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Separating the Communication Complexities of MOD m and MOD p Circuits
 IN PROC. 33RD IEEE FOCS
, 1995
"... We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates ..."
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Cited by 20 (4 self)
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We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, nonprime power number. As a corollary, for all m, we show a function, computable with a depth2 circuit with MODm gates, but not with any depth2 circuit with MOD p gates. Obviously, the kparty protocols are not weaker than the k 0 party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 party protocol with a constant number of communicated bits, while any kparty protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multiparty protocols.
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. ...
SPECTRAL PROPERTIES OF THRESHOLD FUNCTIONS
, 1994
"... We examine the spectra of boolean functions obtained as the sign of a real polynomial of degree d. A tight lower bound on various norms of the lower d levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables ..."
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Cited by 9 (0 self)
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We examine the spectra of boolean functions obtained as the sign of a real polynomial of degree d. A tight lower bound on various norms of the lower d levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables on linear threshold functions. Some conjectures are posed concerning upper and lower bounds on influences of variables in higher order threshold functions.
Computing Elementary Symmetric Polynomials with a SubPolynomial Number of Multiplications
 SIAM Journal on Computing
, 2002
"... Elementary symmetric polynomials S n are used as a benchmark for the boundeddepth arithmetic circuit model of computation. In this work weprovethatS n modulo composite numbers m = p 1 p 2 can be computed with muchfewer multiplications than over any field, if the coefficients of monomials x i 1 ..."
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Cited by 8 (5 self)
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Elementary symmetric polynomials S n are used as a benchmark for the boundeddepth arithmetic circuit model of computation. In this work weprovethatS n modulo composite numbers m = p 1 p 2 can be computed with muchfewer multiplications than over any field, if the coefficients of monomials x i 1 x i 2 i k are allowed to be 1 either mod p 1 or mod p 2 but not necessarily both. More exactly,weprove that for any constant k such a representation of S n can be computed modulo p 1 p 2 using only exp(O( p log n log log n)) multiplications on the most restricted depth3 arithmetic circuits, for min(p 1 #p 2 ) ?k!.
Lower Bounds for (MOD p  MOD m) Circuits
 Proc. 39th IEEE FOCS
, 1998
"... Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circui ..."
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Cited by 1 (0 self)
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Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien (1990), Krause and Pudlak (1994), and others, characterizing symmetric functions computable by small (MOD p ; AND t ; MODm ) circuits. Applying a degreedecreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien (1990), and other related lower bounds for certain (MOD p ; MODm ; AND) circuits. Most of the previous lower bounds on circuits with modular gates used special denitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MODm gates. Our methods are applicable, and our lower bounds are valid, for the most general modular gates as well. 1
A DegreeDecreasing Lemma for (MOD q  MOD p) Circuits
, 2001
"... plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexit ..."
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plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this result
Some Properties of MODm Circuits Computing
"... We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit w ..."
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We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. As a consequence we might think of the strategy for deriving lower bounds on modular circuits: Suppose that a polynomial size constant depth modulo 2p circuit C computes a symmetric function. If we can show that the circuit obtained by applying the procedure given in (i) to the circuit C cannot satisfy the properties described in (ii), then we have a superpolynomial lower bound on the size of a constant depth modulo 2p circuit computing a certain symmetric function.