Equivalence of some Hierarchies of Primitive Recursive Functions

Strictly primitive recursive realizability, I

“…functions according to some measure of complexity other than the number of unbounded recursions-depth of nested recursions, depth of nestings of loops in LOOP programs, bounds on time and space used in computations by Turing machines or register machines, and others-all resulted in the same hierarchy of classes of functions, namely {G,,}. (See [4] and [10] for references.) Finally, it appears that the realizability method studied here is an instance of a more general procedure.…”

“…In particular, Go properly includes the class of elementary functions in the sense of Kalmar. 4 We state some facts about the hierarchy G". Let PR be the class of p.r.…”

“…1t is easily verified that the functions C 1 ' for k > 0 and q > 0 and C/*' for k > 2 can be defined from the remaining initial functions by composition and bounded recursion 4. We write =: to indicate identity by definition.…”