scholarly journals On the sensitivity of cyclically-invariant Boolean functions

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Sourav Chakraborty
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Transitive Group ◽  
60Th Birthday ◽  
Special Issue ◽  
Natural Class ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Class Of Functions ◽  
Block Sensitivity

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper we construct a cyclically invariant Boolean function whose sensitivity is Theta(n(1/3)). This result answers two previously published questions. Turan (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Omega(root n). Kenyon and Kutin (2004) asked whether for a "nice" function the product of 0-sensitivity and 1-sensitivity is Omega(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Omega(n(1/3)). Hence for this class of functions sensitivity and block sensitivity are polynomially related.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals The extended equivalence and equation solvability problems for groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Gabor Horvath ◽  
Csaba Szabo
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Nilpotent Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Solvability Problem ◽  
Np Complete ◽  
Equation Solvability

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.

scholarly journals Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Thomas P. Hayes
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Communication Complexity ◽  
60Th Birthday ◽  
Equal Size ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Zarankiewicz Problem ◽  
Complexity Hierarchy

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

scholarly journals On the metric dimension of Grassmann graphs

2012 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Robert F. Bailey ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
60Th Birthday ◽  
Metric Dimension ◽  
Special Issue ◽  
Resolving Set ◽  
Grassmann Graph ◽  
Algorithms And Complexity ◽  
International Audience

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.

scholarly journals Computing tensor decompositions of finite matrix groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Nurullah Ankaralioglu ◽  
Akos Seress
Keyword(s):  
Tensor Decomposition ◽  
60Th Birthday ◽  
Special Issue ◽  
Matrix Groups ◽  
Tensor Decompositions ◽  
Products Of Groups ◽  
Finite Matrix ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Central Products

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.

scholarly journals On the residual solvability of generalized free products of solvable groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Delaram Kahrobaei ◽  
Stephen Majewicz
Keyword(s):  
Free Product ◽  
Solvable Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Free Products ◽  
Generalized Free Product ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Generalized Free Products

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper, we study the residual solvability of the generalized free product of solvable groups.

scholarly journals Combinatorics, Groups, Algorithms, and Complexity: Conference in honor of Laci Babai's 60th birthday

2010 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Akos Seress ◽  
Mario Szegedy
Keyword(s):  
Theoretical Computer Science ◽  
Ohio State University ◽  
60Th Birthday ◽  
State University ◽  
Special Issue ◽  
Theoretical Computer ◽  
Introductory Article ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Generous Support

International audience Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.

A NOTE ON THE POLYNOMIAL REPRESENTATION OF BOOLEAN FUNCTIONS OVER GF(2)

1999 ◽  
Vol 10 (04) ◽  
pp. 535-542
Author(s):  
RICHARD BEIGEL ◽  
ANNA BERNASCONI
Keyword(s):  
Boolean Function ◽  
Open Problem ◽  
Boolean Functions ◽  
Polynomial Representation ◽  
Maximal Sensitivity ◽  
Input Strings ◽  
The Relationship ◽  
Block Sensitivity

We investigate the representation of Boolean functions as polynomials over the field GF(2), and prove an interesting characteriztion theorem: the degree of a Boolean function over GF(2) is equal to the size of its largest subfunction that takes the value 1 on an odd number of input strings. We then present some properties of odd functions, i.e., functions that take the value 1 on an odd number of strings, and analyze the connections between the problem of the existence of odd functions with very low maximal sensitivity and the long standing open problem of the relationship between the maximal sensitivity and the block sensitivity of Boolean functions.

scholarly journals Monotone Boolean Functions with s Zeros Farthest from Threshold Functions

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kazuyuki Amano ◽  
Jun Tarui
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Learning Algorithm ◽  
Monotone Function ◽  
Threshold Function ◽  
Binary Representation ◽  
Threshold Functions ◽  
Monotone Boolean Function ◽  
Monotone Boolean Functions ◽  
International Audience

International audience Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].

MORE VECTORIAL BOOLEAN FUNCTIONS WITH UNBOUNDED NONLINEARITY PROFILE

2011 ◽  
Vol 22 (06) ◽  
pp. 1259-1269 ◽  
Author(s):  
CLAUDE CARLET
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Multiplicative Inverse ◽  
First Order ◽  
Vectorial Function ◽  
Close Relationship ◽  
Vectorial Functions ◽  
Inverse Functions ◽  
Gowers Norm ◽  
Class Of Functions

The nonlinearity profile of Boolean functions is a generalization of the most important cryptographic criterion, called the (first order) nonlinearity. It is defined as the sequence of the minimum Hamming distances nlr(f) between a given Boolean function f and all Boolean functions in the same number of variables and of degrees at most r, for r ≥ 1. This parameter, which has a close relationship with the Gowers norm, quantifies the resistance to cryptanalyses by low degree approximations of stream ciphers using the Boolean function f as combiner or as filter. The nonlinearity profile can also be defined for vectorial functions: it is the sequence of the minimum Hamming distances between the component functions of the vectorial function and all Boolean functions of degrees at most r, for r ≥ 1. The nonlinearity profile of the multiplicative inverse functions has been lower bounded in a previous paper by the same author. No other example of an infinite class of functions with unbounded nonlinearity profile has been exhibited since then. In this paper, we lower bound the whole nonlinearity profile of the (simplest) Dillon bent function (x,y) ↦ xy2n/2-2, x, y ∈ 𝔽2n/2 and we exhibit another class of functions, for which bounding the whole profile of each of them comes down to bounding the first order nonlinearities of all functions.

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