Polynomial-time hierarchy of computable reals

Abstract

In mathematics, various representations of real numbers have been investigated and all these representations are proved to be mathematically equivalent. Furthermore, it is known that all effective versions of these representations lead to the same class of "computable real numbers". However, when subrecursive (such as primitive recursive) is taken into account, these representations can lead to different notions of "computable real numbers". This paper will look into the polynomial-time version of the problem for computable real numbers under different representations. We will summarize the known results to exhibit the comprehensive hierarchy they form. Our goal is to clarify systematically how the polynomial-time computability depends on the representations of the real numbers.