Limits of Boolean Functions on $\mathbb{F}_p^n$

We study sequences of functions of the form $\mathbb{F}_p^n \to \{0,1\}$ for varying $n$, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions. We also show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Balázs Szegedy [Gowers norms, regularization and limits of functions on abelian groups. 2010. arXiv:1010.6211].

Download Full-text

Additive bases via Fourier analysis

Combinatorics Probability Computing ◽

10.1017/s0963548321000109 ◽

2021 ◽

pp. 1-12

Author(s):

Bodan Arsovski

Keyword(s):

Abelian Group ◽

Fourier Analysis ◽

Vector Space ◽

Abelian Groups ◽

Finite Abelian Group ◽

Probabilistic Method ◽

Additive Basis ◽

Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Download Full-text

DERANDOMIZED LEARNING OF BOOLEAN FUNCTIONS OVER FINITE ABELIAN GROUPS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054101000618 ◽

2001 ◽

Vol 12 (04) ◽

pp. 491-516

Author(s):

M. SITHARAM ◽

TIMOTHY STRANEY

Keyword(s):

Boolean Functions ◽

Partial Information ◽

Abelian Groups ◽

Query Complexity ◽

Character Theory ◽

Finite Abelian Groups ◽

Training Set ◽

Membership Queries ◽

Approximation Classes ◽

Query Model

We employ the Always Approximately Correct or AAC model defined in [35], to prove learnability results for classes of Boolean functions over arbitrary finite Abelian groups. This model is an extension of Angluin's Query model of exact learning. The Boolean functions we consider belong to approximation classes, i.e. functions that are approximable (in various norms) by few Fourier basis functions, or irreducible characters of the domain Abelian group. We contrast our learnability results to previous results for similar classes in the PAC model of learning with and without membership queries. In addition, we discuss new, natural issues and questions that arise when the AAC model is used. One such question is whether a uniform training set is available for learning any function in a given approximation class. No analogous question seems to have been studied in the context of Angluin's Query model. Another question is whether the training set can be found quickly if the approximation class of the function is completely unknown to the learner, or only partial information about the approximation class is given to the learner (in addition to the answers to membership queries). In order to prove the learnability results in this paper we require new techniques for efficiently sampling Boolean functions using the character theory of finite Abelian groups, as well as the development of algebraic algorithms. The techniques result in other natural applications closely related to learning, for example, query complexity of deterministic algorithms for testing linearity, efficient pseudorandom generators, and estimating VC dimensions for classes of Boolean functions over finite Abelian groups.

Download Full-text

Generalization of higher order SAC to vector output Boolean functions

Lecture Notes in Computer Science - Advances in Cryptology — ASIACRYPT '96 ◽

10.1007/bfb0034849 ◽

1996 ◽

pp. 218-231 ◽

Cited By ~ 3

Author(s):

Kaoru Kurosawa ◽

Takashi Satoh

Keyword(s):

Boolean Functions ◽

Higher Order

Download Full-text

On the Higher Order Nonlinearities of Boolean Functions and S-Boxes, and Their Generalizations

Sequences and Their Applications - SETA 2008 - Lecture Notes in Computer Science ◽

10.1007/978-3-540-85912-3_31 ◽

2008 ◽

pp. 345-367 ◽

Cited By ~ 6

Author(s):

Claude Carlet

Keyword(s):

Boolean Functions ◽

Higher Order

Download Full-text

Fourier analysis on finite abelian groups

Graduate Studies in Mathematics - The Distribution of Prime Numbers ◽

10.1090/gsm/203/11 ◽

2019 ◽

pp. 100-109

Keyword(s):

Fourier Analysis ◽

Abelian Groups ◽

Finite Abelian Groups

Download Full-text

FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

Combinatorics, Complexity, and Chance ◽

10.1093/acprof:oso/9780198571278.003.0007 ◽

2007 ◽

pp. 103-129 ◽

Cited By ~ 1

Author(s):

Andrew Goodall

Keyword(s):

Fourier Analysis ◽

Abelian Groups ◽

Finite Abelian Groups

Download Full-text

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

Numerical Linear Algebra with Applications ◽

10.1002/nla.2285 ◽

2020 ◽

Vol 27 (3) ◽

Cited By ~ 2

Author(s):

Yunhui He ◽

Scott MacLachlan

Keyword(s):

Finite Element ◽

Fourier Analysis ◽

Higher Order ◽

Finite Element Discretizations ◽

Element Discretizations

Download Full-text

Distinguishing properties and applications of higher order derivatives of Boolean functions

Information Sciences ◽

10.1016/j.ins.2014.02.108 ◽

2014 ◽

Vol 271 ◽

pp. 224-235 ◽

Cited By ~ 3

Author(s):

Ming Duan ◽

Mohan Yang ◽

Xiaorui Sun ◽

Bo Zhu ◽

Xuejia Lai

Keyword(s):

Boolean Functions ◽

Higher Order ◽

Derivatives Of

Download Full-text

Intersecting Families are Essentially Contained in Juntas

Combinatorics Probability Computing ◽

10.1017/s0963548308009309 ◽

2009 ◽

Vol 18 (1-2) ◽

pp. 107-122 ◽

Cited By ~ 18

Author(s):

IRIT DINUR ◽

EHUD FRIEDGUT

Keyword(s):

Fourier Analysis ◽

Boolean Functions ◽

Simple Description ◽

Discrete Fourier Analysis ◽

Intersecting Family ◽

Intersecting Families ◽

Positive Integers

A family$\J$of subsets of {1, . . .,n} is called aj-junta if there existsJ⊆ {1, . . .,n}, with |J| =j, such that the membership of a setSin$\J$depends only onS∩J.In this paper we provide a simple description of intersecting families of sets. Letnandkbe positive integers withk<n/2, and let$\A$be a family of pairwise intersecting subsets of {1, . . .,n}, all of sizek. We show that such a family is essentially contained in aj-junta$\J$, wherejdoes not depend onnbut only on the ratiok/nand on the interpretation of ‘essentially’.Whenk=o(n) we prove that every intersecting family ofk-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family$\A$there exists an elementi∈ {1, . . .,n} such that the number of sets in$\A$that do not containiis of order$\C {n-2}{k-2}$(which is approximately$\frac {k}{n-k}$times the size of a maximal intersecting family).Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

Download Full-text