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

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 13 - 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.

### Citations

378 |
On a theory of computation and complexity over the real numbers
- Blum, Shub, et al.
- 1989
(Show Context)
Citation Context ...n mention the model proposed by Blum et al., sometimes called real Turing machine, measuring the algebraic complexity of problems independently of real number representation considerations defined in =-=[5]-=- and extended to arbitrary structures in [33]. Several papers have been devoted to understanding complexity classes and their relations in this framework: see monographs [4, 33]. These models concern ... |

350 |
Complexity and Real Computation
- Blum, Cucker, et al.
- 1998
(Show Context)
Citation Context ... considerations defined in [5] and extended to arbitrary structures in [33]. Several papers have been devoted to understanding complexity classes and their relations in this framework: see monographs =-=[4, 33]-=-. These models concern discrete time computability. Models of machines where the time is continuous can also be considered. The first ever built computers were continuous time machines: e.g. Blaise Pa... |

279 | Ordinary Differential Equations
- Arnold
- 1973
(Show Context)
Citation Context ...llow. The following result3 , with previous lemma, is a key to provide upper bounds on the growth of functions of our classes (c.f. Lemma 7). Lemma 2 (Bounding Lemma for Linear Differential Equations =-=[1]-=-) 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 co... |

73 | Recursion theory on the reals and continuous-time computation, Theoret
- Moore
- 1996
(Show Context)
Citation Context ...eas some models were recently proved very (far too?) powerful: using the so-called Zeno’s paradox, some models make it possible to compute non-Turing computable functions in a constant time: see e.g. =-=[23, 7, 3, 19, 15]-=-. Notice that understanding whether there exist analog continuous time models that do not suffer from Zeno’s paradox problems is also closely related to the important problems of finding criteria for ... |

66 | Non-Turing computations via Malament-Hogarth spacetimes
- Etesi, Nemeti
- 2002
(Show Context)
Citation Context ... terms of solutions of polynomial differential equations [36, 34, 22, 16]. Continuous time machines also include analog neural networks [32, 37], hybrid systems [3, 6], or theoretical physical models =-=[31, 19, 15]-=-: see also survey [32]. The relations between all the models are not fully understood. One can say, that the theory of analog computations has not yet experienced the unification that digital discrete... |

59 |
Extension de la Notion de Fonction Récursive aux Fonctions d’une ou Plusieurs Variables Réelles III, Comptes Rendus de l’Académie des sciences Paris 241
- Lacombe
- 1955
(Show Context)
Citation Context ...dels. 1 Introduction Several approaches have been proposed to model computations over real numbers. Recursive analysis or computable analysis, was introduced by Turing [38], Grzegorczyk [17], Lacombe =-=[21]-=-. Many works have been devoted to giving computable foundations to most of the concepts of mathematical analysis in this framework : see e.g. monograph [39]. Alternative views exist. Among them, we ca... |

58 |
Computable functions
- Grzegorczyk
- 1955
(Show Context)
Citation Context ...mputational models. 1 Introduction Several approaches have been proposed to model computations over real numbers. Recursive analysis or computable analysis, was introduced by Turing [38], Grzegorczyk =-=[17]-=-, Lacombe [21]. Many works have been devoted to giving computable foundations to most of the concepts of mathematical analysis in this framework : see e.g. monograph [39]. Alternative views exist. Amo... |

44 | Does General Relativity Allow an Observer to View an Eternity in a Finite Time
- Hogarth
- 1992
(Show Context)
Citation Context ... terms of solutions of polynomial differential equations [36, 34, 22, 16]. Continuous time machines also include analog neural networks [32, 37], hybrid systems [3, 6], or theoretical physical models =-=[31, 19, 15]-=-: see also survey [32]. The relations between all the models are not fully understood. One can say, that the theory of analog computations has not yet experienced the unification that digital discrete... |

33 | Analog computers and recursive functions over the reals
- Graça, Costa
(Show Context)
Citation Context ..., and which motivated Shannon’s General Purpose Analog Computer (GPAC) model [36], whose computational power was characterized algebraically in terms of solutions of polynomial differential equations =-=[36, 34, 22, 16]-=-. Continuous time machines also include analog neural networks [32, 37], hybrid systems [3, 6], or theoretical physical models [31, 19, 15]: see also survey [32]. The relations between all the models ... |

33 |
and L A Rubel, A differentially algebraic replacement theorem
- Lipshitz
(Show Context)
Citation Context ..., and which motivated Shannon’s General Purpose Analog Computer (GPAC) model [36], whose computational power was characterized algebraically in terms of solutions of polynomial differential equations =-=[36, 34, 22, 16]-=-. Continuous time machines also include analog neural networks [32, 37], hybrid systems [3, 6], or theoretical physical models [31, 19, 15]: see also survey [32]. The relations between all the models ... |

31 | Achilles and the tortoise climbing up the arithmetical hierarchy
- Asarin, Maler
- 1998
(Show Context)
Citation Context ...ower was characterized algebraically in terms of solutions of polynomial differential equations [36, 34, 22, 16]. Continuous time machines also include analog neural networks [32, 37], hybrid systems =-=[3, 6]-=-, or theoretical physical models [31, 19, 15]: see also survey [32]. The relations between all the models are not fully understood. One can say, that the theory of analog computations has not yet expe... |

