Relativizations of the P-printable Sets and the Sets with Small Generalized Kolmogorov Complexity (revised) Relativizations of the P-printable Sets and the Sets with Small Generalized Kolmogorov Complexity (revised)

Although tally sets are generally considered to be weak when used as oracles, it is shown here that in relativizing certain complexity classes, they are in fact no less powerful than any other class of sets and are more powerful than the class of recursive sets. More speciically, relativizations of the classes of P-printable sets and sets with small Generalized Kolmogorov complexity (SGK) are studied. It is shown here that all sparse sets are P TALLY-printable and are in SGK TALLY , and that there are self-P-printable sets that are neither P REC-printable nor in SGK REC. There are also sets that are P REC-printable and in SGK REC that are not self-P-printable. Relativizations to various subrecursive oracles are also presented. A restriction on the number of oracle queries is also presented, with the result that relativizing SGK to any oracle with at most O(logn) queries results in a set that is still in SGK.