66 research outputs found
The Wadge Hierarchy of Deterministic Tree Languages
We provide a complete description of the Wadge hierarchy for
deterministically recognisable sets of infinite trees. In particular we give an
elementary procedure to decide if one deterministic tree language is
continuously reducible to another. This extends Wagner's results on the
hierarchy of omega-regular languages of words to the case of trees.Comment: 44 pages, 8 figures; extended abstract presented at ICALP 2006,
Venice, Italy; full version appears in LMCS special issu
Weak index versus Borel rank
We investigate weak recognizability of deterministic languages of infinite
trees. We prove that for deterministic languages the Borel hierarchy and the
weak index hierarchy coincide. Furthermore, we propose a procedure computing
for a deterministic automaton an equivalent minimal index weak automaton with a
quadratic number of states. The algorithm works within the time of solving the
emptiness problem
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Reasoning About Integrity Constraints for Tree-Structured Data
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
Static Analysis of Graph Database Transformations
We investigate graph transformations, defined using Datalog-like rules based
on acyclic conjunctive two-way regular path queries (acyclic C2RPQs), and we
study two fundamental static analysis problems: type checking and equivalence
of transformations in the presence of graph schemas. Additionally, we
investigate the problem of target schema elicitation, which aims to construct a
schema that closely captures all outputs of a transformation over graphs
conforming to the input schema. We show all these problems are in EXPTIME by
reducing them to C2RPQ containment modulo schema; we also provide matching
lower bounds. We use cycle reversing to reduce query containment to the problem
of unrestricted (finite or infinite) satisfiability of C2RPQs modulo a theory
expressed in a description logic
Stackless Processing of Streamed Trees
International audienceProcessing tree-structured data in the streaming model is a challenge: capturing regular properties of streamed trees by means of a stack is costly in memory, but falling back to finite-state automata drastically limits the computational power. We propose an intermediate stackless model based on register automata equipped with a single counter, used to maintain the current depth in the tree. We explore the power of this model to validate and query streamed trees. Our main result is an effective characterization of regular path queries (RPQs) that can be evaluated stacklessly-with and without registers. In particular, we confirm the conjectured characterization of tree languages defined by DTDs that are recognizable without registers, by Segoufin and Vianu (2002), in the special case of tree languages defined by means of an RPQ
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
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