The Speedup Theorem in a Primitive Recursive Framework

Blum's speedup theorem is a major theorem in computational complexity, showing the existence of computable functions for which no optimal program can exist: for any speedup function r there exists a function fr such that for any program computing fr we can find an alternative program computing it with the desired speedup r. The main corollary is that algorithmic problems do not have, in general, a inherent complexity. Traditional proofs of the speedup theorem make an essential use of Kleene's fix point theorem to close a suitable diagonal argument. As a consequence, very little is known about its validity in subrecursive settings, where there is no universal machine, and no fixpoints. In this article we discuss an alternative, formal proof of the speedup theorem that allows us to spare the invocation of the fix point theorem and sheds more light on the actual complexity of the function fr.