Complexité du calcul du développement d'un nombre réel en fractions continues

AbstractIn this note, we are concerned with the use of the continued fraction representation of real numbers in the formal computational theory of recursive analysis. In the recursive case, the most general representation for real numbers is the Cauchy sequence representation. On the one hand, other representations such as the Dedekind cut representation and the binary expansions have been studied and shown not to be as general as the Cauchy sequence representation. On the other hand, the class of real numbers to primitive recursive continued fractions is identical to the class of primitive recursive real numbers in the sense of Cauchy which are recursively irrational. Furthermore, there is no efficient algorithm for implementing addition on real numbers written in the continued fraction form

Espaces métriques rationnellement présentés et complexité, le cas de l'espace des fonctions réelles uniformément continues sur un intervalle compact

AbstractWe define the notion of rational presentation of a complete metric space, in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some representations of the space C[0,1] of uniformly continuous real functions over [0,1] with the usual norm: ||f||∞=Sup{|f(x)|;0⩽x⩽1}. This allows us to have a comparison of global kind between complexity notions attached to these presentations. In particular, we get a generalization of Hoover's results concerning the Weierstrass approximation theorem in polynomial time. We get also a generalization of previous results on analytic functions which are computable in polynomial time