Linear systems over composite moduli

Arkadev Chattopadhyay*, Avi Wigderson

*Corresponding author for this work

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

14 Scopus citations

Abstract

We study solution sets to systems of generalized linear equations of the form ℓi(x1,x2, middot;middot;middot;, xn) ∈ Ai (mod m) where ℓ1,⋯, ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of ℤm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. exp (- Ω(n)), with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ o AND o MODAm (i.e. having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MOD m gates at the base) computing the function MODq. This settles an open problem of Beigel and Maciel [5], for the case of such modulus m. Our technique makes use of the work of Bourgain [6] on estimating exponential sums involving a low-degree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov [15] on arithmetic circuits over finite fields.

Original languageEnglish
Title of host publicationProceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Pages43-52
Number of pages10
DOIs
StatePublished - 2009
Externally publishedYes
Event50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States
Duration: 25 Oct 200927 Oct 2009

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Country/TerritoryUnited States
CityAtlanta, GA
Period25/10/0927/10/09

Keywords

  • Boolean circuit complexity
  • Constant-depth circuits
  • Exponential sums
  • Matrix rigidity
  • Modular gates

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