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 (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the h-monotonic computability of semi-computable real numbers which are 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.