A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas

Authors Suryajith Chillara, Nutan Limaye, Srikanth Srinivasan

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Suryajith Chillara
  • Department of CSE, IIT Bombay, Mumbai, India
Nutan Limaye
  • Department of CSE, IIT Bombay, Mumbai, India
Srikanth Srinivasan
  • Department of Mathematics, IIT Bombay, Mumbai, India

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Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We show explicit separations between the expressive powers of multilinear formulas of small-depth and all polynomial sizes. Formally, for any s = s(n) = n^{O(1)} and any delta>0, we construct explicit families of multilinear polynomials P_n in F[x_1,...,x_n] that have multilinear formulas of size s and depth three but no multilinear formulas of size s^{1/2-delta} and depth o(log n/log log n). As far as we know, this is the first such result for an algebraic model of computation. Our proof can be viewed as a derandomization of a lower bound technique of Raz (JACM 2009) using epsilon-biased spaces.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Algebraic circuit complexity
  • Multilinear formulas
  • Lower Bounds


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