On the Monotonic Computability of Semi-computable Real Numbers

Let h : ℕ → ℚ be a computable function. A real number x is h-monotonically computable if there is a computable sequence (x_s) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by the function h. In this paper we discuss the h-monotonic computability of semi-computable real numbers, i.e., limits of monotone computable sequences of rational numbers. Especially, we show a sufficient and necessary condition for the function h such that the h-monotonic computability is simply equivalent to the normal computability.