We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every De Morgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs.
Our technical contributions include a theorem that shows that the ``expected shrinkage'' result of Hästad (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), using ideas of Impagliazzo, Meka and Zuckerman (FOCS, 2012).
Fixed several typos, fixed a bug from the previous version that appeared in Appendix A (and moved the relevant proof to Section 4), and
improved the formula size lower bounds by using the tight shrinkage result from [Tal14].
We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).
Our technical contributions include a theorem that shows that the ``expected shrinkage'' result of Håstad (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 bit-fixing extractor in the construction of $h$ allows us to simplify a major part of the analysis of Komargodski and Raz.
Fixed a typo in the abstract.
We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).
Our technical contributions include a theorem that shows that the ``expected shrinkage'' result of Hästad (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 bit-fixing extractor in the construction of $h$ allows us to simplify a major part of the analysis of Komargodski and Raz.
