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240,586
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2821 (11 self)
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;enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained
Fully homomorphic encryption using ideal lattices
 In Proc. STOC
, 2009
"... We propose a fully homomorphic encryption scheme – i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result – that, to construct an encryption scheme that permits evaluation of arbitra ..."
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Cited by 664 (17 self)
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that is represented as a lattice), as needed to evaluate general circuits. Unfortunately, our initial scheme is not quite bootstrappable – i.e., the depth that the scheme can correctly evaluate can be logarithmic in the lattice dimension, just like the depth of the decryption circuit, but the latter is greater than
Monotone Circuits for Matching Require Linear Depth
"... We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. ..."
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Cited by 82 (10 self)
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We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.
Almost Optimal Lower Bounds for Small Depth Circuits
, 1986
"... We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when ..."
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Cited by 277 (8 self)
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We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size
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 328 (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
On the Power of SmallDepth Threshold Circuits
, 1990
"... We investigate the power of threshold circuits of small depth. In particular we give functions which require exponential size unweigted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are mone tone functions fk which can be computed in depth k and linear s ..."
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Cited by 121 (2 self)
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size A, Vcircuits but require exponential size to compute by a depth k 1 monotone weighted threshold circuit.
Parallel Prefix Computation
 JOURNAL OF THE ACM
, 1980
"... The prefix problem is to compute all the products x t o x2.... o xk for i ~ k. ~ n, where o is an associative operation A recursive construction IS used to obtain a product circuit for solving the prefix problem which has depth exactly [log:n] and size bounded by 4n An application yields fast, smal ..."
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Cited by 337 (1 self)
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The prefix problem is to compute all the products x t o x2.... o xk for i ~ k. ~ n, where o is an associative operation A recursive construction IS used to obtain a product circuit for solving the prefix problem which has depth exactly [log:n] and size bounded by 4n An application yields fast
Cryptographic Limitations on Learning Boolean Formulae and Finite Automata
 PROCEEDINGS OF THE TWENTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1989
"... In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntact ..."
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Cited by 347 (14 self)
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of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of wellknown publickey cryptosystems to the learning problems. We prove that a polynomialtime learning algorithm for Boolean formulae, deterministic finite automata or constantdepth
vices. Monotone Circuits for Matching Require Linear Depth
, 2003
"... We prove that monotone circuits computing the perfect matching function on nvertex graphs require Ω(n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. ..."
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We prove that monotone circuits computing the perfect matching function on nvertex graphs require Ω(n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.
Monotone Complexity and the Rank of Matrices
, 2001
"... blem, because what is computed in the game is not a function, but only a relation. On the other hand, computing the (two party) communication complexity of a function is usually not dicult, thus the task of proving lower bounds can be considerably simplied, if one could replace the relation in the K ..."
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Cited by 5 (0 self)
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in the KW game by a function. 1 Let us consider the case of monotone circuit depth. The KW game for a monotone function f is dened as follows. One player gets a minterm of f , the other gets a maxterm of f . Think of the min and maxterms as subsets of [n]. The goal of the players is to nd an i 2 [n
Results 1  10
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240,586