Relation of deterministic push down automata and lower elementary recursion Deterministic context free languages are the context free languages that can be accepted by a deterministic push down automata.

Deterministic context free languages can be recognized by a deterministic Turing machine in polynomial time.

The lower elementary functions are a subset of the elementary recursive 
functions.

Lower elementary recursive functions are limited to polynomial time growth.

From this is it correct to derive that deterministic push down automata are of
equal power to the lower elementary recursive functions?

Edit:
My reasoning is that because the deterministic context free languages can be recognized in polynomial time that a deterministic push down automata is implementing a boolean function of up to that complexity. The lower elementary functions are also up to polynomial time. As such are they not equal in power?
No, that doesn't follow at all.  There are problems that can be solved in polynomial time, but that don't correspond to any deterministic context-free language.