The increase of the complexity of functions at an application of the multiple recursion

E. Ya. Dantsin

Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part V, Tome 32 (1972), pp. 12-17

Résumé

The note deals with the hierarchy of classes $C_{\alpha}$, ($\alpha$ being an ordinal) constructed by Löb and Wainer ([I],[2]). If $\alpha{\omega}$ then this hierarchy coincides with Grzegorczyk classification and $\bigcup_{\alpha{\omega^k}}$ is the class of all $k$-recursive functions ($k=1,2,\dots$). The complexity and the rate of growth of functions from $C_{\alpha}$ are characterized by the ordinal $\alpha$. In the present paper it is shown that if $\varphi$ is obtained from functions of $C_{\alpha}$, by one applicatien of the $k$-recursion, then the complexity of $\varphi$ is characterized by the ordinal $\alpha+\omega^{k-1}$, namely $\varphi$ is primitive recursive in $C_{\alpha+\omega^{k-1}}$. In particular for the primitive recursion ($k=1$) $\varphi\in C_{\alpha+1}$.