Abstract
A real number x is called h-monotonically computable (h- mc), for some function h, if there is a computable sequence (x s )∈ℕ of rational numbers such that h(n)∣ x-x n ∣≥∣ x-x m ∣ for any m ≥n.x is called ω -monotonically computable (ω-mc) if it is h-mc for some recursive function h and, for any c ∈ℝ, x is c-mc if it is h-mc for the constant function h ≡ c. In this paper we discuss the properties of c-mc and ω-mc real numbers. Among others we will show a hierarchy theorem of c-mc real numbers that, for any constants c 2 > c 1 ≥1, there is a c 2-mc real number which is not c 1-mc and that there is an ω-mc real number which is not c-mc for any c ∈ ℝ. Furthermore, the class of all ω-mc real numbers is incomparable with the class of weakly computable real numbers which is the arithmetical closure of semi-computable real numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. Ambos-Spies, K. Weihrauch and X. Zheng Weakly computable real numbers. J. of Complexity. 16(2000), 676–690.
C. Calude. A characterization of c.e. random reals. CDMTCS Research Report Series 095, March 1999.
C. Calude, P. Hertling, B. Khoussainov, and Y. Wang, Recursive enumerable reals and Chaitin’s Ω-number, in STACS’98, pp596–606.
C. Calude and P. Hertling Computable approximations of reals: An information-theoretic analysis. Fundamenta Informaticae 33(1998), 105–120.
G. J. Chaitin A theory of program size formally identical to information theory, J. of ACM., 22(1975), 329–340.
P. Martin-Löf The definition of random sequences, Information and Control, 9(1966), 602–619.
R. Rettinger, X. Zheng, R. Gengler and B. von Braunmühl Monotonically computable real numbers. DMTCS’01, July 2–6, 2001, Constançta, Romania.
H. G. Rice Recursive real numbers, Proc. Amer. Math. Soc. 5(1954), 784–791.
R. M. Robinson Review of “R. Peter: ‘Rekursive Funktionen’, Akad. Kiado. Budapest, 1951”, J. Symb. Logic 16(1951), 280.
T. A. Slaman Randomness and recursive enumerability, preprint, 1999.
R. Soare Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin, Heidelberg, 1987.
R. Soare Recursion theory and Dedekind cuts, Trans, Amer. Math. Soc. 140(1969), 271–294.
R. Soare Cohesive sets and recursively enumerable Dedekind cuts, Pacific J. of Math. 31(1969), no. 1, 215–231.
R. Solovay. Draft of a paper (or series of papers) on Chaitin’s work... done for the most part during the period of Sept.–Dec. 1975, unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktoen Heights, New York, May 1975, 215pp.
E. Specker Nicht konstruktive beweisbare Sätze der Analysis, J. Symbolic Logic 14(1949), 145–158
A. M. Turing. On computable number, with an application to the “Entschei-dungsproblem”. Proceeding of the London Mathematical Society, 43(1936), no. 2, 230–265.
K. Weihrauch. An Introduction to Computable Analysis. Texts in Theoretical Computer Science, Springer-Verlag, Heidelberg 2000.
K. Weihrauch & X. Zheng A finite hierarchy of the recursively enumerable real numbers, MFCS’98 Brno, Czech Republic, August 1998, pp798–806.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rettinger, R., Zheng, X. (2001). Hierarchy of Monotonically Computable Real Numbers. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_55
Download citation
DOI: https://doi.org/10.1007/3-540-44683-4_55
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42496-3
Online ISBN: 978-3-540-44683-5
eBook Packages: Springer Book Archive