Distributionally Hard Languages

Cai and Selman [CS] defined a modification of Levin's [Le] notion of average polynomial time and proved, for every P-bi-immune language L and every polynomial-time computable distribution mu with infinite support, that L is not recognizable in polynomial time on the mu -average. We call such languages distributionally hard .Pavan and Selman [PS] proved that there exist distributionally hard sets that are not P-bi-immune if and only P contains P-printable-immune sets. We extend this characterizion to include assertions about several traditional questions about immunity, about finding witnesses for NP-machines, and about existence of one-way functions. Similarly, we address the question of whether NP contains sets that are distributionally hard. Several of our results are implications for which we cannot prove whether or not their converse holds. In nearly all such cases we provide oracles relative to which the converse fails. We use the techniques of Kolmogorov complexity to describe our oracles and to simplify the technical arguments.