Exact pairs for abstract bounded reducibilities

Abstract

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤ r between subsets of the natural numbers where ≤ r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤ r — ideal I and some recursive ≤ r — hard set B for I which is not contained in I, there is some recursive set C where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤ r — cones of B and C where C is not in I. In particular, if the relation ≤ r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets.