Inferability of recursive real-valued functions

This paper presents a method of inductive inference of real-valued functions from given pairs of observed data of (x, h(x)), where h is a target function to be inferred. Each of such observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers which show the approximate value and the error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two sequences of rational numbers, which converges to the real number and converges to zero, respectively. These sequences show an approximate value of the real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals with rational end points which converges to a singleton interval with the real number as both end points. In this paper, we propose a notion of recursive real-valued functions that can enjoy the merits of the both representations of the recursive real numbers. Then we present an algorithm which approximately infers real-valued functions from numerical data with some error bounds, and show that there exists a rich set of real-valued functions which is approximately inferable in the limit from such numerical data. We also discuss the precision of the guesses from the machine when sufficient data have not yet given.