Toward better formula lower bounds: An information complexity approach to the KRW composition conjecture

Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P 6⊈ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds. Karchmer, Raz, and Wigderson [21] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g ο f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P 6⊈ NC 1. As a starting point for studying the composition of functions, they introduced a relation called "the universal relation", and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [12]. An alternative proof was given later by Håstad and Wigderson [18]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it. Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations - communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Original languageEnglish
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)9781450327107
StatePublished - 2014
Externally publishedYes
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: 31 May 20143 Jun 2014

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference4th Annual ACM Symposium on Theory of Computing, STOC 2014
Country/TerritoryUnited States
CityNew York, NY


  • Formula complexity
  • Information complexity

ASJC Scopus subject areas

  • Software


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