Restrictions of nondegenerate Boolean functions and degree lower bounds over different rings

Chia Jung Lee, Satyanarayana V. Lokam, Shi-Chun Tsai, Ming Chuan Yang

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

5 Scopus citations

Abstract

A Boolean function f : {0, 1}n → {0, 1} is called nondegenerate if f depends on all its n variables. We show that, for any nondegenerate function f, there exists a variable xi such that at least one of the restrictions fxi=0 or fxi=1 must depend on all the remaining n - 1 variables. We also consider lower bounds on the degrees of polynomials representing a Boolean function over different rings. Let dq(f) be the degree of the (unique) polynomial over the ring ℤq exactly representing f. For distinct primes pi let m = Πr i=1 pei i. Then, we show that any nondegenerate symmetric Boolean function f must have m · dp1e1(f)⋯dprer(f) > n. We use the existence of nondegenerate subfunctions to prove degree lower bounds on random functions. Specifically, we show that m · dp1e1(f)⋯dprer(f) > lg n - 1 holds for almost all f when f is chosen uniformly at random from all n-variate Boolean functions. Our proof uses the second moment method to show that a random f must almost always contain a nondegenerate symmetric subfunction on at least lg n - 1 variables. It follows that an n-variate nondegenerate symmetric Boolean function can have degree o(√n) over at most one finite field and that almost all f can have degree o(√lg n) over at most one finite field.

Original languageEnglish
Title of host publicationProceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages501-505
Number of pages5
ISBN (Electronic)9781467377041
DOIs
StatePublished - 28 Sep 2015
EventIEEE International Symposium on Information Theory, ISIT 2015 - Hong Kong, Hong Kong
Duration: 14 Jun 201519 Jun 2015

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2015-June
ISSN (Print)2157-8095

Conference

ConferenceIEEE International Symposium on Information Theory, ISIT 2015
CountryHong Kong
CityHong Kong
Period14/06/1519/06/15

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