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Rapport Année : 2004

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

Résumé

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.

Domaines

Autre [cs.OH]
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Dates et versions

inria-00107812 , version 1 (19-10-2006)

Identifiants

  • HAL Id : inria-00107812 , version 1

Citer

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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