On Derandomized Composition of Boolean Functions

Research output: Contribution to journalArticlepeer-review

Abstract

The (block-)composition of two Boolean functions f: { 0 , 1 } m→ { 0 , 1 } , g: { 0 , 1 } n→ { 0 , 1 } is the function f⋄ g that takes asinputs m strings x1, … , xm∈ { 0 , 1 } n and computes (f⋄ g) (x1, … , xm) = f(g(x1) , … , g(xm)). This operation has been used several times in the past for amplifyingdifferent hardness measures of f and g. This comes at a cost: thefunction f⋄ g has input length m· n rather than m or n, which is a bottleneck for some applications. In this paper, we propose to decrease this cost by “derandomizing” the composition: instead of feeding into f⋄ g independent inputs x1, … , xm, we generate x1, … , xm using a shorter seed. We show that this idea can be realized in the particular setting of the composition of functions and universal relations (Gavinsky et al. in SIAM J Comput 46(1):114–131, 2017; Karchmer et al. in Computat Complex 5(3/4):191–204, 1995b). To this end, we provide two different techniques for achieving such a derandomization: a technique based on averaging samplers and a technique based on Reed–Solomon codes.

Original languageEnglish
Pages (from-to)661-708
Number of pages48
JournalComputational Complexity
Volume28
Issue number4
DOIs
StatePublished - 1 Dec 2019

Bibliographical note

Funding Information:
We are grateful to Ronen Shaltiel for explaining to us his paper on derandomized parallel repetition (Shaltiel 2010) which served as an inspiration to this work, for pointers to the extractors? literature, and for numerous valuable discussions. We would also like to thank anonymous referees for comments that improved the presentation of this work. This work was partially supported by the Israel Science Foundation (Grant No. 1445/16).

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

Keywords

  • 68Q15
  • Circuit complexity
  • KRW conjecture
  • Karchmer–Wigderson relations
  • circuit lower bounds
  • communication complexity
  • derandomization
  • formula complexity
  • formula lower bounds

ASJC Scopus subject areas

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

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