluator (see Section 4), which evaluates XPath queries on XML Schema definitions in order to compute the schema paths of the queries. Section 4 also includes a complexity analysis of the approach. Section 5 discusses the satisfiability test of XPath based on the schema paths. We present a comprehensive performance analysis in Section 6. Section 7 deals with further related work. We end up with the summary and conclusions in Section 8.