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13
Uniform ConstantDepth Threshold Circuits for Division and Iterated Multiplication
, 2002
"... this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equival ..."
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Cited by 43 (9 self)
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this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes
The complexity of constructing pseudorandom generators from hard functions
 Computational Complexity
, 2004
"... Abstract. We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constantdepth circuits. We show that, starting from a function f: {0, 1} l → {0, 1} computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a ..."
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Cited by 37 (9 self)
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Abstract. We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constantdepth circuits. We show that, starting from a function f: {0, 1} l → {0, 1} computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant ɛ> 0 such that every circuit of size 2 ɛl fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a PRG: {0, 1} O(log n) → {0, 1} n computable by DLOGTIMEuniform constantdepth circuits of size polynomial in n. Such a PRG implies BP · AC 0 = AC 0 under DLOGTIMEuniformity. On the negative side, we prove that starting from a worstcase hard function f: {0, 1} l → {0, 1} (i.e. there is a constant ɛ> 0 such that every circuit of size 2 ɛl fails to compute f on some input) for every positive constant δ < 1 there is no blackbox construction of a PRG: {0, 1} δn → {0, 1} n computable by constantdepth circuits of size polynomial in n. We also study worstcase hardness amplification, which is the related problem of producing an averagecase hard function starting from a worstcase hard one. In particular, we deduce that there is no blackbox worstcase hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constantdepth circuits cannot compute good extractors and listdecodable codes.
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 30 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
TimeSpace Tradeoffs in the Counting Hierarchy
, 2001
"... We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform ..."
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Cited by 20 (5 self)
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We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform TC 0 circuits for iterated multiplication [9]. Here is an
On approximate majority and probabilistic time
 in Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and ..."
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Cited by 19 (6 self)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�size depth3 circuits for Approximate Majority on n bits have bottom fanin Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomialsize circuits of depth 3. Prior to our work, the only known polynomialsize depth3 circuits for Approximate Majority were nonuniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a randomaccess input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
Hardness vs. Randomness within Alternating Time
, 2003
"... We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f: {0, 1} l → {0, 1} that is mildly hard on average, i.e. every circuit of size 2 Ω(l) fails to compute f on at least a 1/poly(l) fraction of in ..."
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Cited by 10 (0 self)
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We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f: {0, 1} l → {0, 1} that is mildly hard on average, i.e. every circuit of size 2 Ω(l) fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0, 1} O(log n) → {0, 1} n computable in ATIME(O(1), log n) = alternating time O(log n) with O(1) alternations. Such a PRG implies BP · AC0 = AC0 under DLOGTIMEuniformity. On the negative side, we prove a tight lower bound on blackbox PRG constructions that are based on worstcase hard functions. We also prove a tight lower bound on blackbox worstcase hardness amplification, which is the problem of producing an averagecase hard function starting from a worstcase hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and listdecodable codes.
Minimizing DNF Formulas and AC^0 Circuits Given a Truth Table
 IN PROCEEDINGS OF THE 21ST ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2006
"... For circuit classes R, the fundamental computational problem MinR asks for the minimum Rsize of a Boolean function presented as a truth table. Prominent examples of this problem include MinDNF, which asks whether a given Boolean function presented as a truth table has a kterm DNF, and MinCircu ..."
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Cited by 10 (0 self)
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For circuit classes R, the fundamental computational problem MinR asks for the minimum Rsize of a Boolean function presented as a truth table. Prominent examples of this problem include MinDNF, which asks whether a given Boolean function presented as a truth table has a kterm DNF, and MinCircuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that MinDNF is NPcomplete. It is significantly simpler than the known reduction of Masek [30], which is from CircuitSAT. We then give a more complex reduction, yielding the result that MinDNF cannot be approximated to within a factor smaller than (logN) γ, for some constant γ> 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate MinDNF. The question of whether MinDNF can be approximated to within a factor of o(logN) remains open, but we construct an instance of MinDNF on which the solution produced by the greedy algorithm is Ω(logN) larger than optimal. Finally, we turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth two circuits of AND and OR gates. Depth d circuits are denoted by AC0 d. We show that it is hard to approximate the size of AC0 d circuits (for large enough d) under cryptographic assumptions.
On the Power of SmallDepth Computation
, 2009
"... In this work we discuss selected topics on smalldepth computation, presenting a few unpublished proofs along the way. The four chapters contain: 1. A unified treatment of the challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. 2. An unpubl ..."
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Cited by 5 (4 self)
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In this work we discuss selected topics on smalldepth computation, presenting a few unpublished proofs along the way. The four chapters contain: 1. A unified treatment of the challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. 2. An unpublished proof that small boundeddepth circuits (AC 0) have exponentially small correlation with the parity function. The proof is due to Klivans and Vadhan; it builds upon and simplifies previous ones. 3. Valiant’s simulation of logdepth linearsize circuits of fanin 2 by subexponential size circuits of depth 3 and unbounded fanin. To our knowledge, a proof of this result has never appeared in full.
On probabilistic time versus alternating time
 Electronic Colloquium on Computational Complexity
, 2005
"... We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gá ..."
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Cited by 3 (0 self)
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We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime(t) ⊆ ΣcTime(t) for a large constant c> 3 (Ajtai, Adv. in Comp. Complexity Theory ’93). 2. We prove that BPTime(t) � ⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements our result (1), and shows that the running time of the SipserGácsLautemann simulation is optimal, up to a log t factor, for relativizing techniques. (All the results in (1) relativize.) This result is obtained as a corollary from a new circuit lower bound for approximate majority: poly(n)size depth3 circuits for approximate majority have bottom fanin Ω(log n). 3. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on probabilistic Turing machines using space n.9, with random access to input and work tapes, and twoway sequential access to the randombit tape. This is the first lower bound of the form t = n 1+Ω(1) on a model with random access to the input and twoway access to the random bits. 4. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on Turing machines with an input tape and a sequential work tape that is initialized with random bits. This is the first lower bound on a probabilistic extension of the offline Turing machine model with one work tape.
The communication complexity of addition
, 2011
"... Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai ..."
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Cited by 1 (1 self)
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Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (publiccoin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are oneway, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree(k − 1) polynomialthreshold functions in the numberonforehead model. We give a (publiccoin, 2player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).