Speed-Ups by changing the order in which sets are enumerated

Abstract

The purpose of this paper is to show that there are sets for which really large improvements in enumeration times can only be achieved by programs which change the order in which the sets are enumerated and not by programs which merely speed up the enumeration times without changing the order in which the sets are enumerated. We do this by proving, in a suitably general context, the following analogue of the Blum speedup theorem: 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 someone method of enumerating the set in this second order which is much faster thanany method of enumerating the set in the first ordering. The proof itself is one of the first nontrivial applications of priority methods to questions of computational complexity. (See also [8], [6], and [3] for more such applications.)