Computing symmetric functions with AND/OR circuits and a single MAJORITY gate

  • Zhi-Li Zhang
  • David A. Mix Barrington
  • Jun Tarui
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)


Fagin et al. characterized those symmetric Boolean functions which can be computed by small AND/OR circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for d-perceptronsAND/OR circuits of constant depth and unbounded fan-in with a single MAJORITY gate at the output. We show that a symmetric function has small (quasipolynomial, or \(2^{\log ^{O(1)n} }\) size) d-perceptrons iff it has only poly-log many sign changes (i.e., it changes value logO(1)n times as the number of positive inputs varies from zero to n). A consequence of the lower bound is that a recent construction of Beigel is optimal. He showed how to convert a constant-depth unbounded fan-in AND/OR circuit with poly-log many MAJORITY gates into an equivalent d-perceptron — we show that more than poly-log MAJORITY gates cannot in general be converted to one.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Zhi-Li Zhang
    • 1
  • David A. Mix Barrington
    • 1
  • Jun Tarui
    • 2
  1. 1.Computer Science DepartmentUniversity of Massachusetts at AmherstAmherstUSA
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK

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