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 which query their oracle non-adaptively: for every p-random set R, there is a set which 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 [16]. 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 [8], 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 truthtable reducibility