On Incomplete XML Documents with Integrity Constraints

Abstract. We consider incomplete specifications of XML documents in the presence of schema information and integrity constraints. We show that integrity constraints such as keys and foreign keys affect consistency of such specifications. We prove that the consistency problem for incomplete specifications with keys and foreign keys can always be solved in NP. We then show a dichotomy result, classifying the complexity of the problem as NP-complete or PTIME, depending on the precise set of features used in incomplete descriptions.

Frameworks for logically classifying polynomial-time optimisation problems.

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems

Logics for Unranked Trees: An Overview

Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to relate complex properties of trees to the existence of tree automata for those properties. Furthermore, logics differ significantly in their model-checking properties, their automata models, and their behavior on ordered and unordered trees. In this paper we present a survey of logics for unranked trees

Descriptive Complexity of Deterministic Polylogarithmic Time and Space

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. We prove that the inflationary and partial fixed point vartiants of this logic capture PolylogTime and PolylogSpace, respectively. In the course of proving that our logic indeed captures PolylogTime on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of a structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our PolylogTime logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.Comment: Submitted to the Journal of Computer and System Science

Randomisation and Derandomisation in Descriptive Complexity Theory

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures

Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries

Fragments of bag relational algebra: Expressiveness and certain answers

While all relational database systems are based on the bag data model, much of theoretical research still views relations as sets. Recent attempts to provide theoretical foundations for modern data management problems under the bag semantics concentrated on applications that need to deal with incomplete relations, i.e., relations populated by constants and nulls. Our goal is to provide a complete characterization of the complexity of query answering over such relations in fragments of bag relational algebra. The main challenges that we face are twofold. First, bag relational algebra has more operations than its set analog (e.g., additive union, max-union, min-intersection, duplicate elimination) and the relationship between various fragments is not fully known. Thus we first fill this gap. Second, we look at query answering over incomplete data, which again is more complex than in the set case: rather than certainty and possibility of answers, we now have numerical information about occurrences of tuples. We then fully classify the complexity of finding this information in all the fragments of bag relational algebra

On the Hybrid Extension of CTL and CTL+

The paper studies the expressivity, relative succinctness and complexity of satisfiability for hybrid extensions of the branching-time logics CTL and CTL+ by variables. Previous complexity results show that only fragments with one variable do have elementary complexity. It is shown that H1CTL+ and H1CTL, the hybrid extensions with one variable of CTL+ and CTL, respectively, are expressively equivalent but H1CTL+ is exponentially more succinct than H1CTL. On the other hand, HCTL+, the hybrid extension of CTL with arbitrarily many variables does not capture CTL*, as it even cannot express the simple CTL* property EGFp. The satisfiability problem for H1CTL+ is complete for triply exponential time, this remains true for quite weak fragments and quite strong extensions of the logic

Queries with Arithmetic on Incomplete Databases

The standard notion of query answering over incomplete database is that of certain answers, guaranteeing correctness regardless of how incomplete data is interpreted. In majority of real-life databases, relations have numerical columns and queries use arithmetic and comparisons. Even though the notion of certain answers still applies, we explain that it becomes much more problematic in situations when missing data occurs in numerical columns. We propose a new general framework that allows us to assign a measure of certainty to query answers. We test it in the agnostic scenario where we do not have prior information about values of numerical attributes, similarly to the predominant approach in handling incomplete data which assumes that each null can be interpreted as an arbitrary value of the domain. The key technical challenge is the lack of a uniform distribution over the entire domain of numerical attributes, such as real numbers. We overcome this by associating the measure of certainty with the asymptotic behavior of volumes of some subsets of the Euclidean space. We show that this measure is well-defined, and describe approaches to computing and approximating it. While it can be computationally hard, or result in an irrational number, even for simple constraints, we produce polynomial-time randomized approximation schemes with multiplicative guarantees for conjunctive queries, and with additive guarantees for arbitrary first-order queries. We also describe a set of experimental results to confirm the feasibility of this approach