2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 1116
author:  Kazuyuki Amano and Jun Tarui 

title: 
Monotone Boolean Functions with
s
Zeros Farthest from Threshold Functions

keywords:  
abstract: 
Let
T
denote the
t
t
threshold function on the
n
cube:
T
if
t
(x) = 1
{i : x
, and
i
=1} ≥t
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
? We show that the following monotone function
t
)
p
maximizes the distance: For
s
x∈{0,1}
,
n
p
if and only if
s
(x)=0
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=⌈n/2 ⌉
and
s=2
by Blum, Burch, and Langford [BBL98FOCS98], who
considered the problem to analyze the behavior of a
learning algorithm for monotone Boolean functions, and the
previous work for the same
n1
t
and
s
by Amano and Maruoka [AM02ALT02].

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reference: 
Kazuyuki Amano and Jun Tarui (2005), Monotone Boolean
Functions with
s
Zeros Farthest from Threshold Functions, in 2005
European Conference on Combinatorics, Graph Theory and
Applications (EuroComb '05), Stefan Felsner (ed.),
Discrete Mathematics and Theoretical Computer Science
Proceedings AE, pp. 1116

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