Containment of pattern-based queries over data trees

Abstract

We study static analysis, in particular the containment problem, for analogs of conjunctive queries over XML documents. The problem has been studied for queries based on arbitrary patterns, not necessarily following the tree structure of documents. However, many applications force the syntactic shape of queries to be tree-like, as they are based on proper tree patterns. This renders previous results, crucially based on having non-tree-like features, inapplicable. Thus, we investigate static analysis of queries based on proper tree patterns. We go beyond simple navigational conjunctive queries in two ways: we look at unions and Boolean combinations of such queries as well and, crucially, all our queries handle data stored in documents, i.e., we deal with containment over data trees.We start by giving a general Π p 2 upper bound on the containment of conjunctive queries and Boolean combinations for patterns that involve all types of navigation through documents. We then show matching hardness for conjunctive queries with all navigation, or their Boolean combinations with the simplest form of navigation. After that we look at cases when containment can be witnessed by homomorphisms of analogs of tableaux. These include conjunctive queries and their unions over child and next-sibling axes; however, we show that not all cases of containment can be witnessed by homomorphisms. We look at extending tree patterns used in queries in three possible ways: with wildcard, with schema information, and with data value comparisons. The first one is relatively harmless, the second one tends to increase complexity by an exponential, and the last one quickly leads to undecidability.

Pattern logics and auxiliary relations

“…Such a pattern is a small subtree or a subgraph that needs to have a match in the large database. On top of it, one might impose data values [2,8,12,17,34], for instance by means of the data equality predicate [14][15][16]41]. An example of a pattern is the tree in Fig.…”

“…Patterns as those used for querying XML documents with data values are indeed such: one specifies a small connected part of the tree, and then adds an arbitrary data equality relation over it, see [8,12,17,36,41].…”

“…Query languages considered for incomplete XML documents are usually pattern logics [2,8,13,17] or their fragments. As explained in section 5.1, it is crucial for us to know when checking A |= ϕ is equivalent to checking A |= ϕ, i.e., when finding certain answers can be replaced by straightforward query evaluation.…”

“…While some useful queries, in particular some fragments of XPath, can be captured by it [35], more expressive fragments of XPath, and crucially, more expressive queries that can use more complex patterns [3,17,35,36,44] are not captured at all by restricting the number of variables. At the same time, many such querying mechanisms are based on patterns, so perhaps one can recover decidability using pattern logics?…”

“…A key notion in many of them is that of a pattern: essentially a small subgraph, or a subtree, that needs to be matched in a large data set. These often provide the basis for language constructors, for instance, for the navigational language XPath [35,36], more general XML query languages [3,13,17], or path queries over graphs [7,23]. Patterns also provide a standard abstraction of incomplete data in relational, XML, and graph models [1,2,8,9,29].…”