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149
Algebraic methods in the theory of lower bounds for boolean circuit complexity
 IN PROCEEDINGS OF THE 19TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, STOC ’87
, 1987
"... We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fanin circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MOD, where p is a prime require Ezp(O(n’)) gates to calcu ..."
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Cited by 329 (1 self)
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We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fanin circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MOD, where p is a prime require Ezp(O(n’)) gates
Lower Bounds for (MOD p  MOD m) Circuits
 Proc. 39th IEEE FOCS
, 1998
"... Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circui ..."
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Cited by 1 (0 self)
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circuit lower bounds of Barrington, Straubing, Therien (1990), Krause and Pudlak (1994), and others, characterizing symmetric functions computable by small (MOD p ; AND t ; MODm ) circuits. Applying a degreedecreasing technique together with random restriction methods for the AND gates at the bottom
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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nδ. 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
A WeightSize TradeOff for Circuits with MOD m Gates
 In Proc. 26th Ann. ACM Symp. Theor. Comput
, 1994
"... : We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we p ..."
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Cited by 11 (1 self)
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: We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we
Lower Bounds on the Size of Depth 3 Threshold Circuits with AND Gates at the Bottom
 Inform. Process. Lett
, 1993
"... We present a function in ACC 0 such that any depth 3 threshold circuit which computes this function and has AND gates at the bottom must have size n\Omega\Gamma607 n) . Key words. computational complexity; threshold circuits; lower bounds 1. Introduction Constant depth threshold circuits have ..."
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Cited by 25 (1 self)
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We present a function in ACC 0 such that any depth 3 threshold circuit which computes this function and has AND gates at the bottom must have size n\Omega\Gamma607 n) . Key words. computational complexity; threshold circuits; lower bounds 1. Introduction Constant depth threshold circuits have
Upper and Lower Bounds for Some Depth3 Circuit Classes
 In Proc. 12th Ann. IEEE Conf. Comput. Complexity Theory
, 1997
"... We investigate the complexity of depth3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MODm gates at level one. We show that the fanin of the AND gates can be reduced to O(log n) in the case where m is unbounded, and to a constant in the case whe ..."
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Cited by 12 (1 self)
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We investigate the complexity of depth3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MODm gates at level one. We show that the fanin of the AND gates can be reduced to O(log n) in the case where m is unbounded, and to a constant in the case
Lower bounds for circuits with MODm gates
"... Let CC o(n) [m] be the class of circuits that have size o(n) and in which all gates are MODm gates. • We show that CC[m] circuits cannot compute MODq in sublinear size when m, q> 1 are coprime integers. No nontrivial lower bounds were known before on the size of CC[m] circuits of constant dept ..."
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Cited by 9 (4 self)
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Let CC o(n) [m] be the class of circuits that have size o(n) and in which all gates are MODm gates. • We show that CC[m] circuits cannot compute MODq in sublinear size when m, q> 1 are coprime integers. No nontrivial lower bounds were known before on the size of CC[m] circuits of constant
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 12 (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
MOD m Gates do not Help on the Ground Floor
, 1993
"... We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove ..."
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We prove that any depth3 circuit with MOD m gates of unbounded fanin on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we
Lower Bounds on the Size of Depth 3
 Inform. Process. Lett
, 1993
"... We present a function in ACC such that any depth 3 threshold circuit which computes this function and has AND gates at the bottom must have size n 607 . ..."
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We present a function in ACC such that any depth 3 threshold circuit which computes this function and has AND gates at the bottom must have size n 607 .
Results 1  10
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149