research-article

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

Authors:

Valentine Kabanets,

Sajin Koroth,

Zhenjian Lu,

Dimitrios Myrisiotis,

Igor C. OliveiraAuthors Info & Claims

ACM Transactions on Computation Theory (TOCT), Volume 13, Issue 4

Article No.: 23, Pages 1 - 37

https://doi.org/10.1145/3470861

Published: 01 September 2021 Publication History

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Abstract

The class FORMULA[s]∘G consists of Boolean functions computable by size-s De Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and pseudorandom generators (PRGs)) algorithms for FORMULA[n^1.99]∘G, for classes G of functions with low communication complexity. Let R^(k)G be the maximum k-party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following:

•

The Generalized Inner Product function GIP^k_n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for

s=o(n²/k⋅4^k⋅R^(k)(G)⋅log⁡(n/ε)⋅log⁡(1/ε))²).

This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP^k_n against FORMULA[n^1.99]∘PTF^k−1, i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF (polynomial threshold function) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs.

•

There is a PRG of seed length n/2+O(s⋅R⁽²⁾(G)⋅log⁡(s/ε)⋅log⁡(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^1/2⋅s^1/4⋅log⁡(n)⋅log⁡(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45].

•

There exists a randomized 2^n-t #SAT algorithm for FORMULA[s]∘G, where

t=Ω(n\√s⋅log²⁡(s)⋅R⁽²⁾(G))/1/2.

In particular, this implies a nontrivial #SAT algorithm for FORMULA[n^1.99]∘LTF.

•

The Minimum Circuit Size Problem is not in FORMULA[n^1.99]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n^1.99]∘XOR can be PAC-learned in time 2^{O(n/log n)}.

References

[1]

Amir Abboud and Karl Bringmann. 2018. Tighter connections between formula-SAT and shaving logs. In ICALP. 8:1–8:18.

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