Deciding Definability by Deterministic Regular Expressions

International audienceWe investigate the complexity of deciding whether a given regular language can be defined with a deterministic regular expression. Our main technical result shows that the problem is Pspace-complete if the input language is represented as a regular expression or nondeterministic finite automaton. The problem becomes Expspace-complete if the language is represented as a regular expression with counters

Frontiers of tractability for typechecking simple XML transformations

AbstractTypechecking consists of statically verifying whether the output of an XML transformation is always conform to an output type for documents satisfying a given input type. We focus on complete algorithms which always produce the correct answer. We consider top–down XML transformations incorporating XPath expressions and abstract document types by grammars and tree automata. By restricting schema languages and transformations, we identify several practical settings for which typechecking can be done in polynomial time. Moreover, the resulting framework provides a rather complete picture as we show that most scenarios cannot be enlarged without rendering the typechecking problem intractable. So, the present research sheds light on when to use fast complete algorithms and when to reside to sound but incomplete ones

Separability by Short Subsequences and Subwords

The separability problem for regular languages asks, given two regular languages I and E, whether there exists a language S that separates the two, that is, includes I but contains nothing from E. Typically, S comes from a simple, less expressive class of languages than I and E. In general, a simple separator $S$ can be seen as an approximation of I or as an explanation of how I and E are different. In a database context, separators can be used for explaining the result of regular path queries or for finding explanations for the difference between paths in a graph database, that is, how paths from given nodes u_1 to v_1 are different from those from u_2 to v_2. We study the complexity of separability of regular languages by combinations of subsequences or subwords of a given length k. The rationale is that the parameter k can be used to influence the size and simplicity of the separator. The emphasis of our study is on tracing the tractability of the problem

Consistency of injective tree patterns

International audienceTesting if an incomplete description of an XML document is consistent, that is, if it describes a real document conforming to the imposed schema, amounts to deciding if a given tree pattern can be matched injectively into a tree accepted by a fixed automaton. This problem can be solved in polynomial time for patterns that use the child relation and the sibling order, but do not use the descendant relation. For general patterns the problem is in NP, but no lower bound has been known so far. We show that the problem is NP-complete already for patterns using only child and descendant relations. The source of hardness turns out to be the interplay between these relations: for patterns using only descendant we give a polynomial algorithm. We also show that the algorithm can be adapted to patterns using descendant and following-sibling, but combining descendant and next-sibling leads to intractability

Mu-Calculus Based Resolution of XPath Decision Problems

XPath is the standard declarative notation for navigating XML data and returning a set of matching nodes. In the context of XSLT/XQuery analysis, query optimization, and XML type checking, XPath decision problems arise naturally. They notably include XPath containment (whether or not for any tree the result of a particular query is included in the result of a second one), and XPath satisfiability (whether or not an expression yields a non-empty result), in the presence (or the absence) of XML DTDs. In this paper, we propose a unifying logic for XML, namely the alternation-free modal mu-calculus with converse. We show how to translate major XML concepts such as XPath and DTDs into this logic. Based on these embeddings, we show how XPath decision problems can be solved using a state-of-the-art EXPTIME decision procedure for mu-calculus satisfiability. We provide preliminary experiments which shed light, for the first time, on the cost of solving XPath decision problems in practice