  Predicative analysis of recursion schema is a method to characterize
complexity classes like the class FPTIME of polynomial time computable
functions. This analysis comes from the works of Bellantoni and Cook, and
Leivant by data tiering. Here, we refine predicative analysis by using a
ramified Ackermann's construction of a non-primitive recursive function. We
obtain a hierarchy of functions which characterizes exactly functions, which
are computed in O(n^k) time over register machine model of computation. For
this, we introduce a strict ramification principle. Then, we show how to
diagonalize in order to obtain an exponential function and to jump outside
deterministic polynomial time. Lastly, we suggest a dependent typed
lambda-calculus to represent this construction.
