Complexity of the r-query Tautologies in the Presence of a Generic Oracle

Extending techniques in Dowd (Information and Computation vol. 96 (1992)) and those in Poizat (J. Symbolic Logic vol. 51 (1986)), we study computational complexity of rTAUT [A] in the case when A is a generic oracle, where r is a positive integer and rTAUT [A] denotes the collection of all r-query tautologies with respect to an oracle A. We introduce the notion of ceiling-generic oracles, as a generalization of Dowd’s notion of t-generic oracles to arbitrary finitely testable arithmetical predicates. We study how existence of ceiling-generic oracles affects behavior of a generic oracle, by which we show that {X : coNP [X] is not a subset of NP [rTAUT [X]] } is comeager in the Cantor space. Moreover, using ceiling-generic oracles, we present an alternative proof of the fact (Dowd) that the class of all t-generic oracles has Lebesgue measure zero.