Computable functions and generable sets; decidable sets; enumerable sets

Abstract

A computable function is a function which can be computed by an algorithm. Saying “can be computed” we mean (in accordance with chap. 1.1) that when applied to any input the computing algorithm must (1) produce a result equal to the function value for this input if the function is defined on it; and (2) produce no result at all if the function is undefined on this input. Let A be a subset of an aggregate and B be a subset of an aggregate; by Com(A, B) we denote the class of all computable functions from A into B, so Com(A, B) ⊂ F(A, B). If we have an X-Y-representative model for some aggregates X and Y, then, of course, we can formally define Com(X,Y) as the class of all functions from F(X,Y) which can be computed by this model.