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Elementarily Computable Functions Over the Real Numbers and $\mathbb{R}$-Sub-Recursive Functions

Olivier Bournez 1 Emmanuel Hainry 1
1 PROTHEO - Constraints, automatic deduction and software properties proofs
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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. This paper improves several previous partial characterizations and has a dual interest: - Concerning recursive analysis, our results provide %fully 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 and provide new insights for understanding the relations between several analog computational models.
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https://hal.inria.fr/inria-00107812
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Submitted on : Thursday, October 19, 2006 - 9:10:31 AM
Last modification on : Friday, May 13, 2022 - 10:18:05 PM
Long-term archiving on: : Friday, November 25, 2016 - 12:28:13 PM

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  • HAL Id : inria-00107812, version 1

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Olivier Bournez, Emmanuel Hainry. Elementarily Computable Functions Over the Real Numbers and $\mathbb{R}$-Sub-Recursive Functions. [Intern report] A04-R-301 || bournez04f, 2004, 22 p. ⟨inria-00107812⟩

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