Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

02/20/2020
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by   Valentine Kabanets, et al.
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The class FORMULA[s] āˆ˜š’¢ consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class š’¢. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^1.99]āˆ˜š’¢, for classes š’¢ of functions with low communication complexity. Let R^(k)(š’¢) be the maximum k-party NOF randomized communication complexity of š’¢. We show: (1) The Generalized Inner Product function GIP^k_n cannot be computed in FORMULA[s]āˆ˜š’¢ on more than 1/2+Īµ fraction of inputs for s = o ( n^2/(k Ā· 4^k Ā·R^(k)(š’¢) Ā·log (n/Īµ) Ā·log(1/Īµ) )^2). As a corollary, we get an average-case lower bound for GIP^k_n against FORMULA[n^1.99]āˆ˜ PTF^k-1. (2) There is a PRG of seed length n/2 + O(āˆš(s)Ā· R^(2)(š’¢) Ā·log(s/Īµ) Ā·log (1/Īµ) ) that Īµ-fools FORMULA[s] āˆ˜š’¢. For FORMULA[s] āˆ˜ LTF, we get the better seed length O(n^1/2Ā· s^1/4Ā·log(n)Ā·log(n/Īµ)). This gives the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where Īµā‰¤ 1/n. (3) There is a randomized 2^n-t-time #SAT algorithm for FORMULA[s] āˆ˜š’¢, where t=Ī©(n/āˆš(s)Ā·log^2(s)Ā· R^(2)(š’¢))^1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n^1.99]āˆ˜ LTF. (4) The Minimum Circuit Size Problem is not in FORMULA[n^1.99]āˆ˜ XOR. On the algorithmic side, we show that FORMULA[n^1.99] āˆ˜ XOR can be PAC-learned in time 2^O(n/log n).

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