Divergence bounded computable real numbers

AbstractA real x is called h-bounded computable, for some function h:N→N, if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n∈N, at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2-n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h

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 rational numbers converging to x such that the sum s∈N |xs − xs+1 | is finite. In this paper we show that all mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers

h-Monotonically Computable Real Numbers

Let h: N → Q be a computable function. A real number x is called h-monotonically computable (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x h-monotonically in the sense that h(n)|x − xn | ≥ |x − xm | for all n and m> n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h: N → (0, 1)Q, we show that the class h-MC coincides with the classes of computable and semi-computable real numbers if and only if � i∈N (1 − h(i)) = ∞ and the sum i∈N (1 − h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h-MC = SC no matter how fast h converges to 1. Furthermore, for any constant c> 1, if h is increasing and converges to c, then h-MC = c-MC. Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g

On the hierarchy and extension of monotonically computable real numbers

AbstractA 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 numbers 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