Abstract
The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most θ, for a given parameter θ. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N , requires • N 3− o (1) -size de Morgan formulas, improving the recent N 2− o (1) lower bound by Hirahara and Santhanam (CCC, 2017), • N 2− o (1) -size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and • 2 Ω( N 1/( d +1.01)) -size depth- d AC 0 circuits, improving the (implicit, in their work) exponential size lower bound by Allender et al. (SICOMP, 2006). The AC 0 lower bound stated above matches the best-known AC 0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an optimal lower bound of 2 Ω( N ) for MCSP.
Highlights
Given the truth table of some Boolean function f and a size parameter θ, the minimum circuit size problem (MCSP) asks whether f can be computed by a circuit of size at most θ
Can we show that computing MCSP requires depth-d AC0 circuits of size 2N1/(d+O(1)) and de Morgan formulas of size N 3−o(1)? can we show lower bounds for MCSP against some other restricted models that match their state-of-the-art lower bounds? In this paper, we answer these questions in the affirmative
By inspecting the construction, the IMZ pseudorandom generators (PRGs) does not seem to have such a property, and a straightforward implementation seems to require a circuit of size at least s2/3, which yields a weaker lower bound for MCSP
Summary
Given the truth table of some Boolean function f and a size parameter θ, the minimum circuit size problem (MCSP) asks whether f can be computed by a circuit of size at most θ. Two of the most studied restricted computational models in complexity theory are constant-depth circuits (AC0) and de Morgan formulas. For de Morgan formulas, the state-of-the-art lower bound is almost cubic, namely N 3−o(1), for some polynomial-time computable function [7, 18, 19, 5]. Allender et al [2] showed that MCSP, on functions represented as a truth table of length N , cannot be computed by polynomial-size constant-depth AC0 circuits. For de Morgan formulas, Hirahara and Santhanam [9] showed that computing MCSP requires de Morgan formulas of size N 2−o(1) Given these two MCSP lower bounds and the best-known lower bounds against these two models, it is natural to ask whether we can get MCSP lower bounds against small-depth circuits and de Morgan formulas that match the state-of-the-art lower bounds against these models. Can we show that computing MCSP requires depth-d AC0 circuits of size 2N1/(d+O(1)) and de Morgan formulas of size N 3−o(1)? can we show lower bounds for MCSP against some other restricted models that match their state-of-the-art lower bounds? In this paper, we answer these questions in the affirmative
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