1,005 research outputs found
Counting inequivalent monotone Boolean functions
Monotone Boolean functions (MBFs) are Boolean functions satisfying the monotonicity condition for any . The number of MBFs in n variables is
known as the th Dedekind number. It is a longstanding computational
challenge to determine these numbers exactly - these values are only known for
at most 8. Two monotone Boolean functions are inequivalent if one can be
obtained from the other by renaming the variables. The number of inequivalent
MBFs in variables was known only for up to . In this paper we
propose a strategy to count inequivalent MBF's by breaking the calculation into
parts based on the profiles of these functions. As a result we are able to
compute the number of inequivalent MBFs in 7 variables. The number obtained is
490013148
Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids
A prismatoid is a polytope with all its vertices contained in two parallel
facets, called its bases. Its width is the number of steps needed to go from
one base to the other in the dual graph. The first author recently showed that
the existence of counter-examples to the Hirsch conjecture is equivalent to
that of -prismatoids of width larger than , and constructed such
prismatoids in dimension five. Here we show that the same is impossible in
dimension four. This is proved by looking at the pair of graph embeddings on a
2-sphere that arise from the normal fans of the two bases.Comment: This paper merges and supersedes the papers arXiv:1101.3050 (of the
last two authors) and arXiv:1102.2645 (of the first author
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