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Paper:

TR09-040 | 20th April 2009 00:00

On convex complexity measures

TR09-040
Authors: Pavel Hrubes, Stasys Jukna, Alexander Kulikov, Pavel Pudlak
Publication: 6th May 2009 00:44
Keywords:

Abstract:

Khrapchenko's classical lower bound $n^2$ on the formula size of the
parity function~$f$ can be interpreted as designing a suitable
measure of subrectangles of the combinatorial rectangle
$f^{-1}(0)\times f^{-1}(1)$. Trying to generalize this approach we
arrived at the concept of \emph{convex measures}. We prove the
negative result that convex measures are bounded by $O(n^2)$ and
show that several measures considered for proving lower bounds on
the formula size are convex. We also prove quadratic upper bounds on
a class of measures that are not necessarily convex.

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