The set-height of a complex object type is defined to be its level of nesting of the set construct. In a query of the complex object calculus which maps a database D to an output type T,anintermediate type is a type which is used by some variable of the query, but which is not present in D or T.Foreachk, i ≥ 0 we define CALCk,i to be the family of calculus queries mapping from and to types with set-height ≤ k and using intermediate types with set-height ≤ i. In particular, CALC0,0 is the classical relational calculus, and CALC0,1 is equivalent to the family of secondorder (relational) queries. Several results concerning these families of languages are obtained. A primary focus is on the families CALC0,i, which map relations to relations. Upper and lower bounds in terms of hyper-exponential time and space on the complexity of these families are provided. The CALC0,i hierarchy does not collapse with respect to expressive power. The union ∪0≤iCALC0,i is exactly the family of elementary queries, i.e., queries with hyper-exponential complexity. The expressive power of queries from the complex object calculus interpreted using semantics based on the use of arbitrarily large finite or infinite set of invented values is studied. Under these semantics, the expressive power of the relational calculus is not increased, and the CALC0,i hierarchy collapses at CALC0,1. In general, queries with these semantics may not be computable. We also consider an alternative semantics which yields a family of queries equivalent to the computable queries. 1

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We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

The expressiveness and complexity of several calculus-based query languages for complex objects is considered. Unlike previous investigations, we are concerned with the complexity of queries on databases of complex objects, rather than flat databases. This raises new issues specific to complex objects. For instance, it is shown that the way the database makes use of its higher-order types has direct impact on query complexity. The use of fixpoint operators is shown to yield languages well-behaved with respect to complexity and expressiveness. In particular, an extension of the fixpoint queries to complex objects is shown to express precisely the PTIME queries, under the assumption that the database makes "full" use of all its types. Similar results involve range-restricted queries. 1 Introduction Complex objects are increasingly part of advanced database systems. They provide the structural core of object-oriented databases. Several query languages for complex objects have been propo...

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m -complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilinear-time computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including m-complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rst-order de nability, aswell as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

. We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomialsize circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we dis...