Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Authors Mahdi Cheraghchi , Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis

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Author Details

Mahdi Cheraghchi
  • Department of Computing, Imperial College London, London, UK
Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Zhenjian Lu
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Dimitrios Myrisiotis
  • Department of Computing, Imperial College London, London, UK


We thank the anonymous ICALP'19 reviewers for their excellent comments.

Cite AsGet BibTex

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis. Circuit Lower Bounds for MCSP from Local Pseudorandom Generators. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
  • minimum circuit size problem (MCSP)
  • circuit lower bounds
  • pseudorandom generators (PRGs)
  • local PRGs
  • de Morgan formulas
  • branching programs
  • constant depth circuits


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