Staff View
Towards understanding the approximation of Boolean functions by nonclassical polynomials

Descriptive

TitleInfo
Title
Towards understanding the approximation of Boolean functions by nonclassical polynomials
Name (type = personal)
NamePart (type = family)
Bhrushundi
NamePart (type = given)
Abhishek
NamePart (type = date)
1988-
DisplayForm
Abhishek Bhrushundi
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Kopparty
NamePart (type = given)
Swastik
DisplayForm
Swastik Kopparty
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
chair
Name (type = personal)
NamePart (type = family)
Allender
NamePart (type = given)
Eric
DisplayForm
Eric Allender
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Saraf
NamePart (type = given)
Shubhangi
DisplayForm
Shubhangi Saraf
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Hatami
NamePart (type = given)
Hamed
DisplayForm
Hamed Hatami
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
School of Graduate Studies
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (encoding = w3cdtf); (keyDate = yes); (qualifier = exact)
2020
DateOther (encoding = w3cdtf); (qualifier = exact); (type = degree)
2020-05
CopyrightDate (encoding = w3cdtf); (qualifier = exact)
2020
Language
LanguageTerm (authority = ISO 639-3:2007); (type = text)
English
Abstract (type = abstract)
The representation and approximation of Boolean functions by polynomials is an important area of research in theoretical computer science, having numerous applications in circuit complexity, communication complexity, pseudorandomness, quantum computation, learning theory, algorithm design, and explicit combinatorial constructions.

Results of Green and Tao (2009), Lovett et al. (2011), and Tao and Ziegler (2012), on the Inverse Conjecture for the Gowers norm over finite fields of low characteristic, and the subsequent work of Bhowmick and Lovett (2015), suggest that a potential barrier to the resolution of some of the outstanding open problems in this area is the class of nonclassical polynomials and its ability to nontrivially represent and approximate Boolean functions.

Motivated by these works, in this dissertation, we investigate the ability of nonclassical polynomials to approximate Boolean functions with respect to both previously studied and new notions of approximation:
• We introduce and study an agreement-based notion of approximation by polynomials over Z/2k Z. Investigating this notion serves as a proxy for understanding the maximum possible agreement between nonclassical polynomials and Boolean functions. We prove several new results that shed light on this new notion of approximation, and these results help us answer some questions left open in the
work of Bhowmick and Lovett (2015) concerning the approximation of Boolean
functions by nonclassical polynomials in the agreement sense.
• We propose a new notion of point-wise approximation by nonclassical polynomials. Using a result of Green et al. (1992), which itself is an extension of the classic work of Beigel and Tarui (1991), we observe that Boolean functions computable by ACC0 circuits (constant-depth circuits of polynomial size, containing AND, OR, NOT, and MODq gates) are amenable to point-wise approximation by low-degree nonclassical polynomials. Motivated by this new observation, we then explore how well can low-degree nonclassical polynomials point-wise approximate the majority function, in the hope of resolving the longstanding open problem of proving that majority is not computable by ACC0 circuits.

Our results suggest several interesting and promising directions of research. We ex-
plore some of these directions and state concrete open problems along with plausible
approaches to solving them.
Subject (authority = local)
Topic
Nonclassical polynomials
Subject (authority = RUETD)
Topic
Computer Science
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_10846
PhysicalDescription
Form (authority = gmd)
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (viii, 104 pages)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
RelatedItem (type = host)
TitleInfo
Title
School of Graduate Studies Electronic Theses and Dissertations
Identifier (type = local)
rucore10001600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Identifier (type = doi)
doi:10.7282/t3-t449-mj70
Genre (authority = ExL-Esploro)
ETD doctoral
Back to the top

Rights

RightsDeclaration (ID = rulibRdec0006)
The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Bhrushundi
GivenName
Abhishek
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2020-04-27 14:21:40
AssociatedEntity
Name
Abhishek Bhrushundi
Role
Copyright holder
Affiliation
Rutgers University. School of Graduate Studies
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
Back to the top

Technical

RULTechMD (ID = TECHNICAL1)
ContentModel
ETD
OperatingSystem (VERSION = 5.1)
windows xp
CreatingApplication
Version
1.5
ApplicationName
pdfTeX-1.40.16
DateCreated (point = end); (encoding = w3cdtf); (qualifier = exact)
2020-04-27T13:47:46
DateCreated (point = end); (encoding = w3cdtf); (qualifier = exact)
2020-04-27T13:47:46
Back to the top
Version 8.5.5
Rutgers University Libraries - Copyright ©2024