A predicative approach to the classification problem

We harmonize many time-complexity classes DTIMEF(f(n)) (f(n) greater than or equal to n) with the PR functions (at and above the elementary level) in a transfinite hierarchy of classes of functions F-alpha. Class F-alpha is obtained by means of unlimited operators, namely: a variant II of the predicative or safe recursion scheme, introduced by Leivant, and by Bellantoni and Cook, if a is a successor; and constructive diagonalization if alpha is a limit. Substitution (SBST) is discarded because the time complexity classes are not closed under this scheme. F-alpha is a structure for the PR functions finer than epsilon (alpha) to the point that we have F-epsilon0 = epsilon (3) (elementary functions). Although no explicit use is made of hierarchy functions, it is proved that f(n) epsilon F-alpha implies f(n) less than or equal to n(G alpha (n)), where G(alpha) belongs to the slow growing hierarchy (of functions) studied, in particular, by Girard and Wainer.