Divergence bounded computable real numbers

Authors:
Xizhong Zheng;Dianchen Lu;Kejin Bao
Affiliations:
Department of Computer Science, Jiangsu University, Zhenjiang, China and BTU Cottbus, Cottbus, Germany;Department of Mathematics, Jiangsu University, Zhenjiang, China;Department of Computer Science, Jiangsu University, Zhenjiang, China
Venue:
Theoretical Computer Science - Real numbers and computers
Year:
2006

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Abstract

A real x is called h-bounded computable, for some function h: N → N, if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n ∈ at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2-n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h.