A recursion theoretic foundation of computation over real numbers

TL;DR

This paper establishes a foundational framework for computability over real numbers by integrating recursive function theory, machine models, and existing computational theories, providing a unified normal form and characterization of recursive sets.

Contribution

It introduces a new class of computable functions over reals using functional schemes and characterizes them via master-slave machines, combining TTE and BSS models.

Findings

Recursive functions over reals are characterized by master-slave machines.

Recursive subsets of reals are exactly effective Δ²₀ sets.

Normal form theorem for real-number computation models.

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 G\"odel and Kleene. We show that this class of functions can also be characterized by master-slave machines, which are Turing machine like devices. The proof of the characterization gives a normal form theorem in the style of Kleene. Furthermore, this characterization is a natural combination of two most influential theories of computation over real numbers, namely, the type-two theory of effectivity (TTE) (see, for example, Weihrauch) and the Blum-Shub-Smale model of computation (BSS). Under this notion of computability, the recursive (or computable) subsets of real numbers are exactly effective $\Delta^0_2$ sets.