Weakly Computable Real Numbers

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semicomputable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and ...

On the Turing Degrees of Weakly Computable Real Numbers

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference doe...

M2-computable real numbers

Monotonically Computable Real Numbers

A real number x is called k-monotonically computable (k-mc), for constant k > 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k · |x−xn| ≥ |x−xm| for any m > n and x is monotonically computable (mc) if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence (xs)s∈N of rat...

H-monotonically Computable Real Numbers

Xizhong Zheng∗1,2, Robert Rettinger3, and George Barmpalias4 1 Theoretische Informatik, BTU Cottbus, 03044 Cottbus, Germany. [email protected] 2 Dept. of Comput. Sci., Jiangsu University, Zhenjiang 212013, China 3 Theoretische Informatik II, FernUniversität Hagen, 58084 Hagen, Germany. [email protected] 4 School of Mathematics University of Leeds, Leeds LS2 9JT, U.K...