Area number x is called k-monotonically computable , for constant k > 0, if there is a computable sequence 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 if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence s ∈ ℕ of (...) rational numbers converging to x such that the sum equation image|xs — xs + 1| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers. (shrink)

Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > 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 → ℚ, we show that the class h-MC coincides (...) with the classes of computable and semi-computable real numbers if and only if Σi∈ℕ) = ∞and the sum Σi∈ℕ) is a computable real number, respectively. On the other hand, if h ≥ 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. (shrink)

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore how (...) the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (shrink)