On the definitions of some complexity classes of real numbers

Abstract

Three formulations of computable real numbers based on the concepts of Cauchy sequences, Dedekind cuts, and the binary expansions have been carefully studied. We extend the definitions to recursively enumerable real numbers, and some complexity-bounded classes of real numbers. In particular, polynomial time and nondeterministic polynomial time computable real numbers are defined and compared. Although the three definitions are equivalent for the class of recursive real numbers, they are not equivalent for other classes of real numbers. It seems that the definition based on the concept of Cauchy sequences is the most general one.