Separations by random oracles and “Almost” classes for generalized reducibilities

Abstract

Given two binary relations ⩽_r and ⩽_s on 2^ω which are closed under finite variation (of their set arguments) and a set X chosen randomly by independent tosses of a fair coin, one might ask for the probability that the lower cones $\left\{ {A \subseteq \omega :A \leqslant _r X} \right\} and \left\{ {A \subseteq \omega :A \leqslant _3 X} \right\}$w.r.t. ⩽_r and ⩽_s are different. By closure under finite variation, the Kolmogorov 0–1 Law yields immediately that this probability is either 0 or 1; in the case it is 1, the relations are said to be separable by random oracles. Again by closure under finite variation, the probability that a randomly chosen set X is in the upper cone of a fixed set A w.r.t. ⩽_r is either 0 or 1. Almost _r is the class of sets for which the upper cone w.r.t. ⩽_r 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^ω which can be defined by a countable set of total continuous functionals on 2^ω in the same way as the usual resource bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept 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. From the results for generalized reducibilities, one obtains 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.

Both authors were partially supported by the EU network Complexity, Logic and Recursion Theory (COLORET), EU Contract No. ERBCHRXCT930415.