Separations by Random Oracles and “Almost” Classes for Generalized Reducibilities

Abstract

Let ≤r and ≤s be two binary relations on 2 N which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2 N , which is obtained by choosing elements of 2 N by independent tosses of a fair coin. Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤r X, differs from the lower ≤s-cone of X. By closure under finite variation, the Kolmogorov 0-1 law yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles. Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2 N which can be defined by a countable set of total continuous functionals on 2 N in the same way as the usual resourcebounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises all natural resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations.Mathematics Subject Classification: 03D15, 03D30, 03D75, 68Q10, 68Q15.

Exact Pairs for Abstract Bounded Reducibilities

“…Effective reduction covers are effective in two respects due to being effective listings of effective functionals. In [12], results about almost classes and random separations are shown in an abstract setting for reducibilities which have reduction covers which might fail to satisfy these effectivity conditions, that is, for binary relations on 2" which can be defined by a countable set of continuous, but not necessarily partial recursive functionals.…”