Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation

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Abstract

One of the major open problems in complexity theory is proving super-logarithmiclower bounds on the depth of circuits (i.e., P⊈ NC1).Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4):191–204, 1995)suggested to approach this problem by proving that depth complexitybehaves ``as expected'' with respect to the composition of functions f ◊ g.They showed that the validity of this conjecture would imply that P⊈ NC1. As a way to realize this program, Edmonds et al. (Computational Complexity10(3):210–246, 2001) suggested to study the ``multiplexor relation'' MUX.In this paper, we present two results regarding this relation:○The multiplexor relation is ``complete'' for the approach of Karchmeret al. in the following sense: if we could prove (a variant of) theirconjecture for the composition f ◊ MUX for every function f, thenthis would imply P⊈ NC1.○A simpler proof of a lower bound for the multiplexor relation dueto Edmonds et al. Our proof has the additional benefit of fittingbetter with the machinery used in previous works on the subject.

Original languageEnglish
Article number4
JournalComputational Complexity
Volume29
Issue number1
DOIs
StatePublished - 1 Jun 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

Keywords

  • 68Q15
  • Address function
  • Circuit Lower Bounds
  • Circuit complexity
  • Communication complexity
  • Depth complexity
  • Depth lower bounds
  • KRW conjecture
  • Karchmer–Wigersion relations
  • Multiplexer
  • Multiplexor

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (all)
  • Computational Theory and Mathematics
  • Computational Mathematics

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