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34
Majority Gates vs. General Weighted Threshold Gates
 Computational Complexity
, 1992
"... . In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth c ..."
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Cited by 87 (6 self)
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. In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following: 1. A single threshold gate with weights cannot in general be replaced by a polynomial fanin unweighted threshold gate of parity gates. 2. On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depth d polynomial size threshold circuit with weights can be computed by a depth d + 1 polynomial size unweighted threshold circuit, where d is an arbitrary fixed integer. 3. A polynomial fanin threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size...
On the Computational Power of Depth 2 Circuits with Threshold and Modulo Gates
, 2000
"... We investigate the computational power of depth two circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, thresholdMOD r circuits) and circuits with two levels of MOD gates (MOD p MOD q circuits). In particular, we will show ..."
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Cited by 55 (4 self)
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We investigate the computational power of depth two circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, thresholdMOD r circuits) and circuits with two levels of MOD gates (MOD p MOD q circuits). In particular, we will show the following results. (i) For all prime numbers p and integers q; r, it holds that if p divides r but not q then all thresholdMOD q circuits for MOD r have exponentially many nodes. (ii) For all integers r, all problems computable by depth two fAND;OR;NOTg circuits of polynomial size have thresholdMOD r circuits with polynomially many edges. (iii) There is a problem computable by depth 3 fAND;OR;NOTgcircuits of linear size and constant bottom fanin which for all r needs thresholdMOD r circuits with exponentially many nodes. (iv) For p; r different primes, and q 2; k positive integers, where r does not divide q; every MOD p k MOD q circuit for MOD r has e...
Representing Boolean Functions As Polynomials Modulo Composite Numbers
 Computational Complexity
, 1994
"... . Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), wher ..."
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Cited by 53 (6 self)
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. Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm degree of the MODm function is 1; we show that the MODm degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...
Spectral Methods for Matrix Rigidity with Applications to SizeDepth Tradeoffs and Communication Complexity
 In Proc. 36th
, 1996
"... The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider ..."
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Cited by 45 (3 self)
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The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. Two applications of such restricted variants are given. First, we show that our lower bound on a variant of rigidity implies lower bounds on sizedepth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain restricted forms of rigidity to conclude several separation results such as separating the analogs of PH and PSPACE in communication complexity theory. Our res...
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 bounds. Al ..."
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Cited by 30 (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.
Lower bounds for approximations by low degree polynomials over Zm
, 2001
"... Abstract We use a Ramseytheoretic argument to obtain the firstlower bounds for approximations over Zm by nonlinearpolynomials: ffl A degree2 polynomial over Zm (m odd) mustdiffer from the parity function on at least a ..."
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Cited by 27 (0 self)
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Abstract We use a Ramseytheoretic argument to obtain the firstlower bounds for approximations over Zm by nonlinearpolynomials: ffl A degree2 polynomial over Zm (m odd) mustdiffer from the parity function on at least a
On Small Depth Threshold Circuits
 PROCEEDINGS OF THE 3RD SWAT SCANDINAVIAN WORKSHOP ON ALGORITHM THEORY, HELSINKI, FINLAND (LNCS 621
, 1992
"... In this talk we will consider various classes defined by small depth polynomial size circuits which contain threshold gates and parity gates. We will describe various inclusions between many classes defined in this way and also classes whose definitions rely upon spectral properties of Boolean fu ..."
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Cited by 20 (2 self)
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In this talk we will consider various classes defined by small depth polynomial size circuits which contain threshold gates and parity gates. We will describe various inclusions between many classes defined in this way and also classes whose definitions rely upon spectral properties of Boolean functions.
Improved Lower Bounds on the Rigidity of Hadamard Matrices
, 1998
"... We show that at least =r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank below r. This improves upon the previously known bound ) [6]. If we additionally know that these changes are bounded from above in their absolute values by so ..."
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Cited by 18 (0 self)
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We show that at least =r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank below r. This improves upon the previously known bound ) [6]. If we additionally know that these changes are bounded from above in their absolute values by some n=r, we prove another bound on their number. This improves upon the bound ) from [13].
A WeightSize TradeOff for Circuits with MOD m Gates
 In Proc. 26th Ann. ACM Symp. Theor. Comput
, 1994
"... : We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we p ..."
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Cited by 10 (1 self)
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: We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that log(wM ) = \Omega\Gamma n), where w is the sum of the absolute values of the weights, and M is the maximum fanin of the AND gates on level 2. Setting all weights to 1, we have got a tradeoff between the numbers of the MOD m gates and the AND gates. By our knowledge, this is the first tradeoff result involving hardtohandle MOD m gates. In contrast, with n AND gates at the bottom and a single MOD 2 gate at the top one can compute the inner product function. The lowerbound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fanin. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates. ...
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 10 (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...