An Analog Characterization of Elementarily Computable Functions over the Real Numbers

  • Olivier Bournez
  • Emmanuel Hainry
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Emmanuel Hainry
    • 1
  1. 1.LORIA/INRIAVillers lès NancyFrance

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