Toward better formula lower bounds: The composition of a function and a universal relation

Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Research output: Contribution to journalArticlepeer-review


One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., P ⊈ 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 superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [Comput. Complexity, 5 (1995), pp. 191-204] suggested approaching this problem by proving the following conjecture: given two Boolean functions f and g, the depth complexity of the composed function g ⋄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 ⊈ NC1. As a starting point for studying the composition of functions, they introduced a relation called "the universal relation" and suggested studying the composition of universal relations. This suggestion proved fruitful, and an analogue of the Karchmer-Raz-Wigderson (KRW) conjecture for the universal relation was proved by Edmonds et al. [Comput. Complexity, 10 (2001), pp. 210-246]. An alternative proof was given later by Håstad and Wigderson [in Advances in Computational Complexity Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 13, AMS, Providence, RI, 1993, pp. 119-134]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still an open question. 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.

Original languageEnglish
Pages (from-to)114-131
Number of pages18
JournalSIAM Journal on Computing
Issue number1
StatePublished - 2017

Bibliographical note

Funding Information:
The first author was partially supported by grant P202/12/G061 of GA ?R and by RVO: 67985840. Part of this work was done while the first author was visiting the CQT at the National University of Singapore, and was partially funded by the Singapore Ministry of Education and the NRF. The second and fourth authors were partially supported by NSF grant CCF-1412958. The third author was supported by a Simons Society junior fellowship.

Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.


  • Communication complexity
  • Formula
  • Information complexity
  • KRW conjecture
  • Karchmer-Wigderson relations
  • Lower bounds

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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