Two Logical Hierarchies of Optimization Problems over the Real Numbers

  • Uffe Flarup Hansen
  • Klaus Meer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


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 [9],[8]). More precisely, based on a real analogue of Fagin’s theorem [9] we deal with two classes MAX-NP and MIN-NP of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that MAX-NP decomposes into four natural subclasses, whereas MIN-NP decomposes into two. This gives a real number analogue of a result by Kolaitis and Thakur [10] in the Turing model. Our proofs mainly use techniques from [13]. Finally, approximation issues are briefly discussed.


Feasible Solution Polynomial System Real Zero Logical Framework Satisfying Assignment 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Uffe Flarup Hansen
    • 1
  • Klaus Meer
    • 1
  1. 1.Department of Mathematics and Computer ScienceSyddansk UniversitetOdense MDenmark

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