Rationally presented metric spaces and complexity, the case of the space of uniformly continuous real functions on a compact interval

TL;DR

This paper introduces rational presentations of metric spaces to analyze the algorithmic complexity of the space of uniformly continuous functions on [0,1], extending results on polynomial-time computability of classical theorems.

Contribution

It develops a framework for rational presentations of metric spaces and applies it to analyze the complexity of function spaces, generalizing known polynomial-time results.

Findings

Generalization of Hoover's polynomial-time Weierstrass approximation theorem

Extension of polynomial-time computability results to analytic functions

Comparison of complexity notions for different presentations of function spaces

Abstract

We define the notion of {\em 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 presentations of the space $\czu$ of uniformly continuous real functions over [0,1] with the usual norm: $\norme{f}_{\infty} = {\bf Sup} \{ \abs{f(x)} ; \;0 \leq x \leq 1\}.$ This allows us to have a comparison of a global kind between complexity notions attached to these presentations. In particular, we get a generalisation of Hoover's results concerning the {\sl Weierstrass approximation theorem in polynomial time}. We get also a generalisation of previous results on analytic functions which are computable in polynomial time.