Speed-ups by changing the order in which sets are enumerated (Preliminary Version)

Abstract

In a suitably general context, the following analogue of the Blum Speed-up Theorem is proven: There are some infinite sets which are so difficult to enumerate that, given any order for enumerating the set, there is some other order, and some one method of enumerating the set in this second order which is much faster than any method of enumerating the set in the first ordering. It may be possible to interpret this result as a statement about the relative merits of “hardware” vs. “programming” speed-ups. The proof itself is one of the first nontrivial applications of priority methods to questions of computational complexity. As such, it perhaps represents an advance in bringing the results and techniques of contemporary “pure” recursion theory to bear on questions of computational complexity. In this paper we shall prove, in a suitably general context, the following analogue of the Blum Speed-up Theorem, [B1]: There are some infinite sets which are so difficult to enumerate that, given any order for enumerating the set, there is some other order, and some one method of enumerating the set in this second order which is much faster than any method of enumerating the set in the first ordering.