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15
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.
Pseudorandomness for regular branching programs via fourier analysis
 In APPROXRANDOM
, 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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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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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
Pseudorandomness for multilinear readonce algebraic branching programs
 in any order. Electronic Colloquium on Computational Complexity (ECCC
"... We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. F ..."
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We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. Furthermore, our result has no known analogue in the model of readonce oblivious boolean branching programs with unknown order, as despite recent work (eg. [BPW11, IMZ12, RSV13]) there is no known pseudorandom generator for this model with subpolynomial seedlength (for unboundedwidth branching programs). This result extends and generalizes the result of Forbes and Shpilka [FS12b] that obtained a nO(lgn)time algorithm when given the order. We also extend and strengthen the work of Agrawal, Saha and Saxena [ASS12] that gave a blackbox algorithm running in time exp((lg n)Ω(d)) for setmultilinear formulas of depth d. We note that the model of multilinear ROABPs contains the model of setmultilinear algebraic branching programs, which itself contains the model of setmultilinear formulas of arbitrary depth. We obtain our results by recasting,
Polynomial Identity Testing of ReadOnce Oblivious Algebraic Branching Programs
, 2014
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are readonce and oblivious. This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shp ..."
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are readonce and oblivious. This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work there was no known such blackbox algorithm. The main result of this work gives the first quasipolynomial sized hitting set for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg2 S), this is analogous (in the terminology of boolean pseudorandomness) to a seedlength of lg2 S, which is the seed length of the pseudorandom generators of Nisan [Nis92] and ImpagliazzoNisanWigderson [INW94] for readonce oblivious
NegationLimited Formulas
"... Understanding the power of negation gates is crucial to bridge the exponential gap between monotone and nonmonotone computation. We focus on the model of formulas over the De Morgan basis and consider it in a negationlimited setting. We prove that every formula that contains t negation gates can b ..."
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Understanding the power of negation gates is crucial to bridge the exponential gap between monotone and nonmonotone computation. We focus on the model of formulas over the De Morgan basis and consider it in a negationlimited setting. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. Moreover, we show that averagecase lower bounds for monotone formulas can be extended to get averagecase lower bounds for formulas with few negations. Using the averagecase lower bound for polynomialsize monotone formulas of Rossman (CCC ’15), we obtain an averagecase lower bound for polynomialsize formulas with n1/2−o(1) negations, where n is the input size. Recently, circuits with few negations have drawn much attention in various areas of theoretical computer science. Specifically, Blais et al. (ECCC ’14) studied the uniformdistribution learnability of circuits with few negations, and Guo et al. (TCC ’15) proved lower bounds on the
Pseudorandomness and fourier growth bounds for width 3 branching programs, CoRR abs/1405.7028
, 2014
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Progress on Polynomial Identity Testing  II
, 2013
"... We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years. ..."
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We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.
HITTINGSETS FOR ROABP AND SUM OF SETMULTILINEAR CIRCUITS
"... Abstract. We give a nO(logn)time (n is the input size) blackbox polynomial identity testing algorithm for unknownorder readonce oblivious arithmetic branching programs (ROABP). The best timecomplexity known for blackbox PIT for this class was nO(log 2 n) due to ForbesSaptharishiShpilka (STOC 2 ..."
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Abstract. We give a nO(logn)time (n is the input size) blackbox polynomial identity testing algorithm for unknownorder readonce oblivious arithmetic branching programs (ROABP). The best timecomplexity known for blackbox PIT for this class was nO(log 2 n) due to ForbesSaptharishiShpilka (STOC 2014). Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the timecomplexity for the unknown order ROABP with the known order ROABP (due to ForbesShpilka (FOCS 2013)) and also with the depth3 setmultilinear circuits (due to AgrawalSahaSaxena (STOC 2013)). Our proof is simpler and involves a new technique called basis isolation. The depth3 model has recently gained much importance, as it has become a stepping stone to understanding general arithmetic circuits. Multilinear depth3 circuits are known to have exponential lower bounds but no polynomial time blackbox identity tests. In this paper, we take a step towards designing such hittingsets. We give the first subexponential whitebox PIT for the sum of constantly many setmultilinear depth3 circuits. To achieve this, we define the notions of distance and base sets. Distance, for a multilinear depth3 circuit (say, in n variables and k product gates), measures how far are the variable partitions corresponding to the product gates, from being a mere refinement of each other. The 1distance circuits strictly contain the setmultilinear model, while ndistance captures general multilinear depth3. We design a hittingset in time (nk)O( ∆ logn) for ∆distance. Further, we give an extension of our result to models where the distance is large (close to n) but it is small when restricted to certain base sets (of variables). We also explore a new model of readonce arithmetic branching programs (ROABP) where the factormatrices are invertible (called invertiblefactor ROABP). We design a hittingset in time poly(nw 2) for widthw invertiblefactor ROABP. Further, we could do without the invertibility restriction when w = 2. Previously, the best result for width2 ROABP was quasipolynomial time (ForbesSaptharishiShpilka, STOC 2014). 1. Introduction. The problem of Polynomial Identity Testing