Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds

Or Meir, Avi Wigderson

Research output: Contribution to journalArticlepeer-review


Consider a random sequence of n bits that has entropy at least n−k, where k≪ n. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query ≈nk other coordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions. As an application of this result, we prove a new result on depth-3 circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265, 1990), and, in particular, it is a “top-down” proof (Håstad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.

Original languageEnglish
Pages (from-to)145-183
Number of pages39
JournalComputational Complexity
Issue number2
StatePublished - 1 Jun 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.


  • 68Q15
  • Certificate complexity
  • Circuit complexity
  • Circuit complexity lower bounds
  • Decision tree complexity
  • Information theoretic
  • Query complexity
  • Sensitivity

ASJC Scopus subject areas

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


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