We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Fur-ther we show that there are functions computable in polynomial size and depth k but requires ex-ponential size when the depth is restricted to k-1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.

We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over GF (2)). This paper shows how to learn in polynomial time any function that can be approximated (in norm L 2 ) by a polynomially sparse function (i.e., a function with only polynomially many non-zero Fourier coefficients). The authors demonstrate that any function f whose L 1 -norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial L 1 -norm can be learned deterministically. The algorithm can a...

Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and linear size ^ � _-circuits but require exponential size to compute by a depth k; 1 monotone weighted threshold circuit. Key words. Circuit complexity, monotone circuits, threshold circuits, lower bounds Subject classi cations. 68Q15, 68Q99 1.

We consider the problem of approximating a Boolean function f : f0; 1g n ! f0; 1g by the sign of an integer polynomial p of degree k. For us, a polynomial p(x) predicts the value of f(x) if, whenever p(x) 0, f(x) = 1, and whenever p(x) ! 0, f(x) = 0. A low-degree polynomial p is a good approximator for f if it predicts f at almost all points. Given a positive integer k, and a Boolean function f , we ask, "how good is the best degree k approximation to f?" We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the case f is parity. Minsky and Papert [10] proved that a perceptron can not compute parity; our bounds indicate exactly how well Yale University, Dept. of Computer Science, P.O. Box 208285, New Haven CT 06520-8285. y Email: aspnes-james@cs.yale.edu. z Email: beigel-richard@cs.yale.edu. Supported in part by NSF grants CCR-8808949 and CCR-8958528. x Carnegie-Mellon University, Schoo...

We show very efficient constructions for a pseudo-random generator and for a universal one-way hash function based on the intractability of the subset sum problem for certain dimensions. (Pseudo-random generators can be used for private key encryption and universal one-way hash functions for signature schemes). The increase in efficiency in our construction is due to the fact that many bits can be generated/hashed with one application of the assumed one-way function. All our construction can be implemented in NC using an optimal number of processors. Part of this work done while both authors were at UC Berkeley and part when the second author was at the IBM Almaden Research Center. Research supported by NSF grant CCR 88 - 13632. A preliminary version of this paper appeared in Proc. of the 30th Symp. on Foundations of Computer Science, 1989. 1 Introduction Many cryptosystems are based on the intractability of such number theoretic problems such as factoring and discrete logarit...

We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including (\Delta + 1)-coloring constantdegree graphs and 5-coloring planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs. 1 Introduction Some problems for which trivial sequential algorithms exist appear to be much harder to solve in a parallel framework. When converting a sequential algorithm to a parallel one, at each step of the parallel algorithm we have to choose a set of operations which may be executed in parallel. Often, we have to choose these operations from a large set A preliminary version of this paper appear...

To appear in Izvestiya of the RAN Abstract We show that if strong pseudorandom generators exist then the statement "ff encodes a circuit of size n(log * n) for SATISFIABILITY " is not refutable in S22 (ff). For refutation in S12 (ff), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constant-depth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolation-like theorems for certain "split versions " of classical systems of Bounded Arithmetic introduced in this paper.