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Modèles de calcul sur les réels, résultats de comparaison

Emmanuel Hainry 1
1 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : Computation on the real numbers can be modelised in several different ways.
There indeed exist a lot of different computation models on the reals.
However, there exist few results for comparing those models, and most of these
results are incomparability results. The case of computation over the real
numbers hence is quite different from that of computation over integer numbers
where Church-Turing's thesis claims that all reasonable models compute exactly
the same functions.

The results presented in this document are twofold.
One, we show that recursively computable functions (in the sense of
computable analysis) can be shown equivalent to some adequately defined
subclass of R-recursive functions, and also to GPAC-computable functions
with GPAC-computable roughly meaning computable through a converging
GPAC. Hence we get a machine independent characterization of recursively
computable functions, and a bridge between type 2-machines and GPAC.
Two, more than an analog characterization of recursively enumerable
functions, we show that the limit operator we defined can be used to provide
an analog characterization of elementarily computable functions and
En-computable functions for n?3, where En
represents the levels of Grzegorczyk's hierarchy.

Those results can be seen as a first step toward a unification of computable
functions over the reals.
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Emmanuel Hainry. Modèles de calcul sur les réels, résultats de comparaison. Autre [cs.OH]. Institut National Polytechnique de Lorraine - INPL, 2006. Français. ⟨tel-00142675⟩

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