Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

摘要

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 , for a given parameter . 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 N 1 ( d +2 01) -size depth- d A C 0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006). The A C 0 lower bound stated above matches the best-known A C 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 almost optimal lower bound of 2 N 1 ? o (1) for MCSP.