this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The Blum-Shub-Smale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N> 0. • In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.

Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. In contrast to the preceding # Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.

We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical model-theoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra - a class of n-ary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their model-theory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,

We show that extremely accurate approximation to the prefix sums of a sequence of n integers can be computed deterministically in O(log log n) time using O(n= log log n) processors in the Common CRCW PRAM model. This complements randomized approximation methods obtained recently by Goodrich, Matias and Vishkin and improves previous deterministic results obtained by Hagerup and Raman. Furthermore, our results completely match a lower bound obtained recently by Chaudhuri. Our results have many applications. Using them we improve upon the best known time bounds for deterministic approximate selection and for deterministic padded sorting. 1 Introduction The computation of prefix sums is one of the most basic tools in the design of fast parallel algorithms (see Blelloch [9] and J'aJ'a [33]). Prefix-sums can be computed in O(logn) time and linear work in the EREW PRAM model (Ladner and Fischer [34]) and in O(log n= log log n) and linear work in the Common CRCW PRAM model (Cole and Vishkin...

We investigate the problem of how to extend constraint query languages with aggregate operators. We deal with standard relational aggregation, and also with aggregates specific to spatial data, such as volume. We study several approaches, including the addition of a new class of approximate aggregate operators which allow an error tolerance in the computation. We show how techniques based on VC-dimension can be used to give languages with approximation operators, but also show that these languages have a number of shortcomings. We then give a set of results showing that it is impossible to get constraint-based languages that admit definable aggregation operators, both for exact operators and for approximate ones. These results are quite robust, in that they show that closure under aggregation is problematic even when the class of functions permitted in constraints is expanded. This motivates a different approach to the aggregation problem. We introduce a language FO + Poly+S...

Constant-depth polynomial-size threshold circuits are usually classified according to their total depth. For example, the best known threshold circuits for iterated multiplication and division have depth four and three, respectively. In this paper, the complexity of threshold circuits is investigated from a different point of view: explicit AND, OR gates are allowed in the circuits, and a threshold circuit is said to have majority-depth d if no path traverses more than d threshold gates. It is then shown that iterated multiplication can be computed by polynomial-size threshold circuits of total depth five but of majority-depth three. Circuits of depth four and majority-depth two are obtained for division and powering. These results rely on a careful implementation of iterated addition and Chinese remaindering. In addition, a simple symbolic calculus for composing circuit classes is developed: this notation allows for a concise and elegant presentation of the results. 3 List of symbol...

Constant-depth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC .

A language L over an alphabet A is said to have a neutral letter if there is a letter e # A such that inserting or deleting e's from any word in A # does not change its membership (or non--membership) in L. The presence of a neutral letter affects the definability of a language in first--order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first--order logic with linear order, then it is not definable in first--order logic with any set N of numerical predicates. We investigate this conjecture in detail, showing that it fails already for N = {+, #}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, lg (m) , for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for N = {+}, for the fragment BC(# 1 ) of first--order logic, and for binary alphabets. # Supported by NSF grant CCR-9988260. + Supported by NSF grant CCR-9877078. # Supported by NSERC and FCAR. 1

We prove a lower bound of ω(n k/4) on the size of constantdepth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(n k/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff. Our k-clique lower bound derives from a stronger result of independent interest. Suppose fn: {0, 1} (n2) − → {0, 1} is a sequence of functions computed by constant-depth circuits of size O(n t). Let G be an Erdős-Rényi random graph with vertex set {1,..., n} and independent edge probabilities n −α where α ≤ 1 2t−1. Let A be a uniform random k-element subset of {1,..., n} (where k is any constant independent of n) and let KA denote the clique supported on A. We prove that fn(G) = fn(G ∪ KA) asymptotically almost surely. These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The m-variable fragment of first-order logic, denoted by FO m, consists of the first-order sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO 1 ⊂ FO 2 ⊂ · · · ⊂ FO m ⊂ · · · is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO 3 is less expressive than full first-order logic on finite ordered graphs.