We represent a set of possible worlds using an incomplete information database. The representation techniques that we study range from the very simple Codd-table (a relation over constants and uniquely occurring variables called nulls) to much more complex mechanisms involving views of conditioned-tables (programs applied to Coddtables augmented by equality and inequality conditions). (1) We provide matching upper and lower bounds on the data-complexity of testing containment, membership, and uniqueness for sets of possible worlds. We fully classify these problems with respect to our representations. (2) We investigate the data-complexity of querying incomplete information databases for both possible and certain facts. For each fixed positive existential query on conditioned-tables we present a polynomial time algorithm solving the possible fact problem. We match this upper bound by two NP-completeness lower bounds, when the fixed query contains either negation or recursion ...

We introduce the natural class SP 2 containing those languages which maybe ex-P pressed in terms of two symmetric quanti ers. This class lies between 2 and P 2 \ P 2 and naturally generates a \symmetric " hierarchy corresponding to the polynomial-time hierarchy. Wedemonstrate, using the probabilistic method, new containment theorems for BPP. Weshow that MA (and hence BPP) lies within SP 2, improving the con-P structions of [10, 8] (which show that BPP

We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.

| Research in non-monotonic reasoning has focused largely on the idea of representing knowledge about the world via rules that are generally true but can be defeated. Even if relational databases are nowadays the main tool for storing very large sets of data, the approach of using non-monotonic AI formalisms as relational database query languages has been investigated to a much smaller extent. In this work we propose a novel application of Reiter's default logic by introducing a default query language (DQL) for nite relational databases, which is based on default rules. The main result of this paper is that DQL is as expressive as SO 98 , the existential-universal fragment of secondorder logic. This result is not only of theoretical importance: We exhibit queries {which are useful in practice{ that can be expressed with DQL and can not with other query languages based on non-monotonic logics such as DATALOG with negation under the stable model semantics. In particular, we show that DQ...

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: We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using first-order formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties. To appear in Information and Computation. Research partially supported by NSF Grants CCR8905038 and CCR-9108631. y e-mail addresses: kolaitis@cse.ucsc.edu, thakur@cse.ucsc.edu z supersedes Technical report UCSC-CRL-90-48 1 Introduction and Summary of Results It is well known that optimization problems had a major influence on the development of the theory of NP-co...

We investigate the complexity of query processing in the logical data model (LDM). We use two measures: data complexity, which is complexity with respect to the size of the data, and expression complexity, which is complexity with respect to the size of the expressions denoting the queries. Our investigation shows that while the operations of product and union are essentially first-order operations, the power set operation is inherently a higher-order operation and is exponentially expensive. We define a hierarchy of queries based on the depth of nesting of power set operations and show that this hierarchy corresponds to a natural hierarchy of Turing machines that run in multiply exponential time.

. We consider uniform subclasses of the nonuniform complexity classes defined by Karp and Lipton [23] via the notion of advice functions. These subclasses are obtained by restricting the complexity of computing correct advice. We also investigate the effect of allowing advice functions of limited complexity to depend on the input rather than on the input's length. Among other results, using the notions described above, we give new characterizations of (a) NP NP"SPARSE , (b) NP with a restricted access to an NP oracle and (c) the odd levels of the boolean hierarchy. As a consequence, we show that every set that is nondeterministically truth-table reducible to SAT in the sense of Rich [35] is already deterministically truth-table reducible to SAT. Furthermore, it turns out that the NP reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy. Key words. nonuniform complexity classes, advice classes, optimization functions, restric...

This paper examines an extension of Horn logic in which rules can add entries to a database hypothetically. Several researchers have developed logical systems along these lines, but the complexity and expressibility of such logics is only now being explored. It has been shown, for instance, that the data-complexity of these logics is PSPACE-complete in the function-free, predicate case. This paper extends this line of research by developing syntactic restrictions with lower complexity. These restrictions are based on two ideas from Horn-clause logic: linear recursion and stratified negation. In particular, a notion of stratification is developed in which negation-as-failure alternates with linear recursion. The complexity of such rulebases depends on the number of layers of stratification. The result is a hierarchy of syntactic classes which corresponds exactly to the polynomial-time hierarchy of complexity classes. In particular, rulebases with k strata are data-complete for \Sigma P...

In the exponential case circuits of bounded depth characterize the polynomial hierachy. Using the notion of an unambiguous circuit we give a uniform framework to relate the various types of unambiguous polynomial hierarchies and to explain their differences.