@misc {IOPORT.04016888,
    author       = {Wainer, S.S.},
    title        = {The ``slow-growing'' $\Pi \sp 1\sb 2$ approach to hierarchies.},
    howpublished = {Recursion theory, Proc. AMS-ASL Summer Inst., Ithaca/N.Y. 1982, Proc. Symp. Pure Math. 42, 487-502 (1985).},
    year         = {1985},
    abstract     = {[For the entire collection see Zbl 0556.00008.] The recursion-theoretic hierarchies assign ordinal notations to functions so as to reflect their computational complexity. This paper gives a new approach such that each Kleene-computation in the maximal type structure over the integers N appears as a natural collapse (under the slow-growing function G) of an identical computation defined over abstract ordinal notations $\Omega$. Kleene's 0 appears as the collapse of a $\Pi\sp 1\sb 2$-complete set of ``notations over notations''. The collapse function G is the map from $\Omega$ to $N\sp N$ and defined as $G\sb x(0)=0$; $G\sb x(\alpha +\sb 01)=G\sb x(\alpha)+1$, and $G\sb x(Sup \alpha\sb x)=G\sb x(\alpha\sb x)$. We may extend G to higher type and apply Kleene's schemes S1-S9 (with S5 modified and S10 added) to G-continuous functions over $\Omega$, which would provide refined measure of the complexity of the recursion. The hierarchy is also generated to the usual construction in Rec($\Omega)$ analogous to those in Rec(N). As a result, the transfinite Grzegorczyk hierarchy is the collapse of the Bachmann hierarchy and the first non-elementary function in this hierarchy occurs at the first ``elementary-closed'' ordinal $\epsilon\sb 0$, and the first nonprimitive recursive function occurs at the first prime-closed ordinal.},
    reviewer     = {Moh Shawkwei},
    identifier   = {04016888},
}