A recursion theoretic foundation of computation over real numbers

Abstract

Abstract We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel (1931, 1934) and Kleene (1936, Math. Ann., 112, 727–742). We show that this class of functions can also be characterized by MS-machines, which are Turing machine-like devices. The proof of the characterization gives a normal form theorem in the style of Kleene (1936, Math. Ann., 112, 727–742). Furthermore, this characterization is a natural combination of two most influential theories of computation over real numbers, namely the type-two theory of effectivity (see, e.g. Weihrauch (2000, Springer)) and the Blum–Shub–Smale (1989, Bull. Amer. Math. Soc. (N.S.), 21, 1–46) model of computation. Under this notion of computability, the recursive (or computable) subsets of real numbers are exactly effective $\varDelta ^0_2$ sets.