Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity

Irit Dinur, Or Meir

Research output: Contribution to journalArticlepeer-review


One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de Morgan formulas. Karchmer et al. (Comput Complex 5(3/4):191–204, 1995b) suggested to approach this problem by proving that formula complexity behaves “as expected” with respect to the composition of functions f⋄ g. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et al. (Comput Complex 10(3):210–246, 2001), who proved an analogue of the conjecture for the composition of “universal relations.” In this work, we extend the argument of Edmonds et al. (2001) further to f⋄ g where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Håstad’s work on random restrictions (Håstad in SIAM J Comput 27(1):48–64, 1998), our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer & Wigderson in (SIAM J Discrete Math 3(2):255–265, 1990). In addition, our proof gives a new structural result, which roughly says that the naive way for computing f⋄ g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n3-o(1) due to Håstad (1998).

Original languageEnglish
Pages (from-to)375-462
Number of pages88
JournalComputational Complexity
Issue number3
StatePublished - 1 Sep 2018

Bibliographical note

Publisher Copyright:
© 2017, Springer International Publishing AG.


  • 68Q17
  • Communication Complexity
  • KRW conjecture
  • Karchmer–Wigderson games
  • Karchmer–Wigderson relations
  • de-Morgan formulas
  • lower bounds

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


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