Paweł Parys

Generalization of Binary Search: Searching in Trees and Forest-Like Partial Orders

We extend the binary search technique to searching in trees. We consider two models of queries: questions about vertices and questions about edges. We present a general approach to this sort of problem, and apply it to both cases, achieving algorithms constructing optimal decision trees. In the edge query model the problem is identical to the problem of searching in a special class of tree-like posets stated by Ben-Asher et al. (1999). Our upper bound on computation time, O(n3 ), improves the previous best known O(n4 log3n). In the vertex query model we show how to compute an optimal strategy much faster, in O(n) steps. We also present an almost optimal approximation algorithm for another class of tree-like (and forest-like) partial orders

On the Significance of the Collapse Operation

We show that deterministic collapsible pushdown automata of second level can recognize a language which is not recognizable by any deterministic higher order pushdown automaton (without collapse) of any level. This implies that there exists a tree generated by a second level collapsible pushdown system (equivalently: by a recursion scheme of second level), which is not generated by any deterministic higher order pushdown system (without collapse) of any level (equivalently: by any safe recursion scheme of any level). As a side effect, we present a pumping lemma for deterministic higher order pushdown automata, which potentially can be useful for other applications.

XPath evaluation in linear time

We consider a fragment of XPath where attribute values can only be tested for equality. We show that for any fixed unary query in this fragment, the set of nodes that satisfy the query can be calculated in time linear in the document size.

Collapse Operation Increases Expressive Power of Deterministic Higher Order Pushdown Automata

We show that collapsible deterministic second level pushdown automata can recognize more languages than deterministic second level pushdown automata (without collapse). This implies that there exists a tree generated by a second level recursion scheme which is not generated by any second level safe recursion scheme.

The MSO+U Theory of (N, <) Is Undecidable

We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.

A Pumping Lemma for Pushdown Graphs of Any Level

We present a pumping lemma for the class of epsilon-contractions of pushdown graphs of level n, for each n. A pumping lemma was proposed by Blumensath, but there is an irrecoverable error in his proof; we present a new proof. Our pumping lemma also improves the bounds given in the invalid paper of Blumensath.

The (Almost) Complete Guide to Tree Pattern Containment

Tree pattern queries are being investigated in database theory for more than a decade. They are a fundamental and flexible query mechanism and have been considered in the context of querying tree structured as well as graph structured data. We revisit their containment, validity, and satisfiability problem, both with and without schema information. We present a comprehensive overview of what is known about the complexity of containment and develop new techniques which allow us to obtain tractability- and hardness results for cases that have been open since the early work on tree pattern containment. For the tree pattern queries we consider in this paper, it is known that the containment problem does not depend on whether patterns are evaluated on trees or on graphs. This means that our results also shed new light on tree pattern queries on graphs.

Minimization of Tree Pattern Queries

We investigate minimization of tree pattern queries that use the child relation, descendant relation, node labels, and wildcards. We prove that minimization for such tree patterns is Sigma2P-complete and thus solve a problem first attacked by Flesca, Furfaro, and Masciari in 2003. We first provide an example that shows that tree patterns cannot be minimized by deleting nodes. This example shows that the M-NR conjecture, which states that minimality of tree patterns is equivalent to their nonredundancy, is false. We then show how the example can be turned into a gadget that allows us to prove Sigma2P-completeness.

Two-way cost automata and cost logics over infinite trees

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

A non-deterministic recursion scheme recognizes a language of fi-nite trees. This very expressive model can simulate, among others, higher-order pushdown automata with collapse. We show decidability of the diagonal problem for schemes. This result has several interesting consequences. In particular, it gives an algorithm that computes the downward closure of languages of words recognized by schemes. In turn, this has immediate application to separability problems and reachability analysis of concurrent systems.Categories and Subject Descriptors 500 [Theory of computation]: Grammars and context-free languages; 500 [Theory of computation]: Tree languages; 500 [Theory of computation]: Regular languages