An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Bournez, Olivier; Hainry, Emmanuel

doi:10.1007/978-3-540-27836-8_25

Cited by 8 publications

(8 citation statements)

References 25 publications

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“…M ϕx computes m(n + 1) and writes it on the oracle tape, 2. the oracle responds with d = ϕ x (m(n + 1)), 3. M ϕx computes and outputs e = ψ(d, n + 1).…”

Section: Definition 6 (Modulus Of Continuity) Assume a Functionmentioning

confidence: 99%

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A survey of recursive analysis and Moore’s notion of real computation

Gomaa

2011

Nat Comput

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The theory of analog computation aims at modeling computational systems that evolve in a continuous manner. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore's class of recursive real functions. Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zerofinding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GP AC. In this article we give a detailed exposition of recursive analysis and Moore's class and the relationships between them.

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“…M ϕx computes m(n + 1) and writes it on the oracle tape, 2. the oracle responds with d = ϕ x (m(n + 1)), 3. M ϕx computes and outputs e = ψ(d, n + 1).…”

Section: Definition 6 (Modulus Of Continuity) Assume a Functionmentioning

confidence: 99%

“…(Limit schema [3,4]) Assume a function g : R × D → R such that D ⊆ R k is a product of compact intervals. Assume g satisfies the following:…”

Section: Enriching Lmentioning

confidence: 99%

A survey of recursive analysis and Moore’s notion of real computation

Gomaa

2011

Nat Comput

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“…Some authors avoid this trouble by studying only operators preserving totality, so that partial functions never come into discussion. Campagnolo and Moore [5] take this path by considering linear differential recursion in place of dr. For classes defined by this operator, some relationships with digital computation are known [2,4].…”

Section: Linear Differential Recursionmentioning

confidence: 99%

Differential recursion

Kawamura

2009

ACM Trans. Comput. Logic

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Moore introduced a class of real-valued "recursive" functions by analogy with Kleene's formulation of the standard recursive functions. While his concise definition inspired a new line of research on analog computation, it contains some technical inaccuracies. Focusing on his "primitive recursive" functions, we pin down what is problematic and discuss possible attempts to remove the ambiguity regarding the behavior of the differential recursion operator on partial functions. It turns out that in any case the purported relation to differentially algebraic functions, and hence to Shannon's model of analog computation, fails.

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“…There is no agreed analog counterpart of the Church-Turing thesis; relating the models is crucial to understand the differences between the various computing capabilities. For example, Bournez et al related Moore's recursion theory on R [Moo96], computable analysis [Wei00] and the general purpose analog computer [BH04,BCGH06]. The aim of this paper is to link two analog models of computation.…”

Section: Introductionmentioning

confidence: 99%

Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model

Durand-Lose

2007

Lecture Notes in Computer Science

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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Cited by 8 publications

References 25 publications

A survey of recursive analysis and Moore’s notion of real computation

A survey of recursive analysis and Moore’s notion of real computation

Differential recursion

Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model

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