## Recursive analysis characterized as a class of real recursive functions (2006)

Venue: | Fundamenta Informaticae |

Citations: | 18 - 8 self |

### BibTeX

@ARTICLE{Bournez06recursiveanalysis,

author = {Olivier Bournez and Emmanuel Hainry},

title = {Recursive analysis characterized as a class of real recursive functions},

journal = {Fundamenta Informaticae},

year = {2006},

volume = {74},

pages = {2006}

}

### OpenURL

### Abstract

Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.

### Citations

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Citation Context ...some basic functions, and closed by composition, linear integration, minimalization and limit schema. 1 Introduction Recursive analysis, also called computable analysis, has been introduced by Turing =-=[Tur36]-=-, Grzegorczyk [Grz55], Lacombe [Lac55]. It has shown to provide a very robust concept of computability, that enables to discuss most arguments of mathematical analysis from the computability point of ... |

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Citation Context ...real numbers, and the set of non-negative real numbers respectively. Given x ∈ R n , we write −→ x to emphasize that x is a vector. Lemma 1 (Bounding Lemma for Linear Differential Equations (see e.g. =-=[Arn92]-=-)) For linear differential equation −→ x ′ = A(t) −→ x , if A is defined and continuous on interval I = [a, b], where a ≤ 0 ≤ b, then, for all −→ x 0, the solution of −→ x ′ = A(t) −→ x with initial c... |

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Citation Context ...combe [Lac55]. It has shown to provide a very robust concept of computability, that enables to discuss most arguments of mathematical analysis from the computability point of view: see e.g. monograph =-=[Wei00]-=-. In this framework, a function f : R → R over the reals is considered as computable, if there is some computable functional, or Type 2 machine, that maps any sequence quickly converging to some x to ... |

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Citation Context ...ctions with several analog models. Indeed, the classes from Campagnolo, Costa and Moore, are inspired from a class of functions over the reals, called real recursive functions, introduced by Moore in =-=[Moo96]-=-. Real recursive functions have been shown (see [Moo96] with corrections from Graça and Costa in [GC03]) to be strongly connected to functions computable by the General Purpose Analog Computer (GPAC) ... |

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Citation Context ...mposition, linear integration, minimalization and limit schema. 1 Introduction Recursive analysis, also called computable analysis, has been introduced by Turing [Tur36], Grzegorczyk [Grz55], Lacombe =-=[Lac55]-=-. It has shown to provide a very robust concept of computability, that enables to discuss most arguments of mathematical analysis from the computability point of view: see e.g. monograph [Wei00]. In t... |

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Citation Context ... and closed by composition, linear integration, minimalization and limit schema. 1 Introduction Recursive analysis, also called computable analysis, has been introduced by Turing [Tur36], Grzegorczyk =-=[Grz55]-=-, Lacombe [Lac55]. It has shown to provide a very robust concept of computability, that enables to discuss most arguments of mathematical analysis from the computability point of view: see e.g. monogr... |

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Citation Context ... turn an abstraction of some systems that really existed [TLK76, Bus31, Bow96], or is an abstraction of easy realizable systems using today’s electronic. Extensions of the GPAC have been discussed in =-=[Rub93]-=- and [Mil95]. Fourth, these results show that the provided class of functions does not exhibit super-Turing phenomena such as [Moo96, Bou99, AM98, Hog92, EN02], and benefits from all the robustness re... |

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Citation Context ...i92, Odi99]: given a set F of functions and a set O of operators on functions (an operator is an operation that maps one or more functions to a new function), [F; O] will denote the closure of F by O =-=[Clo98]-=-. Proposition 1 (Classical settings: see e.g. [Ros84, Odi92, Odi99]) Let f be a function from N k to N for k ∈ N. Function f is • elementar iff it belongs to E = [0, S, U, +, ⊖; COMP, BSUM, BPROD]; • ... |

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Citation Context ...x , y) is a (n + 1) × (n + 1) matrix with elements in L. Class L includes common functions like +,sin,cos,−,×,exp, or constants r for all r ∈ Q (see [Cam01, CMC02]), but contains only total functions =-=[CMC02]-=-: Proposition 3 ([CMC02]) All functions from L are continuous, defined everywhere, and of class C 2 . Actually, observing the proofs from [Cam01, CMC02], schema LI can be strengthened as follows: Prop... |

28 |
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Citation Context ...rsive functions have been shown (see [Moo96] with corrections from Graça and Costa in [GC03]) to be strongly connected to functions computable by the General Purpose Analog Computer (GPAC) of Shannon =-=[Sha41]-=-. GPAC is in turn an abstraction of some systems that really existed [TLK76, Bus31, Bow96], or is an abstraction of easy realizable systems using today’s electronic. Extensions of the GPAC have been d... |

