I am currently reading the paper "On the computational complexity of Algorithms" by J. Hartmanis and R. E. Stearns ( http://www.jstor.org/pss/1994208 )
It includes a proof that any $T$-computable sequence is also $\lceil kT \rceil$-computable, where $k$ is a computable, positive real number, $T$ a time-function and $\lceil kT \rceil$ denotes the smallest integer $m$ such that $m \ge kT$.
The proof idea is to use a (multitape) TM that has a head that can read 2 symbols at once, while storing internally the symbol to the left and right of that head, so that the new TM can simulate 2 steps of a normal (multitape) TM in only once step. By doing the same trick on the new TM using induction, one can prove 1/2k speed-up.
I have two questions, which revolve around the same thing:
First of all, can I extend this idea and use a head so large that reads the entire input at once? That would require to save a super-constant number of symbols internally in the TM. Would that be a violation of the model?
Reflecting to my first question, whatever the answer is : Are there "complexity" classes that restraint the TM's control logic somehow? e.g. allowing up to a specific number of states or for a specific structure of the automaton?