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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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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
New correlation bounds for GF(2) polynomials using Gowers uniformity
 Electronic Colloquium on Computational Complexity
, 2006
"... We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on th ..."
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Cited by 8 (4 self)
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We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on the previous exp � −Ω � n/8 d� � bound by Bourgain (C. R. Acad. Sci. Paris, 2005) and Green et al. (C. R. Acad. Sci. Paris, 2005). II We exhibit a polynomialtime computable function on n bits that has correlation at most exp � −Ω � n/2 d� � with any GF (2) polynomial of degree d. Previous to our work the best correlation bound for an explicit function was exp � −Ω � n / � d · 2 d�� �, which follows from (Chung and Tetali; SIAM J. Discrete Math., 1993). III We derive an ‘XOR Lemma ’ for lowdegree GF (2) polynomials: We show that if a function f has correlation at most 1 − 4 −d with any GF (2) polynomial of degree d (and Prx[f(x) = 1] ≈ 1/2) then the XOR of m independent copies of f has correlation at most exp � −Ω � m/4 d� � with any GF (2) polynomial of degree d. Our results rely on a measure of the ‘complexity ’ of a function due to Gowers (Geom.
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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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
Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds
"... Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argum ..."
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Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argument can prove that a problem requires circuits of superpolynomial size, even for some very restricted classes of circuits (under reasonable cryptographic assumptions). This barrier is so daunting, that some researchers have decided to focus their attentions elsewhere. Yet the goal of proving circuit lower bounds is of such importance, that some in the community have proposed concrete strategies for surmounting the obstacle. This lecture will discuss some of these strategies, and will dwell at length on a recent approach proposed by Michal Koucky and the author.
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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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.
Uniqueness of Optimal Mod 3 Circuits for Parity
"... Abstract. In this paper, we prove that the quadratic polynomials modulo 3 with the largest correlation with parity are unique up to permutation of variables and constant factors. As a consequence of our result, we completely characterize the smallest MAJ◦ MOD3 ◦ AND2 circuits that compute parity, wh ..."
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Abstract. In this paper, we prove that the quadratic polynomials modulo 3 with the largest correlation with parity are unique up to permutation of variables and constant factors. As a consequence of our result, we completely characterize the smallest MAJ◦ MOD3 ◦ AND2 circuits that compute parity, where a MAJ ◦ MOD3 ◦ AND2 circuit is one that has a majority gate as output, a middle layer of MOD3 gates and a bottom layer of AND gates of fanin 2. We also prove that the suboptimal circuits exhibit a stepped behavior: any suboptimal circuits of this class that compute parity must have size at least a factor of 2 √ times the optimal size. This verifies, for the special case of 3 m = 3, two conjectures made in [5] for general MAJ ◦ MODm ◦ AND2 circuits for any odd m. The correlation and circuit bounds are obtained by studying the associated exponential sums, based on some of the techniques developed in [7]. 1.
Correlation bounds for polynomials over {0, 1}
, 2009
"... This article is a unified treatment of the stateoftheart on the fundamental challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. It discusses longstanding results and recent developments, related proof techniques, and connections with ps ..."
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This article is a unified treatment of the stateoftheart on the fundamental challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. It discusses longstanding results and recent developments, related proof techniques, and connections with pseudorandom generators. It also suggests several research directions.