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Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipoly ..."
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Cited by 51 (8 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Exact threshold circuits
 In IEEE Conf. on Computational Complexity (CCC
, 2010
"... Abstract—We initiate a systematic study of constant depth Boolean circuits built using exact threshold gates. We consider both unweighted and weighted exact threshold gates and introduce corresponding circuit classes. We next show that this gives a hierarchy of classes that seamlessly interleave wit ..."
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Abstract—We initiate a systematic study of constant depth Boolean circuits built using exact threshold gates. We consider both unweighted and weighted exact threshold gates and introduce corresponding circuit classes. We next show that this gives a hierarchy of classes that seamlessly interleave with the wellstudied corresponding hierarchies defined using ordinary threshold gates. A major open problem in Boolean circuit complexity is to provide an explicit superpolynomial lower bound for depth two threshold circuits. We identify the class of depth two exact threshold circuits as a natural subclass of these where also no explicit lower bounds are known. Many of our results can be seen as evidence that this class is a strict subclass of depth two threshold circuits — thus we argue that efforts in proving lower bounds should be directed towards this class.
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