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13 | Elementarily computable functions over the real numbers and R-subrecursive functions, Theoretical Computer Science 348
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(Show Context)
Citation Context ...that ∂f ∂y (−→ x , y0) �= 0 in the corresponding root y0. Then function g : R k → R that maps −→ x to the corresponding root y0 is defined over D and also of class C k . Lemma 3 (Basic fact (see e.g. =-=[BH05]-=-)) Let F : R×V ⊂ R k+1 → R l be a function of class C 1 , and β( −→ x ) : V → R, K( −→ x ) : V → R be some continuous functions. Assume that for all t and −→ x , � ∂F ∂t (t, −→ x )� ≤ K( −→ x ) exp(−t... |

13 | µ-Recursion and infinite limits - Mycka |

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11 | Classical Recursion Theory (Volume II), volume 143 - Odifreddi - 1999 |

10 |
An analog characterization of elementarily computable functions over the real numbers
- Bournez, Hainry
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(Show Context)
Citation Context ...second step: mix these constructions with the ones of [BH04a, BH05] to get a characterization of the whole class of computable functions over the reals. This is done by extending the constructions of =-=[BH04a]-=-, and in particular provides extensions of [BH04a, BH05] that allow to talk about functions defined on non-compact domains. Indeed, computable functions over the reals are characterized in an algebrai... |

10 |
Computability over topological structures
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(Show Context)
Citation Context ...to define computability in the sense of recursive analysis in a machine independent way, avoiding to talk about higher order Turing machines, or functionals, nor less natural characterization such as =-=[Bra03]-=-. Second, that proves that the study of mathematical concepts through recursive analysis can be investigated by talking in terms of these algebraic classes and operators, providing a rather natural co... |

7 |
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- Daniel
(Show Context)
Citation Context ...om a class of functions over the reals, called real recursive functions, introduced by Moore in [Moo96]. Real recursive functions have been shown (see [Moo96] with corrections from Graça and Costa in =-=[GC03]-=-) to be strongly connected to functions computable by the General Purpose Analog Computer (GPAC) of Shannon [Sha41]. GPAC is in turn an abstraction of some systems that really existed [TLK76, Bus31, B... |

6 | The differential analyser - Bush |

6 | F.: An Analog Characterization of the Subrecursive Functions - Campagnolo, Moore, et al. - 2000 |

5 | Infinite limits and R-recursive functions - Mycka |

5 |
Cours de Mathématiques Spéciales, Tome 3, Topologie et éléments d’analyse
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(Show Context)
Citation Context ...condition −→ x (0) = −→ x 0 is defined and unique on I. Furthermore, we know that the solution satisfies �−→ x (t)� ≤ �−→ x 0� exp( sup �A(τ)�t). τ∈[0,t] Lemma 2 (Implicit Functions Theorem (see e.g. =-=[RDO95]-=-)) Let f : D × I ⊂ R k+1 → R where D × I is a product of closed intervals be a function of class 1 C k , for k ≥ 1. Assume that for all −→ x ∈ D, the equation f( −→ x , y) = 0 has exactly one solution... |

3 |
Real recursive functions and real extentions of recursive functions
- Bournez, Hainry
- 2004
(Show Context)
Citation Context ...s over the integers, and second to understand how and whether it could be arranged with the arguments of [BH04a, BH05], to provide such a characterization. The first step was solved recently in paper =-=[BH04b]-=-. This journal paper presents detailed proofs of the claims of [BH04b], with several extensions. In particular, it characterizes also non-total functions. More importantly, it proves that this is inde... |

3 | Programmable VLSI Extended Analog Computer for Cyclotron Beam Control
- Mills
- 1995
(Show Context)
Citation Context ...traction of some systems that really existed [TLK76, Bus31, Bow96], or is an abstraction of easy realizable systems using today’s electronic. Extensions of the GPAC have been discussed in [Rub93] and =-=[Mil95]-=-. Fourth, these results show that the provided class of functions does not exhibit super-Turing phenomena such as [Moo96, Bou99, AM98, Hog92, EN02], and benefits from all the robustness results that h... |

2 |
Lameiras Campagnolo. Computational Complexity of Real Valued Recursive Functions and Analog Circuits
- Manuel
- 2001
(Show Context)
Citation Context ...mber and also that the chosen bound is arbitrary and could be replaced by another function converging fast toward 0. In particular, our notion of computability is equivalent to the one of [Wei00], or =-=[Cam01]-=-. 3 Formally, a function f over the integers can be considered as functional f : (V, −→ n ) ↦→ f( −→ n ). Similarly, an operator Op on functions f1, . . . , fm over the integers can be extended to an ... |

2 | Classical Recursion Theory, Volume I, volume 125 - Odifreddi - 1992 |