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Discrepancy and the power of bottom fanin in depththree circuits
 In Proc. of the 48th Symposium on Foundations of Computer Science (FOCS
, 2007
"... We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent wor ..."
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Cited by 26 (2 self)
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We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fanin at the base layer i.e. circuits of type MAJ ◦ SYMM ◦ ANY O(1) cannot simulate the circuit class AC 0 in subexponential size. Further, even if the fanin of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC 0 in quasipolynomial size. This is in contrast to the classical result of Yao and BeigelTarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC 0 in quasipolynomial size when the bottom fanin is increased to polylogarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain [7] for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented by polynomials of small degree over Zm, when m, q ≥ 2 are coprime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of [7, 14]. It is known that such estimates imply that circuits of type MAJ ◦ MODm ◦ ANDɛ log n cannot compute the MODq function in subexponential size. It remains a major open question to determine if such circuits can simulate ACC 0 in polynomial size when the bottom fanin is increased to polylogarithmic size. 1
The correlation between parity and quadratic polynomials mod 3
 J. Comput. System Sci
"... We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fanin two at the inputs must be of s ..."
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Cited by 17 (4 self)
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We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fanin two at the inputs must be of size 2 Ω(n). This is the first result of this type for general mod 3 subcircuits with ANDs of fanin greater than 1. This yields an exponential improvement over a longstanding result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight. 1
Incomplete quadratic exponential sums in several variables
 GR10] [Gre04] [GRS05] [Rad09] [Smo87] Frederic Green and Amitabha
, 2010
"... Abstract. We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1 ..."
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Cited by 7 (3 self)
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Abstract. We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1
On the correlation between parity and modular polynomials
 Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bo ..."
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Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bounds on the size necessary to compute parity by depth3 circuits of certain form. Our technique is based on a new representation of the correlation using exponential sums. Our results include Goldmann’s result [Go] on the correlation between parity and degree one polynomials as a special case. Our general expression for representing correlation can be used to derive the bounds of Cai, Green, and Thierauf [CGT] for symmetric polynomials, using ideas of the [CGT] proof. The classes of polynomials for which we obtain exponentially small upper bounds include polynomials of large degree and with a large number of terms, that previous techniques did not apply to. 1
Lower Bounds for Circuits with Few Modular Gates using Exponential Sums
 Electronic Colloquium on Computational Complexity
, 2005
"... We prove that any AC 0 circuit augmented with ɛ log 2 n MODm gates and with a MAJORITY gate at the output, require size n Ω(log n) to compute MODl, when l has a prime factor not dividing m and ɛ is sufficiently small. We also obtain that the MOD2 function is hard on the average for AC 0 circuits of ..."
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We prove that any AC 0 circuit augmented with ɛ log 2 n MODm gates and with a MAJORITY gate at the output, require size n Ω(log n) to compute MODl, when l has a prime factor not dividing m and ɛ is sufficiently small. We also obtain that the MOD2 function is hard on the average for AC 0 circuits of size n ɛ log n augmented with ɛ log 2 n MODm gates, for every odd integer m and any sufficiently small ɛ. As a consequence, for every odd integer m, we obtain a pseudorandom generator, based on the MOD2 function, for circuits of size S containing ɛ log S MODm gates. Our results are based on recent bounds of exponential sums that were previously introduced for proving lower bounds for MAJ◦MODm ◦ ANDd circuits.
An improved bound on correlation between polynomials over Zm and MODq
 Electronic Colloquium on Computational Complexity
, 2006
"... achatt3cs.mcgill.ca Let m, q> 1 be two integers that are coprime and A be any subset of Zm. Let P be any multivariate polynomial of degree d in n variables over Zm. We show that the MODq boolean function on n variables has correlation at most exp(−Ω(n/(m2 m−1)d)) with the boolean function f def ..."
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Cited by 4 (0 self)
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achatt3cs.mcgill.ca Let m, q> 1 be two integers that are coprime and A be any subset of Zm. Let P be any multivariate polynomial of degree d in n variables over Zm. We show that the MODq boolean function on n variables has correlation at most exp(−Ω(n/(m2 m−1)d)) with the boolean function f defined by f(x) = 1 iff P (x) ∈ A for all x ∈ {0, 1}n. This improves on the bound of exp(−Ω(n/(m2m)d)) obtained in the breakthrough work of Bourgain [3] and Green et al. [9]. Our calculation is also slightly shorter than theirs. Our result immediately implies the bound of exp(−Ω(n/4d)) for the special case of m = 2. This bound was first reported in the recent work of Viola [11]. [11] states that it is not clear how to extend their method to general m. 1
Linear systems over composite moduli
 In IEEE FOCS
"... We study solution sets to systems of generalized linear equations of the form ℓi(x1, x2, · · · , xn) ∈ Ai (mod m) where ℓ1,...,ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of Zm, and m is a composite integer that is a product of two distinct primes, like 6. Our main ..."
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We study solution sets to systems of generalized linear equations of the form ℓi(x1, x2, · · · , xn) ∈ Ai (mod m) where ℓ1,...,ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of Zm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. exp ( − Ω(n) ) , with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constantdepth circuits with modular gates. We derive the first exponential lower bound on the size of depththree circuits of type MAJ ◦ AND ◦ MOD A m (i.e. having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MODm gates at the base) computing the function MODq. This settles a decadeold open problem of Beigel and Maciel [5], for the case of such modulus m. Our technique makes use of the work of Bourgain [6] on estimating exponential sums involving a lowdegree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov [15] on arithmetic circuits over finite fields.
ARTICLE IN PRESS Journal of Number Theory ( ) –
, 2004
"... www.elsevier.com/locate/jnt Incomplete quadratic exponential sums in several variables ..."
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www.elsevier.com/locate/jnt Incomplete quadratic exponential sums in several variables