A recursion theoretic foundation of computation over real numbers

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