Open AccessJournal Article
Recursive Analysis Characterized as a Class of Real Recursive Functions
Olivier Bournez,Emmanuel Hainry +1 more
TLDR
It is proved 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.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.read more
Citations
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Book ChapterDOI
A Survey on Continuous Time Computations
TL;DR: An overview of theories of continuous time computation is provided, which allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuously time analog models.
Journal ArticleDOI
Polynomial differential equations compute all real computable functions on computable compact intervals
TL;DR: It is shown that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals, and that all real computable functions over compact intervals can be defined by GPACs.
Journal ArticleDOI
Liste des publications
TL;DR: Bournez et al. as mentioned in this paper showed that pure nash equilibria can be learned efficiently in potential games by a wide class of fully distributed algorithms, including stochastic distributed algorithms.
Book ChapterDOI
A Survey on Analog Models of Computation.
Olivier Bournez,Amaury Pouly +1 more
TL;DR: A survey on analog models of computations, which considers both approaches, often intertwined, with a point of view mostly oriented by computation theory.
Book ChapterDOI
The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
TL;DR: In this paper, it was shown that GPACs are equivalent to systems of polynomial differential equations, and that all real computable functions can be defined by such models.
References
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Ordinary differential equations
TL;DR: In this article, the Poincare-Bendixson theory is used to explain the existence of linear differential equations and the use of Implicity Function and fixed point Theorems.
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Ordinary differential equations
TL;DR: The fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ODEs was published by as discussed by the authors, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.
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On Computable Numbers, with an Application to the Entscheidungsproblem
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Ordinary Differential Equations
TL;DR: In this paper, differential equations on manifolds have been used to prove linear systems and linear systems are proven to be linear systems with differential equations on manifolds.