Circuits with Medium Fan-In

Authors Pavel Hrubes, Anup Rao

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Pavel Hrubes
Anup Rao

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Pavel Hrubes and Anup Rao. Circuits with Medium Fan-In. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We consider boolean circuits in which every gate may compute an arbitrary boolean function of k other gates, for a parameter k. We give an explicit function $f:{0,1}^n -> {0,1} that requires at least Omega(log^2(n)) non-input gates when k = 2n/3. When the circuit is restricted to being layered and depth 2, we prove a lower bound of n^(Omega(1)) on the number of non-input gates. When the circuit is a formula with gates of fan-in k, we give a lower bound Omega(n^2/k*log(n)) on the total number of gates. Our model is connected to some well known approaches to proving lower bounds in complexity theory. Optimal lower bounds for the Number-On-Forehead model in communication complexity, or for bounded depth circuits in AC_0, or extractors for varieties over small fields would imply strong lower bounds in our model. On the other hand, new lower bounds for our model would prove new time-space tradeoffs for branching programs and impossibility results for (fan-in 2) circuits with linear size and logarithmic depth. In particular, our lower bound gives a different proof for a known time-space tradeoff for oblivious branching programs.
  • Boolean circuit
  • Complexity
  • Communication Complexity


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