PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Two logical hierarchies of optimization problems over the real numbers
Klaus Meer and Uffe Flarup
Mathematical Logic Quarterly Volume 52, Number 1, pp. 37-50, 2006.


We 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.

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EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Learning/Statistics & Optimisation
Theory & Algorithms
ID Code:2479
Deposited By:Klaus Meer
Deposited On:22 November 2006