Separations by Random Oracles and \ Almost " Classes for Generalized

Let r and s be two binary relations on 2 N which are meant as reducibilities. Let both relations be closed under nite 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, diiers from the lower scone of X. By closure under nite 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 nite 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 upperr-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 deened by a countable set of total continuous functionals on 2 N in the same way as the usual resource-bounded reducibilities are deened 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 speciic machine model or even eeectivity. The results on generalized reducibilities yields corollaries about speciic resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations.