Some Relations between Classes of Low Computational Complexity

Evidently F is 1-bounded (respectively 2-bounded) if F(x1,..., xm) is bounded for all x1, . . . ,xm by a linear (respectively a polynomial) function of Max{xl 5. . . , xm}. We shall prove that if a 0-bounded function can be defined using 2-bounded primitive recursion it can also be defined using simultaneous 0-bounded recursions. By a class %> of number-theoretic functions we shall always mean a collection which contains the successor function, the case function C {Cxyuv = x if u = v, Cxyuv = y\fu^v) and which is closed under explicit definition. #„. denotes the set of relations whose characteristic functions are in 6. In [2] Grzegorczyk introduced the hierarchy S of primitive recursive functions. For n = 0 , 1 , 2, S is, roughly speaking, the smallest class closed under n-bounded recursion. This statement becomes exact if we enlarge S° to the class S by adding Max {xj, x2} as an initial function. It is an idiosyncracy of Grzegorczyk's definition that M a x ^ 0 ; thus S° is not a class. But S° and S' contain the same relations: