Real numbers, continued fractions and complexity classes

Abstract

We study some representations of real numbers. We compare these representations, on the one hand from the viewpoint of recursive functionals, and of complexity on the other hand. The impossibility of obtaining some functions as recursive functionals is, in general, easy. This impossibility may often be explicited (and reinforced) in terms of complexity: - existence of a sequence of low complexity whose image is not a recursive sequence, - existence of objects of low complexity but whose images have arbitrarily high time- complexity (often, the ‘low complexity’ is linear time or polynomial time). Moreover, some representations of real numbers that are equivalent from the viewpoint of recursive functionals, are very distinct from the viewpoint of complexity. We make a particular study of representations via continued fractions (dfc). We precise exactly what part of information available in the x's dfc is equivalent to the information available in its Dedekinds cut. We show that the sum of two reals whose dfcs are polynomial-time computable may be a real whose dfc has time complexity arbitrarily high. This work confirms that the unique representation of real numbers suitable for the ordinary calculus is via explicit Cauchy sequences of rationals.