Cubic Formula Size Lower Bounds Based on Compositions with Majority

Authors Anna Gál, Avishay Tal, Adrian Trejo Nuñez

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Author Details

Anna Gál
  • The University of Texas at Austin, Austin, TX, USA
Avishay Tal
  • Stanford University, Palo Alto, CA, USA
Adrian Trejo Nuñez
  • The University of Texas at Austin, Austin, TX, USA

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Anna Gál, Avishay Tal, and Adrian Trejo Nuñez. Cubic Formula Size Lower Bounds Based on Compositions with Majority. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We define new functions based on the Andreev function and prove that they require n^{3}/polylog(n) formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the majority function (or its negation) on the middle slices of the Boolean cube, as well as iterated compositions of such functions. As a consequence, we obtain n^{3}/polylog(n) lower bounds on the (non-monotone) formula size of an explicit monotone function by combining the monotone address function with the majority function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Circuit complexity
  • formula lower bounds
  • random restrictions
  • KRW conjecture
  • composition


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