On a relative computability notion for real functions

Abstract

For any class of total functions in the set of natural numbers, we define what it means for a real function to be conditionally computable with respect to this class. This notion extends a notion of relative uniform computability of real functions introduced in a previous paper co-authored by Andreas Weiermann. If the given class consists of recursive functions then the conditionally computable real functions are computable in the usual sense. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. All elementary functions of calculus turn out to be conditionally computable with respect to one of the small subrecursive classes. © 2011 Springer-Verlag.