Primitive Recursiveness of Real Numbers under Different Representations

In mathematics, various representations of real numbers have been investigated, such as nested interval representations, Cauchy sequences, ... • The effective versions of these representations are equivalent in the sense that they define the same notion of computability of real numbers. • However, the primitive recursive (p.r., for short) versions of these representations can lead to different notions of p.r. real numbers. • We summarize the known results about the primitive recursiveness of real numbers for different representations as well as show some new relationships. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. The Encoding of Real Numbers In order to compute a real number, we need to represent it in some way as input or output. The concrete means to represent a real number: • by decimal expansion, • by Cauchy sequences, • by Dedekind cut, • by continued fraction, • by nested intervals, • ... The Definition of Computable Real Numbers The original definition of computable real numbers [Turing, 1936]: “the ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means” Turing-Church Thesis: “Finite means = Procedure of a Turing Machine”. Namely, x is computable if there is a computable function f : N → {0, 1, · · · , 9} such that