Small Bias Requires Large Formulas

Author Andrej Bogdanov

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Andrej Bogdanov
  • Department of Computer Science and Engineering and , Institute of Theoretical Computer Science and Communications, Chinese University of Hong Kong.

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Andrej Bogdanov. Small Bias Requires Large Formulas. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 22:1-22:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We demonstrate that known explicit lower bounds on (1) the size of general Boolean formulas, (2) the size of De Morgan formulas, and (3) correlation against small De Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best-known explicit formula lower bounds in all these models. On the other hand, we give a construction of a small-biased function that is tight with respect to lower bound (1) for the relevant range of parameters. We interpret this construction as a natural-type barrier against substantially stronger lower bounds for general formulas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
  • formula lower bounds
  • natural proofs
  • pseudorandomness


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