On computational speed-up

Let F be any effective mapping from total functions on the integers to total functions. Composition and iterated composition are examples of such mappings. The "operator speed-up" theorem in this paper establishes the existence of a computable function f such that for any program computing f(x) in p1(x) steps for all x, there is another program computing f(x) in p2(x) steps and F(p2) ≪ P1 almost everywhere. Thus, there is no best program for f. The notions of "program" and "number of steps" are treated axiomatically, so that the theorem is independent of any particular model of a computing machine. An example of speed-up for Turing machines is considered.