Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Computable approximations of reals: an information-theoretic analysis
Fundamenta Informaticae
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Computable analysis: an introduction
Computable analysis: an introduction
Weakly computable real numbers
Journal of Complexity
A Finite Hierarchy of the Recursively Enumerable Real Numbers
MFCS '98 Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science
Hierarchy of Monotonically Computable Real Numbers
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Recursively Enumerable Reals and Chaitin Omega Numbers
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
Hierarchy of Monotonically Computable Real Numbers
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
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A real number x is called h-monotonically computable (h- mc), for some function h, if there is a computable sequence (xs)s∈N of rational numbers such that h(n)|x - xn| ≥ |x - xm| 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-mcif it is h-mcfor 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 c2 c1 ≥ 1, there is a c2-mcr eal number which is not c1-mca nd that there is an ω-mc real number which is not c-mc for any c ∈ R. Furthermore, the class of all ω-mcr eal numbers is incomparable with the class of weakly computable real numbers which is the arithmetical closure of semi-computable real numbers.