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12
Lower bounds for approximations by low degree polynomials over Zm
, 2001
"... We use a Ramseytheoretic argument to obtain the first lower bounds for approximations over Zm by nonlinear polynomials: ffl A degree2 polynomial over Zm (m odd) must differ from the parity function on at least a ..."
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Cited by 28 (0 self)
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We use a Ramseytheoretic argument to obtain the first lower bounds for approximations over Zm by nonlinear polynomials: ffl A degree2 polynomial over Zm (m odd) must differ from the parity function on at least a
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
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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Cited by 6 (2 self)
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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 and Symmetric Gates
 In International Conference on Automata, Languages, and Programming (ICALP
, 2005
"... circuits augmented with s MODm gates must have size n\Omega ( 1s log 1r1 n) to compute MAJORITY or MODl, if l has a prime factor not dividing m. For proving the latter result we introduce a new notion of representation of boolean function by polynomials, for which we obtain degree lower bounds that ..."
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circuits augmented with s MODm gates must have size n\Omega ( 1s log 1r1 n) to compute MAJORITY or MODl, if l has a prime factor not dividing m. For proving the latter result we introduce a new notion of representation of boolean function by polynomials, for which we obtain degree lower bounds that are of independent interest. 1 Introduction Strong size lower bounds have been obtained for several classes of circuits. Inparticular constant depth circuits ( AC0 circuits) require exponential size to compute even simple functions such as PARITY [14]. More generally, if we also allow
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.
Size and Energy of Threshold Circuits Computing Mod Functions
"... Abstract. Let C be a threshold logic circuit computing a Boolean function MODm : {0, 1} n → {0, 1}, where n ≥ 1 and m ≥ 2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the ..."
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Abstract. Let C be a threshold logic circuit computing a Boolean function MODm : {0, 1} n → {0, 1}, where n ≥ 1 and m ≥ 2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the socalled PARITY function, and MODn+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈ {0, 1} n . In the paper, we prove that a very simple inequality n/(m − 1) ≤ s e holds for every circuit C computing MODm. The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MODm, and yields a lower bound e = Ω((log n − log m)/ log log n) on e if s = O(polylog(n)). We actually obtain a general result on the socalled generalized mod function, from which the result on the ordinary mod function MODm immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.
Some Properties of MODm Circuits Computing Simple Functions
"... 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.
Linear Systems Over Finite Abelian Groups
"... We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1,...,xn) ≡ ℓi,1x1 + ·· · + ℓi,nxn ∈ Ai for i = 1,...,t and each Ai ⊂ G, ℓi,j is an element of G and xi’s are Boolean variables. Our main result shows that the subset of the Boolean cube that satis ..."
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We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1,...,xn) ≡ ℓi,1x1 + ·· · + ℓi,nxn ∈ Ai for i = 1,...,t and each Ai ⊂ G, ℓi,j is an element of G and xi’s are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MODq boolean function, when the order of G and q are coprime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS’09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential boundsonthesizeofbooleandepthfour circuitsoftheformMAJ◦AND◦ANY O(1)◦ MODm for computing the MODq function, when m,q are coprime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity’97). 1
Techniques for Analyzing the Computational Power of ConstantDepth Circuits and SpaceBounded Computation
, 2006
"... The subject of computational complexity theory is to analyze the difficulty of solving computational problems within different models of computation. Proving lower bounds is easier in less powerful models and proving upper bounds is easier in the more powerful models. This dissertation studies techn ..."
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The subject of computational complexity theory is to analyze the difficulty of solving computational problems within different models of computation. Proving lower bounds is easier in less powerful models and proving upper bounds is easier in the more powerful models. This dissertation studies techniques for analyzing the power of models of computation which are at the frontier of currently existing methods. First, we study the power of certain classes of depththree circuits. The power of such circuits is largely not understood and studying them under further restrictions has received a lot of attention. We prove exponential lower bounds on the size of certain depththree circuits computing parity. Our approach is based on relating the lower bounds to correlation between parity and modular polynomials, and expressing the correlation with exponential sums. We show a new expression for the exponential sum which involves a certain affine space corresponding to the polynomial. This technique gives a unified treatment and generalization of bounds which include the results of Goldmann on linear polynomials and Cai, Green, and Thierauf on symmetric polynomials. We obtain bounds on the exponential sums for classes of polynomials of large degree and with a large number of terms, which previous techniques did not apply to.