Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

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 theta, for a given parameter theta. 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^{Omega (N^{1/(d+2.01)})}-size depth-d AC^0 circuits, improving the superpolynomial 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 almost optimal lower bound of 2^{N^{1-o(1)}} for MCSP.

[1]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[2]  Igor S. Sergeev,et al.  Complexity of computation in finite fields , 2013 .

[3]  Noam Nisan,et al.  Randomness is Linear in Space , 1996, J. Comput. Syst. Sci..

[4]  Shachar Lovett,et al.  Correlation bounds for poly-size AC 0 circuits with n 1-o(1) symmetric gates , 2011 .

[5]  Shuichi Hirahara,et al.  Non-Black-Box Worst-Case to Average-Case Reductions within NP , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[6]  Igor Carboni Oliveira,et al.  Hardness Magnification for Natural Problems , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[7]  Avishay Tal,et al.  Tight bounds on The Fourier Spectrum of AC0 , 2017, Electron. Colloquium Comput. Complex..

[8]  Jin-Yi Cai,et al.  Circuit minimization problem , 2000, STOC '00.

[9]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[10]  Boris A. Trakhtenbrot,et al.  A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms , 1984, Annals of the History of Computing.

[11]  Eric Allender,et al.  Power from random strings , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[12]  Russell Impagliazzo,et al.  Pseudorandomness from Shrinkage , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[13]  Or Meir,et al.  Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity , 2016, Electron. Colloquium Comput. Complex..

[14]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[15]  Madhur Tulsiani,et al.  Improved Pseudorandom Generators for Depth 2 Circuits , 2010, APPROX-RANDOM.

[16]  Luca Trevisan,et al.  A Derandomized Switching Lemma and an Improved Derandomization of AC0 , 2013, 2013 IEEE Conference on Computational Complexity.

[17]  Shachar Lovett,et al.  Pseudorandom Generators from Polarizing Random Walks , 2018, Electron. Colloquium Comput. Complex..

[18]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[19]  Rahul Santhanam,et al.  On the Average-Case Complexity of MCSP and Its Variants , 2017, CCC.

[20]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

[21]  Russell Impagliazzo,et al.  Learning Algorithms from Natural Proofs , 2016, CCC.

[22]  Igor Carboni Oliveira,et al.  Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness , 2016, CCC.

[23]  Avi Wigderson,et al.  Deterministic approximate counting of depth-2 circuits , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[24]  Emanuele Viola,et al.  Pseudorandom bits for constant depth circuits with few arbitrary symmetric gates , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[25]  Avi Wigderson,et al.  Deterministic simulation of probabilistic constant depth circuits , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[26]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[27]  Rocco A. Servedio,et al.  Luby-Veličković-Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits , 2018, APPROX-RANDOM.

[28]  Johan Håstad The Shrinkage Exponent of de Morgan Formulas is 2 , 1998, SIAM J. Comput..

[29]  R. Santhanam,et al.  27 : 2 Hardness Magnification near State-OfThe-Art Lower Bounds , 2019 .

[30]  Avishay Tal,et al.  Formula lower bounds via the quantum method , 2017, STOC.

[31]  Avishay Tal,et al.  Shrinkage of De Morgan Formulae by Spectral Techniques , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.