## Elementarily computable functions over the real numbers and R-sub-recursive functions (2005)

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 16 - 5 self |

### BibTeX

@ARTICLE{Bournez05elementarilycomputable,

author = {Olivier Bournez and Emmanuel Hainry},

title = {Elementarily computable functions over the real numbers and R-sub-recursive functions},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {2005},

volume = {348},

pages = {2005}

}

### OpenURL

### 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 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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Citation Context ...riations on schemas First, we can note that it is possible to change a bit our schemata in order to have a more natural LIM schema. The price to pay is a less natural LI schema, that we called CLI in =-=[8]-=-. Formally, we define CLI as follows: Definition 7 (CLI schema) From g,h and c, with h differentiable and first derivatives of h bounded by c, CLI(g, h, c) is any solution defined on a product of clos... |

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Citation Context ...lest class of functions containing some basic functions, and closed by composition, differential equation solving (called integration), and minimization. This class of functions, also investigated in =-=[24, 25, 26, 27, 28, 29]-=-, can be related to GPAC computable functions: see [23], corrected by [16]. Putting aside possible objections about the physical feasibility of the µ-operator considered in paper [23], the original de... |