Relation To Primitive Recursive Functions
The definition of is the same as that of the primitive recursive functions, RP, except that recursion is limited ( for some j in ) and the functions are explicitly included in . Thus the Grzegorczyk hierarchy can be seen as a way to limit the power of primitive recursion to different levels.
It is clear from this fact that all functions in any level of the Grzegorczyk hierarchy are primitive recursive functions (i.e. ) and thus:
It can also be shown that all primitive recursive functions are in some level of the hierarchy (Rose 1984; Gakwaya 1997), thus
and the sets partition the set of primitive recursive functions, RP.
Read more about this topic: Grzegorczyk Hierarchy
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