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11
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 56 (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...
Lower Bounds on Arithmetic Circuits via Partial Derivatives
 COMPUTATIONAL COMPLEXITY
, 1995
"... In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lo ..."
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Cited by 38 (6 self)
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In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials and iterated matrix products.
The Communication Complexity of Threshold Gates
 In Proceedings of “Combinatorics, Paul Erdos is Eighty
, 1994
"... We prove upper bounds on the randomized communication complexity of evaluating a threshold gate (with arbitrary weights). For linear threshold gates this is done in the usual 2 party communication model, and for degreed threshold gates this is done in the multiparty model. We then use these upp ..."
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Cited by 29 (1 self)
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We prove upper bounds on the randomized communication complexity of evaluating a threshold gate (with arbitrary weights). For linear threshold gates this is done in the usual 2 party communication model, and for degreed threshold gates this is done in the multiparty model. We then use these upper bounds together with known lower bounds for communication complexity in order to give very easy proofs for lower bounds in various models of computation involving threshold gates. This generalizes several known bounds and answers several open problems.
Some Problems Involving RazborovSmolensky Polynomials
, 1991
"... Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polyno ..."
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Cited by 11 (2 self)
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Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polynomials over general finite rings or finite abelian groups. Here we pose a number of conjectures on the behavior of such polynomials over rings and groups, and present some partial results toward proving them. 1. Introduction 1.1. Polynomials and Circuit Complexity The representation of Boolean functions as polynomials over the finite field Z 2 = f0; 1g dates back to early work in switching theory [?]. A formal language L can be identified with the family of functions f i : Z i 2 ! Z 2 , where f i (x 1 ; : : : ; x i ) = 1 iff x 1 : : : x i 2 L. Each of these functions can be written as a polynomial in the variables x 1 ; : : : ; x n . We can consider algebraic formulas or circuits with...
On Relations Between Counting Communication Complexity Classes
"... We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of compl ..."
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Cited by 6 (2 self)
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We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of complexity classes initiated by Babai, Frankl, and Simon [Proc. 27th IEEE FOCS 1986, pp. 337347] and continued in several papers (e.g., Halstenberg and Reischuk [Journ. of Comput. and Syst. Sci. 41(1990), pp. 402429], Karchmer et al. [Journ. of Comput. and Syst. Sci. 49(1994), pp. 247257] More precisely, it will be shown that the communication complexity classes MOD p P cc and MOD q P cc are incomparable with regard to inclusion, for all pairs of distinct prime numbers p and q. The same is true for PP cc and MODmP cc , for any number m 2. Moreover, nondeterminism and modularity are incomparable to a large extend. On the other hand, if m = p l 1 1 \Delta : : : \Delta p l r r ...
Approximation From Linear Spaces And Applications To Complexity
"... . We develop an analytic framework based on linear approximation and duality and point out how a number of apparently diverse complexity related questions  on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean func ..."
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Cited by 3 (2 self)
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. We develop an analytic framework based on linear approximation and duality and point out how a number of apparently diverse complexity related questions  on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean functions  fit neatly into this framework. This isolates the analytic content of these problems from their combinatorial content and clarifies the close relationship between the analytic structure of questions. (1) We give several results that convert a statement of nonapproximability from spaces of functions to statements of approximability. We point how that crucial portions of a significant number of the known complexityrelated results can be unified and given shorter and cleaner proofs using these general theorems. (2) We give several new complexityrelated applications, including circuit complexity lower bounds, and results concerning pseudorandomness, learning, and combinator...
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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Cited by 3 (1 self)
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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.
A note on a theorem of Barrington, Straubing and Thérien
, 1996
"... We show that the result of Barrington, Straubing and Thérien [5] provides, as a direct corollary, an exponential lower bound for the size of depthtwo MOD 6 circuits computing the AND function. This problem was solved, in a more general way, by Krause and Waack [8]. We point out that all known lower ..."
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Cited by 2 (0 self)
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We show that the result of Barrington, Straubing and Thérien [5] provides, as a direct corollary, an exponential lower bound for the size of depthtwo MOD 6 circuits computing the AND function. This problem was solved, in a more general way, by Krause and Waack [8]. We point out that all known lower bounds rely on the special form of the MOD 6 gate occurring at the bottom of the circuits, so that in fact, proving a lower bound for "general" MOD 6 circuits of depth two is still an open question.
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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Cited by 1 (1 self)
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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
Factoring Polynomials Modulo Composites
, 1997
"... This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . ..."
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This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Important Tools in Z n [x] 4 2.1 The Z n [x] phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . 5 2.3 Irreducibility criteria in Z p k [x] . . . . . . . . . . . . . . . . . . . 7 2.4 Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 A naive approach to factoring . . . . . . . . . . . . . . . . . . . . 11 3 The Case of Small Discriminants 12 3.1 The padic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The correspondence to factoring over the padics . . . . ....