A modified version of the classical µ -operator as well as thefirst value operator and the operator of inverting unaryfunctions, applied in combination with the composition offunctions and starting from the primitive recursive functions,generate all arithmetically representable functions. Moreover, thenesting levels of these operators are closely related to thestratification of the arithmetical hierarchy. The same is shownfor some further function operators known from computability and complexitytheory. The close relationships between nesting levels of operators andthe stratification of the hierarchy also hold for suitablerestrictions of the operators with respect to the polynomialhierarchy if one starts with the polynomial-time computablefunctions. It follows that questions around P vs. NP andNP vs. coNP can equivalently be expressed by closureproperties of function classes under these operators. The polytime version of the first value operator can be used toestablish hierarchies between certain consecutive levels withinthe polynomial hierarchy of functions, which are related togeneralizations of the Boolean hierarchies over the classes $\mbox{$\Sigma^p_{k}$}$ .