On the Weak Computability of Continuous Real Functions

TL;DR

This paper investigates various classes of weakly computable continuous real functions, introduces new classifications, and analyzes their properties and closure under arithmetic operations and composition.

Contribution

It defines new classes of weakly computable functions, compares two different definitions, and studies their properties and algebraic closure.

Findings

Two definitions of weakly computable functions are not equivalent.

Weakly computable functions are closed under arithmetic operations.

Weakly computable functions are closed under composition.

Abstract

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.