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Two logical hierarchies of optimization problems over the real numbers
Klaus Meer and Uffe Flarup
Mathematical Logic Quarterly
Volume 52,
Number 1,
,
2006.
AbstractWe introduce and study certain classes of optimization problems
over the real numbers. The classes are defined
by logical means, relying on metafinite model theory
for so called $\R$-structures (see \cite{Graedel-Meer}, \cite{Graedel-Gurevich}).
More precisely, based on a real analogue of Fagin's theorem \cite{Graedel-Meer}
we deal with two classes
$\maxnr$ and $\minnr$
of maximization and minimization problems, respectively, and figure
out their intrinsic logical structure.
It is proven that $\maxnr$ decomposes into four natural subclasses, whereas
$\minnr$ decomposes into two. This gives a real number
analogue of a result by Kolaitis and Thakur \cite{Kolaitis}
in the Turing model.
Our proofs mainly use techniques from \cite{Meer}.
Finally, approximation issues are briefly discussed. [Edit]
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