29 | An analog characterization of the Grzegorczyk hierarchy
- Campagnolo, Moore, et al.
(Show Context)
Citation Context ...C 2 by C k−1 in the statements of the theorems. 11s7 Upper bounds We now prove the upper bound L ∗ ⊂ E(R). As one may expect, this direction of the proof has many similarities with the proof L ⊂ E in =-=[12, 13]-=-: main differences lie in the presence of non-total functions and of schema LIM. We first discuss the domain of the considered functions. Lemma 6 All functions from L ∗ are of class C 2 and defined on... |

27 | Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy
- Bournez
- 1999
(Show Context)
Citation Context ...ower was characterized algebraically in terms of solutions of polynomial differential equations [36, 34, 22, 16]. Continuous time machines also include analog neural networks [32, 37], hybrid systems =-=[3, 6]-=-, or theoretical physical models [31, 19, 15]: see also survey [32]. The relations between all the models are not fully understood. One can say, that the theory of analog computations has not yet expe... |

26 |
Computational Complexity of Real Valued Recursive Functions and Analog Circuits
- Campagnolo
- 2002
(Show Context)
Citation Context ...ne can say, that the theory of analog computations has not yet experienced the unification that digital discrete time computations have experienced through Turing work and the so-called Church thesis =-=[13, 32]-=-. This however becomes a crucial matter since the progress of electronics makes the construction of some of the machines realistic, whereas some models were recently proved very (far too?) powerful: u... |

24 | Robust undecidability of timed and hybrid systems
- Henzinger, Raskin
- 2000
(Show Context)
Citation Context ...ime models that do not suffer from Zeno’s paradox problems is also closely related to the important problems of finding criteria for so-called robustness for continuous (hybrid) time models: see e.g. =-=[18, 2]-=-. In [23], Moore introduced a class of functions over the reals inspired from the classical characterization of computable functions over integers: observing that the continuous analog of a primitive ... |

20 | A.: Perturbed Turing machines and hybrid systems - Asarin, Bouajjani - 2001 |

17 |
Complexité algorithmique des systèmes dynamiques continus et hybrides
- Bournez
- 1999
(Show Context)
Citation Context ...eas some models were recently proved very (far too?) powerful: using the so-called Zeno’s paradox, some models make it possible to compute non-Turing computable functions in a constant time: see e.g. =-=[23, 7, 3, 19, 15]-=-. Notice that understanding whether there exist analog continuous time models that do not suffer from Zeno’s paradox problems is also closely related to the important problems of finding criteria for ... |

16 |
Analyse numérique et équations différentielles. pug
- Demailly
- 1991
(Show Context)
Citation Context ...s a closed interval. Proof By structural induction • This is clear for basic functions (1, 0, −1, U, and θ3). • Composition preserves this property. • Linear differential equations preserve class C 2 =-=[1, 14]-=-. They also preserve the domain property by definition. • If g = LIM(f, K, β), from definition of LIM schema, this is clear. We propose to introduce the following notation: given a ∈ R, let ρa be the ... |

13 |
µ-Recursion and infinite limits
- Mycka
(Show Context)
Citation Context ...o goes to R n and h( −→ x , y) is a n × n matrix with elements in L. Now, we suggest to add a limit operator. Remark 4 The idea of adding a limit operator has already been investigated in papers like =-=[28, 24]-=-. However, since we are interested in R-sub-recursive functions, and not to build a whole hierarchy above recursive functions as in [28, 24], our limit schema will not be as general: as the LI schema ... |

12 |
technological enthusiasm and british technological skepticism in the age of the analog brain
- S
- 1996
(Show Context)
Citation Context ... Scientifique, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, FRANCE - Emmanuel.Hainry@loria.fr 1sAnalyzer [20], that gave birth to a real machine, built in 1931 at the MIT to solve differential equations =-=[9]-=-, and which motivated Shannon’s General Purpose Analog Computer (GPAC) model [36], whose computational power was characterized algebraically in terms of solutions of polynomial differential equations ... |

12 |
On an instrument for calculating the integral of the product of two given functions
- Thomson
(Show Context)
Citation Context ...edex, FRANCE - Olivier.Bournez@loria.fr † INPL, LORIA (UMR 7503 CNRS-INPL-INRIA-Nancy2-UHP) Campus Scientifique, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, FRANCE - Emmanuel.Hainry@loria.fr 1sAnalyzer =-=[20]-=-, that gave birth to a real machine, built in 1931 at the MIT to solve differential equations [9], and which motivated Shannon’s General Purpose Analog Computer (GPAC) model [36], whose computational ... |

10 |
Computability over topological structures
- Brattka
- 2003
(Show Context)
Citation Context ...d to classical definitions of these classes in recursive analysis, involving discussions about higher-order (type 2) Turing machines (see e.g. [39]), or compared to characterizations in the spirit of =-=[10]-=-. In Section 2, we start by some mathematical preliminaries. In Section 3, we recall some notions from classical recursion theory. We present basic definitions of recursive analysis in Section 4. Prev... |

6 | F.: An Analog Characterization of the Subrecursive Functions
- Campagnolo, Moore, et al.
- 2000
(Show Context)
Citation Context ...ssible to use a “compression trick” (another incarnation of Zeno’s paradox) to simulate in a bounded time an unbounded number of discrete transitions in order to recognize arithmetical reals [23]. In =-=[11, 12, 13]-=-, Campagnolo, Costa and Moore propose to consider the (betterdefined) subclass L of R-recursive functions corresponding to the smallest class of functions containing some basic functions and closed by... |

6 |
The computational power of continuous dynamic systems
- Mycka, Costa
(Show Context)
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... |

2 |
recursive functions and real extentions of recursive functions
- Real
- 2004
(Show Context)
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... |

2 |
Analog computation and beyond
- Mycka, Costa
- 2006
(Show Context)
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... |