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41
The Polynomial Method in Circuit Complexity
 In Proceedings of the 8th IEEE Structure in Complexity Theory Conference
, 1993
"... The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polyno ..."
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Cited by 70 (4 self)
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The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of ANDOR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...
Counting Classes: Thresholds, Parity, Mods, and Fewness
, 1996
"... Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable ..."
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Cited by 61 (13 self)
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Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k iP = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [28]). 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomialtime bo...
Superpolynomial Size SetSystems with Restricted Intersections mod 6 and Explicit Ramsey Graphs
 Combinatorica
, 1999
"... We construct a system H of exp(c log 2 n= log log n) subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all nonprimepower moduli m in place of m = 6. This result is in sharp contrast w ..."
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Cited by 36 (5 self)
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We construct a system H of exp(c log 2 n= log log n) subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all nonprimepower moduli m in place of m = 6. This result is in sharp contrast with results of Frankl and Wilson (1981) for prime power moduli and gives strong negative answers to questions by Frankl and Wilson (1981) and Babai and Frankl (1992). We use our setsystem H to give an explicit Ramseygraph construction, reproducing the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981). Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994). 1 Introduction Generalizing the RayChaudhuriWilson theorem [8], Frankl and Wilson [6] proved the following intersection theorem, one of the most important results in extremal set theory: Department of Computer Science, Eotvos Un...
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
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Cited by 30 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
NP Might Not Be As Easy As Detecting Unique Solutions
, 1998
"... We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the fi ..."
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Cited by 23 (6 self)
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We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NPcomplete sets are polynomialtime isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP complete under reductions that make one query to L but not under traditional manyone reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the sur...
A Lower Bound On The Mod 6 Degree Of The Or Function
 Computational Complexity
, 1995
"... We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in boolean variables over Zm . In particular, we say that a polynomial P weakly represents a boolean function f (both have n variables) if for any inputs x and y in f0; ..."
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Cited by 21 (1 self)
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We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in boolean variables over Zm . In particular, we say that a polynomial P weakly represents a boolean function f (both have n variables) if for any inputs x and y in f0; 1g n we have P (x) 6= P (y) whenever f(x) 6= f(y).
On the Correlation of Symmetric Functions
 MATH. SYSTEMS THEORY
, 1996
"... The correlation between two Boolean functions of n inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any two symmetric Boolean functions. As a consequence of ..."
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Cited by 17 (6 self)
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The correlation between two Boolean functions of n inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any two symmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an exponentially small correlation (in n) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Mod q gate over ANDgates of small fanin, where q is odd and the function computed by the sum of the ANDgates is symmetric, is bounded by 2 \Gamma\Omega\Gamma n) . In addition, we find that for a large class of symmetric functions the correlation with parity is identically zero for infinitely many n. We characterize exactly those symmetric Boolean functions having this property.
Low Rank CoDiagonal Matrices and Ramsey Graphs
 Gur03] V. Guruswami. Better Extractors for Better Codes? Electronic Colloquium on Computational Complexity (ECCC
"... We examine n×n matrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1/(p−1) − O(1)). If m is a nonprimepower integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n ..."
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Cited by 14 (3 self)
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We examine n×n matrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1/(p−1) − O(1)). If m is a nonprimepower integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n log log n). We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs. Keywords: composite modulus, explicit Ramseygraph constructions, matrices over rings, codiagonal matrices 1
Exponential Sums and Circuits with a Single Threshold Gate and ModGates
 Theory Comput. Systems
, 1999
"... Consider circuits consisting of a threshold gate at the top, Modm gates at the next level (for a fixed m), and polylog fanin AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for r ..."
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Cited by 13 (1 self)
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Consider circuits consisting of a threshold gate at the top, Modm gates at the next level (for a fixed m), and polylog fanin AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for restricted versions of this model, in which each Modm  ofAND subcircuit is a symmetric function of the inputs to that subcircuit. We show that if q is a prime not dividing m, the Mod q function requires exponential size circuits of this type. This generalizes recent results and techniques of Cai, Green and Thierauf [CGT] (which held only for q = 2) and Goldmann (which held only for depth two threshold over Modm circuits). As a further generalization of the [CGT] result, the symmetry condition on the Modm subcircuits can be relaxed somewhat, still resulting in an exponential lower bound. The basis of the proof is to reduce the problem to estimating an exponential sum, which generalizes the ...