This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

The representation of functions as low-degree polynomials over various rings has provided many insights in the theory of small-depth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of AND-OR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...

. A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomial-time verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE ([Condon et al., Proc. 25th ACM Symposium on Theory of Computing, 1993, pp. 304--315]). In this paper, we restrict attention to RPCDS's, which are PCDS's in which Player 0 follows a very simple strategy: On each turn, Player 0 chooses uniformly at random from the set of legal moves. We prove the following result. Theorem: L has an RPCDS in which the verifier flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE. This new characterization of PSPACE is used to show that certain stochastic PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. Exam...

In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1

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

It has been shown previously that sorting n items into n locations with a polynomial number of processors requires Ω(log n/log log n) time. We sidestep this lower bound with the idea of Padded Sorting, or sorting n items into n + o(n) locations. Since many problems do not rely on the exact rank of sorted items, a Padded Sort is often just as useful as an unpadded sort. Our algorithm for Padded Sort runs on the Tolerant CRCW PRAM and takes Θ(log log n/log log log n) expected time using n log log log n/log log n processors, assuming the items are taken from a uniform distribution. Using similar techniques we solve some computational geometry problems, including Voronoi Diagram, with the same processor and time bounds, assuming points are taken from a uniform distribution in the unit square. Further, we present an Arbitrary CRCW PRAM algorithm to solve the Closest Pair problem in constant expected time with n processors regardless of the distribution of points. All of these algorithms achieve linear speedup in expected time over their optimal serial counterparts. 1 Research done while at the University of Michigan and supported by an AT&T Fellowship.

In this paper, we are concerned with randomized OBDDs and randomized read-ktimes branching programs. We present an example of a Boolean function which has polynomial size randomized OBDDs with small, one-sided error, but only nondeterministic read-once branching programs of exponential size. Furthermore, we discuss a lower bound technique for randomized OBDDs with two-sided error and prove an exponential lower bound of this type. Our main result is an exponential lower bound for randomized read-k-times branching programs with two-sided error. 1 Introduction Branching programs are a theoretically and practically interesting data structure for the representation of Boolean functions. In complexity theory, among other problems, lower bounds for the size of branching programs for explicitly defined functions and the relations of the various branching program models are investigated. A branching program (BP) on the variable set fx 1 ; : : : ; x n g is a directed acyclic graph with one sour...

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 prove that constant depth circuits of size n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC can be recognized by a family of depth three . The size bound n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold both for the uniform and the nonuniform case.

We develop a general approach for improving the efficiency of a computationally bounded receiver interacting with a powerful and possibly malicious sender. The key idea we use is that of delegating some of the receiver’s computation to the (potentially malicious) sender. This idea was recently introduced by Goldwasser et al. [14] in the area of program checking. A classic example of such a sender-receiver setting is interactive proof systems. By taking the sender to be a (potentially malicious) prover and the receiver to be a verifier, we show that (p-prover) interactive proofs with k rounds of interaction are equivalent to (p-prover) interactive proofs with k + O(1) rounds, where the verifier is in NC 0. That is, each round of the verifier’s computation can be implemented in constant parallel time. As a corollary, we obtain interactive proof systems, with (optimally) constant soundness, for languages in AM and NEXP, where the verifier runs in constant parallel-time. Another, less immediate sender-receiver setting arises in considering error correcting codes. By taking the sender to be a (potentially corrupted) codeword and the receiver to be a decoder, we obtain explicit families of codes that are locally (list-)decodable by constant-depth circuits of size polylogarithmic in the length of the codeword. Using the tight connection between locally list-decodable codes and average-case complexity, we obtain a new, more efficient, worst-case to average-case reduction for languages in EXP.