Results 1  10
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14
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.
A Satisfiability Algorithm for AC 0
, 2011
"... We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is cons ..."
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Cited by 15 (2 self)
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We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is constant. Let d denote the depth of the circuit and cn denote the number of gates. This algorithm runs in time C2 n(1−µc,d) where C  is the size of the circuit for µc,d ≥ 1/O[lg c + d lg d] d−1 with probability at least 1 − 2 −n. As a result, we get improved exponential time algorithms for AC 0 circuit satisfiability and for counting solutions. In addition, we get an improved bound on the correlation of AC 0 circuits with parity. As an important component of our analysis, we extend the H˚astad Switching Lemma to handle multiple kcnfs and kdnfs. 1
Mining circuit lower bound proofs for metaalgorithms
, 2013
"... We show that circuit lower bound proofs based on the method of random restrictions yield nontrivial compression algorithms for “easy ” Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an nvariate Boolean function f co ..."
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Cited by 6 (0 self)
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We show that circuit lower bound proofs based on the method of random restrictions yield nontrivial compression algorithms for “easy ” Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an nvariate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2n/n. We get nontrivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (readonce) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of “easy ” functions, which are useful both for proving circuit lower bounds and for designing “metaalgorithms” (such as CircuitSAT). For (de Morgan) formulas, such structural characterization is provided by the “shrinkage under random restrictions ” results [Sub61, H̊as98], strengthened to the “highprobability ” version by [San10, IMZ12, KR13]. We give a new, simple proof of the “highprobability ” version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n2. We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [KR13] of the averagecase lower bound against small (de Morgan) formulas. Finally, we show that the existence of any nontrivial compression algorithm for a circuit class C ⊆ P/poly would imply the circuit lower bound NEXP 6 ⊆ C; a similar implication is independently proved also by Williams [Wil13]. This complements Williams’s result [Wil10] that any nontrivial CircuitSAT algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP.
Approximating AC0 by small height decision trees and a deterministic algorithm for #AC0SAT
 In Proceedings of the TwentySeventh Annual IEEE Conference on Computational Complexity
, 2012
"... We show how to approximate any function in AC0 by decision trees of much smaller height than its number of variables. More precisely, we show that any function in n variables computable by an unbounded fanin circuit of AND, OR, and NOT gates that has size S and depth d can be approximated by a deci ..."
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We show how to approximate any function in AC0 by decision trees of much smaller height than its number of variables. More precisely, we show that any function in n variables computable by an unbounded fanin circuit of AND, OR, and NOT gates that has size S and depth d can be approximated by a decision tree of height n − βn to within error exp(−βn), where β = β(S, d) = 2−O(d log4/5 S). Our proof is constructive and we use its constructivity to derive a deterministic algorithm for #AC0SAT with multiplicative factor savings over the naive 2nS algorithm of 2−Ω(βn), when applied to any ninput AC0 circuit of size S and depth d. Indeed, in the same running time we can deterministically construct a decision tree of size at most 2n−βn that exactly computes the function given by such a circuit. Recently, Impagliazzo, Matthews, and Paturi derived an algorithm for #AC0SAT with greater savings over the naive algorithm but their algorithm is only randomized rather than deterministic. The main technical result we prove to show the above is that for every family F of kDNF formulas in n variables and every 1 < C = C(n) ≤ logpoly(k) F, one can construct a distribution on restrictions that each set at most n/C variables such that, except with probability at most 2−n/(2 O(k)C log F), after application of the restriction, all formulas in F simultaneously reduce to logpoly(k) Fjuntas where an sjunta is a function whose value depends on only s of its inputs. Previously, Ajtai showed simultaneous approximations for kDNF formulas by juntas related to the one we show but with a dependence on exp(k) rather than poly(k), resulting in a weaker heightapproximation tradeoff than ours.
A satisfiability algorithm for sparse depth two threshold circuits
 In Proceedings of the 54th Annual Symposium on the Foundations of Computer Science (FOCS 2013
, 2013
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Averagecase lower bounds for formula size
 Electronic Colloquium on Computational Complexity (ECCC
, 2012
"... We give an explicit function h: {0, 1} n → {0, 1} such that any deMorgan formula + ɛ fraction of the inputs, where ɛ is of size O(n 2.499) agrees with h on at most 1 2 exponentially small (i.e. ɛ = 2−nΩ(1)). We also show, using the same technique, that any boolean formula of size O(n1.999) over the ..."
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Cited by 4 (1 self)
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We give an explicit function h: {0, 1} n → {0, 1} such that any deMorgan formula + ɛ fraction of the inputs, where ɛ is of size O(n 2.499) agrees with h on at most 1 2 exponentially small (i.e. ɛ = 2−nΩ(1)). We also show, using the same technique, that any boolean formula of size O(n1.999) over the complete basis, agrees with h on at most 1 2 + ɛ fraction of the inputs, where ɛ is exponentially small (i.e. ɛ = 2−nΩ(1)). Our construction is based on Andreev’s Ω(n2.5−o(1) ) formula size lower bound that was proved for the case of exact computation [And87]. 1
Improved AverageCase Lower Bounds for DeMorgan Formula Size Matching WorstCase Lower Bound
"... We give a function h: {0, 1} n → {0, 1} such that every deMorgan formula of size n3−o(1) /r2 agrees with h on at most a fraction of 1 2 + 2−Ω(r) of the inputs. This improves the previous averagecase lower bound of Komargodski and Raz (STOC, 2013). Our technical contributions include a theorem that ..."
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Cited by 3 (0 self)
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We give a function h: {0, 1} n → {0, 1} such that every deMorgan formula of size n3−o(1) /r2 agrees with h on at most a fraction of 1 2 + 2−Ω(r) of the inputs. This improves the previous averagecase lower bound of Komargodski and Raz (STOC, 2013). Our technical contributions include a theorem that shows that the “expected shrinkage” result of H˚astad (SIAM J. Comput., 1998) actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), combining ideas of both Impagliazzo, Meka and Zuckerman (FOCS, 2012) and Komargodski and Raz. In addition, using a bitfixing extractor in the construction of h allows us to simplify a major part of the analysis of Komargodski and Raz. 1 1
On the Limits of Sparsification
, 2012
"... Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural ..."
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Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural question is whether an analogous structural result holds for CNFs or even for broader nonuniform classes such as constantdepth circuits or Boolean formulae. We prove a very strong negative result in this connection: For every superlinear function f(n), there are CNFs of size f(n) which cannot be written as a disjunction of 2 n−εn CNFs each having a linear number of clauses for any ε> 0. We also give a hierarchy of such nonsparsifiable CNFs: For every k, there is a k ′ for which there are CNFs of size n k′ which cannot be written as a subexponential size disjunction of CNFs of size n k. Furthermore, our lower bounds hold not just against CNFs but against an arbitrary family
Towards NP−P via Proof Complexity and Search
, 2009
"... This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP. ..."
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This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP.