Computability of Real Numbers

Abstract

In scientific computation and engineering real numbers are typically approximated by rational numbers which approximate, in principle, the real numbers up to any given precision. This means that they are treated as the limits of computable sequences of rational numbers where an effective error estimation is often expected. Although such approximations exist for many constants, they do not exist for all constants. More precisely, approximations with an effective control over the approximation error exist only for the computable real numbers. As long as we are interested in approximations of real numbers, the weakest condition we can ask for is that a real number can be approximated by a computable sequence of rational numbers without any information about the approximation error. These real numbers are called computably approximable. By relaxing and varying conditions on the knowledge one has about the approximation, one gets several natural classes between these two classes of real numbers. In this paper we review the most natural classes which have been investigated over the past decades with respect to this viewpoint. Most of these classes can be characterized in different ways, partly by purely mathematical properties and partly by their computational properties.