The Global Power of Additional Queries to p-Random Oracles

We consider separations of reducibilities by random sets. First, we show a result on polynomial time-bounded reducibilities that query their oracle nonadaptively: for every p-random set R, there is a set that is reducible to R with k + 1 queries but is not reducible to any other p-random set with at most k queries. This result solves an open problem stated in a recent survey paper by Lutz and Mayordomo [EATCS Bulletin, 68 (1999), pp. 64–80]. Second, we show that the separation result above can be transferred from the setting of polynomial time-bounds to a setting of rec-random sets and recursive reducibilities. This extends the main result of Book, Lutz, and Martin [Inform. and Comput., 120 (1995), pp. 49–54] who, by using different methods, showed a similar separation with respect to Martin-Löf-random sets. Moreover, in both settings we obtain similar separation results for truth-table versus bounded truth-table reducibility.