On the hierarchy and extension of monotonically computable real numbers

  • Authors:

  • Robert Rettinger;Xizhong Zheng

  • Affiliations:

  • Theoretische Informatik II, FernUniversität Hagen, 58084 Hagen, Germany;Theoretische Informatik, Brandenburgische Technische Universität Cottbus, 03044 Cottbus, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

A real number x is called h-monotonically computable (h-mc for short), for some function h: N → N, if there is a computable sequence (xs) of rational numbers converging to x such that h(n)|x - xn| ≥ |x - xm| for all m n. x is called ω-monotonically computable (ω-mc) if it is h-mc for some computable function h. Thus, the class of ω-mc real numbers is an extension of the class of monotonically computable real nambers introduced in (Math. Logic Quart. 48(3) (2002) 459), where only constant functions h ≡ c are considered and the corresponding real numbers are called c-monotonically computable. In (Math. Logic Quart. 48(3) (2002) 459) it is shown that the classes of c-mc real numbers form a proper hierarchy inside the class of weakly computable real numbers which is the arithmetical closure of the 1-mc real numbers. In this paper, we show that this hierarchy is dense, i.e., for any real numbers c2 c1 ≥ 1, there is a c2- mc real number which is not c1-mc and there is also an ω-mc real number which is not c-mc for any c ∈ R. Furthermore, we show that the class of all ω-mc real numbers is incomparable with the class of weakly computable real numbers.