Results 1  10
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15
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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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.
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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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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Local reductions
, 2013
"... We reduce nondeterministic time T ≥ 2 n to a 3SAT instance φ of size φ  = T ·log O(1) T such that there is an explicit circuit C that on input an index i of logφ bits outputs the ith clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC 1. Even in th ..."
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Cited by 3 (1 self)
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We reduce nondeterministic time T ≥ 2 n to a 3SAT instance φ of size φ  = T ·log O(1) T such that there is an explicit circuit C that on input an index i of logφ bits outputs the ith clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC 1. Even in the simpler setting of φ  = poly(T) the previous best result was C in AC 0. More generally, for any time T ≥ n and parameter r ≤ n we obtain log 2φ  = max(logT,n/r)+O(logn)+O(loglogT) and each output bit of C is a decision tree of depth O(logr). As an application, we simplify the proof of Williams ’ ACC 0 lower bound, and tighten his connection between satisfiability algorithms and lower bounds.
Ironic Complicity: Satisfiability Algorithms and Circuit Lower
"... The 1980's was a golden period for Boolean circuit complexity lower bounds. There were major breakthroughs. For example, Razborov's exponential size lower bound for monotone Boolean circuits computing the Clique function and the RazborovSmolensky superpolynomial size lower bounds for cons ..."
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The 1980's was a golden period for Boolean circuit complexity lower bounds. There were major breakthroughs. For example, Razborov's exponential size lower bound for monotone Boolean circuits computing the Clique function and the RazborovSmolensky superpolynomial size lower bounds for constantdepth circuits with MODp gates for prime p. These results made researchers optimistic of progress on big lower bound questions and complexity class separations. However, in the last two decades, this optimism gradually turned into despair. We still do not know how to prove superpolynomial lower bounds for constantdepth circuits with MOD6 gates for a function computable in exponential time. Ryan Williams ' exciting lower bound result of 2011, that nondeterministic exponential time does not have polynomialsize unbounded fanin constantdepth circuits with MODm gates for any composite m, has renewed optimism in the area. The best part is that his approach is potentially applicable to other lower bound questions. In this wonderful article, Rahul Santhanam explores this theme of connections between improved SAT algorithms and circuit lower bounds.
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
Compression of Boolean Functions
, 2013
"... We consider the problem of compression for “easy ” Boolean functions: 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 tha ..."
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We consider the problem of compression for “easy ” Boolean functions: 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 both positive and negative results. On the positive side, we show that several circuit classes for which lower bounds are proved by a method of random restrictions: • AC0, • (de Morgan) formulas, and • (readonce) branching programs, allow nontrivial compression for circuits up to the size for which lower bounds are known. On the negative side, we show that compressing functions from any class C ⊆ P/poly implies superpolynomial lower bounds against C for a function in NEXP; we also observe that compressing monotone functions of polynomial circuit complexity or functions computable by largesize AC0 circuits would also imply new superpolynomial circuit lower bounds. Finally, we apply the ideas used for compression to get zeroerror randomized #SATalgorithms for de Morgan and completebasis formulas, as well as branching programs, on n variables of about quadratic size that run in expected time 2n/2n ϵ, for some ϵ> 0 (dependent on the size of the formula/branching program). ∗Research partially supported by an NSERC Discovery grant. †Research partially supported by an NSERC Discovery grant. 1