## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 11-16

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 11-16

author: Kazuyuki Amano and Jun Tarui Monotone Boolean Functions with s Zeros Farthest from Threshold Functions Let T t denote the t -threshold function on the n -cube: T t (x) = 1 if |{i : x i =1}| ≥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∈{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=⌈n/2 ⌉ 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]. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. 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. 11-16 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0103.ps.gz (58 K) dmAE0103.ps (142 K) dmAE0103.pdf (144 K)

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