Abstract
In effective analysis, various classes of real numbers are discussed. For example, the classes of computable, semi-computable, weakly computable, recursively approximable real numbers, etc. All these classes correspond to some kind of (weak) computability of the real numbers. In this paper we discuss mathematical closure properties of these classes under the limit, effective limit and computable function. Among others, we show that the class of weakly computable real numbers is not closed under effective limit and partial computable functions while the class of recursively approximable real numbers is closed under effective limit and partial computable functions.
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References
K. Ambos-Spies A note on recursively approximable real numbers, Research Report on Mathematical Logic, University of Heidelberg, No. 38, September 1998.
C. Calude, P. Hertling, B. Khoussainov, and Y. Wang, Recursive enumerable reals and Chaintin’s-number, in STACS’98, pp596–606.
C. Calude. A characterization of c.e. random reals. CDMTCS Research Report Series 095, March 1999.
G. S. Ceitin A pseudofundamental sequence that is not equivalent to a monotone one. (Russian) Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 20 1971 263–271, 290.
G. J. Chaitin A theory of program size formally identical to information theory, J. of ACM., 22(1975), 329–340.
A. Grzegorczyk. On the definitions of recursive real continuous functions, Fund. Math. 44(1957), 61–71.
Ker-I Ko Reducibilities of real numbers, Theoret. Comp. Sci. 31(1984) 101–123.
Ker-I Ko Complexity Theory of Real Functions, Birkhäuser, Berlin, 1991.
M. Pour-El & J. Richards Computability in Analysis and Physics. Springer-Verlag, Berlin, Heidelberg, 1989.
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 done for the most part during the period of Sept.-Dec. 1975, unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktoen Heights, New York, May 1975.
E. Specter Nicht konstruktive beweisbare Sätze der Analysis, J. Symbolic Logic 14(1949), 145–158
K. Weihrauch Computability. EATCS Monographs on Theoretical Computer Science Vol. 9, Springer-Verlag, Berlin, Heidelberg, 1987.
K. Weihrauch. An Introduction to Computable Analysis. Springer-Verlag, 2000. (to appear).
K. Weihrauch & X. Zheng A finite hierarchy of the recursively enumerable real numbers, MFCS’98 Brno, Czech Republic, August 1998, pp798–806.
K. Weihrauch & X. Zheng Arithmetical hierarchy of ral numbers. in MFCS’99, Szklarska Poreba, Poland, September 1999, pp 23–33.
X. Zheng. Binary enumerability of real numbers. in Computing and Combinatorics, Proc. of COCOON’99, Tokyo, Japan. July 26–28, 1999, pp300–309.
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Zheng, X. (2000). Closure Properties of Real Number Classes under Limits and Computable Operators. In: Du, DZ., Eades, P., Estivill-Castro, V., Lin, X., Sharma, A. (eds) Computing and Combinatorics. COCOON 2000. Lecture Notes in Computer Science, vol 1858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44968-X_17
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DOI: https://doi.org/10.1007/3-540-44968-X_17
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