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computational complexity

, Volume 10, Issue 3, pp 210–246 | Cite as

Communication complexity towards lower bounds on circuit depth

  • J. Edmonds
  • R. Impagliazzo
  • S. Rudich
  • J. Sgall

Abstract.

Karchmer, Raz, and Wigderson (1995) discuss the circuit depth complexity of n-bit Boolean functions constructed by composing up to d = log n/log log n levels of k = log n-bit Boolean functions. Any such function is in AC1 . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires \( dk \in \Omega({\rm log}^2 n/{\rm log\,log}\,n) \) depth. This would separate AC1 from NC1. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit depth lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dkO(d 2(k log k)1/2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring \( \Omega(k) \) circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds.

Keywords. Communication complexity, Boolean complexity, lower bounds. 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • J. Edmonds
    • 1
  • R. Impagliazzo
    • 2
  • S. Rudich
    • 3
  • J. Sgall
    • 4
  1. 1.Department of Computer Science, York University, North York, Ontario M3J 1P3, Canada, e-mail: jeff@cs.yorku.caCA
  2. 2.Department of Computer Science, UC San Diego, La Jolla CA 92093, U.S.A., e-mail: russell@cs.ucsd.eduUS
  3. 3.Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A., e-mail: rudich@theory.cs.cmu.eduUS
  4. 4.Mathematical Institute, AS CR, Zitná 25, 115 67 Praha 1, Czech Republic, e-mail: sgall@math.cas.cz, http://www.math.cas.cz/~sgall/CZ

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