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Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates
 SIAM Journal on Computing
, 2005
"... We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ..."
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We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ’93) that achieves the same stretch but only fools circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan’s generator for constant depth circuits (Combinatorica ’91) (but Nisan’s generator has a much bigger stretch). In particular, we conclude that every function computable by uniform poly(n)size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME 2no(1)�. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function f: {0, 1} n → {0, 1} that is very hard on average for constantdepth circuits of size nɛ·log n with ɛ log 2 n arbitrary symmetric gates, and plugging it into the NisanWigderson pseudorandom generator construction (FOCS ’88). The proof of the averagecase hardness of this function is a modification of arguments by Razborov and Wigderson (IPL ’93), and Hansen and Miltersen (MFCS ’04), and combines H˚astad’s switching lemma (STOC ’86) with a multiparty communication complexity lower bound by Babai, Nisan and
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.
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.
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
"... We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e AC0◦MODm circuits. We show that the following holds for several types of gates G: by adding a gate of type G at the output, it is possible to obtain an equivalent probabi ..."
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We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e AC0◦MODm circuits. We show that the following holds for several types of gates G: by adding a gate of type G at the output, it is possible to obtain an equivalent probabilistic depth 2 circuit of quasipolynomial size consisting of a gate of type G at the output and a layer of modular counting gates, i.e G ◦MODm circuits. The types of gates G we consider are modular counting gates and thresholdstyle gates. For all of these, strong lower bounds are known for (deterministic) G ◦MODm circuits. 1