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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
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
Strong ETH and Resolution via Games and the Multiplicity of Strategies
"... We consider a restriction of the Resolution proof system in which at most a fixed number of variables can be resolved more than once along each refutation path. This system lies between regular Resolution, in which no variable can be resolved more than once along any path, and general Resolution whe ..."
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We consider a restriction of the Resolution proof system in which at most a fixed number of variables can be resolved more than once along each refutation path. This system lies between regular Resolution, in which no variable can be resolved more than once along any path, and general Resolution where there is no restriction on the number of such variables. We show that when the number of reresolved variables is not too large, this proof system is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely for large n and k we show that there are unsatisfiable kCNF formulas which require Resolution refutations of size 2(1−k)n, where n is the number of variables and k = Õ(k−1/5), whenever in each refutation path we only allow at most Õ(k−1/5)n variables to be resolved multiple times. However, these reresolved variables along different paths do not need to be the same. Prior to this work, the strongest proof system shown to be consistent with SETH was regular Resolution [Beck and Impagliazzo, STOC’13]. This work strengthens that result and gives a different and conceptually simpler gametheoretic proof for the case of regular Resolution. 1