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25
The polynomial method in circuit complexity,
 Proc. of 8th Annual Structure in Complexity Theory Conference
, 1993
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The Communication Complexity of Threshold Gates
 In Proceedings of “Combinatorics, Paul Erdos is Eighty
, 1994
"... We prove upper bounds on the randomized communication complexity of evaluating a threshold gate (with arbitrary weights). For linear threshold gates this is done in the usual 2 party communication model, and for degreed threshold gates this is done in the multiparty model. We then use these upp ..."
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Cited by 41 (1 self)
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We prove upper bounds on the randomized communication complexity of evaluating a threshold gate (with arbitrary weights). For linear threshold gates this is done in the usual 2 party communication model, and for degreed threshold gates this is done in the multiparty model. We then use these upper bounds together with known lower bounds for communication complexity in order to give very easy proofs for lower bounds in various models of computation involving threshold gates. This generalizes several known bounds and answers several open problems.
Simulating Threshold Circuits by Majority Circuits
 SIAM JOURNAL ON COMPUTING
, 1994
"... We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomialsize depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10 ..."
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Cited by 35 (0 self)
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We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomialsize depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10] that a nonuniform simulation exists. Our construction answers two open questions posed in [10]: we give an explicit construction whereas [10] uses a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (the simulation in [10] gives polynomialsize circuits only when d is constant).
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bound ..."
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Cited by 34 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates
 SIAM Journal on Computing
, 2005
"... We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ..."
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Cited by 29 (12 self)
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We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ’93) that achieves the same stretch but only fools circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan’s generator for constant depth circuits (Combinatorica ’91) (but Nisan’s generator has a much bigger stretch). In particular, we conclude that every function computable by uniform poly(n)size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME 2no(1)�. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function f: {0, 1} n → {0, 1} that is very hard on average for constantdepth circuits of size nɛ·log n with ɛ log 2 n arbitrary symmetric gates, and plugging it into the NisanWigderson pseudorandom generator construction (FOCS ’88). The proof of the averagecase hardness of this function is a modification of arguments by Razborov and Wigderson (IPL ’93), and Hansen and Miltersen (MFCS ’04), and combines H˚astad’s switching lemma (STOC ’86) with a multiparty communication complexity lower bound by Babai, Nisan and
Computational Complexity Of Neural Networks: A Survey
, 1994
"... . We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks fr ..."
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Cited by 25 (6 self)
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. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks from examples of their behavior. CR Classification: F.1.1 [Computation by Abstract Devices]: Models of Computationneural networks, circuits; F.1.3 [Computation by Abstract Devices ]: Complexity Classescomplexity hierarchies Key words: Neural networks, computational complexity, threshold circuits, associative memory 1. Introduction The currently again very active field of computation by "neural" networks has opened up a wealth of fascinating research topics in the computational complexity analysis of the models considered. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks...
Improved separations between nondeterministic and randomized multiparty communication
 in Proc. of the 12th Intl. Workshop on Randomization and Computation (RANDOM
"... We exhibit an explicit function f: {0,1} n → {0,1} that can be computed by a nondeterministic numberonforehead protocol communicating O(logn) bits, but that requires n Ω(1) bits of communication for randomized numberonforehead protocols with k = δ · logn players, for any fixed δ < 1. Recent b ..."
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Cited by 19 (2 self)
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We exhibit an explicit function f: {0,1} n → {0,1} that can be computed by a nondeterministic numberonforehead protocol communicating O(logn) bits, but that requires n Ω(1) bits of communication for randomized numberonforehead protocols with k = δ · logn players, for any fixed δ < 1. Recent breakthrough results for the SetDisjointness function (Lee
Oneway multiparty communication lower bound for pointer jumping with applications
, 2007
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A Separation of NP and coNP in Multiparty Communication Complexity
 THEORY OF COMPUTING
, 2010
"... We prove that coNP � MA and in particular NP ̸ = coNP in the numberonforehead model of multiparty communication complexity for up to k = (1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k → {0,1} with conondeterministic complexity O(l ..."
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Cited by 13 (3 self)
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We prove that coNP � MA and in particular NP ̸ = coNP in the numberonforehead model of multiparty communication complexity for up to k = (1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k → {0,1} with conondeterministic complexity O(logn) and MerlinArthur complexity nΩ(1). The problem was open for k >= 3.
Upper and Lower Bounds for Some Depth3 Circuit Classes
 In Proc. 12th Ann. IEEE Conf. Comput. Complexity Theory
, 1997
"... We investigate the complexity of depth3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MODm gates at level one. We show that the fanin of the AND gates can be reduced to O(log n) in the case where m is unbounded, and to a constant in the case whe ..."
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Cited by 12 (1 self)
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We investigate the complexity of depth3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MODm gates at level one. We show that the fanin of the AND gates can be reduced to O(log n) in the case where m is unbounded, and to a constant in the case where m is constant. We then use these upper bounds to derive exponential lower bounds for this class of circuits. In the unbounded m case, this yields a new proof of a lower bound of Grolmusz; in the constant m case, our result sharpens his lower bound. In addition, we prove an exponential lower bound if OR gates are also permitted on level two and m is a constant prime power. 1 Introduction About ten years ago, Furst, Saxe and Sipser [FSS] and Ajtai [Aj] showed that polynomialsize AC 0 circuits could not compute the parity function. It was hoped that this seminal result would be the first in a series of lower bounds for increasingly larger classes of circuits and that this would lead to th...