Subclasses of computable real valued functions

Abstract

This paper deals with the computability of sequences of reals and sequences of functions within the framework of Grzegorczyk's hierarchy, which is in the Number 1 of the addendum to open problems in [4]. We combine the two concepts, computability of sequences of real valued functions and Grzegorczyk's hierarchy of recursive number theoretic functions, together, and examine the computability on real valued functions restricted to primitive recursion and below. Related approaches taken in the literature are the study of primitively recursive reals and real valued functions, and the study of polynomial time computable functions. The notions of (ε ^r) primitive computability for sequences of reals and real valued functions are introduced; relations between (ε ^r) primitive computabilities of sequences of reals and sequences of real valued functions are mathematically proved; some basic properties are studied.