 
Summary: DistributionallyHard Languages
Lance Fortnow A. Pavan Alan L. Selman
Abstract
Cai and Selman [CS99] defined a modification of Levin's notion of average poly
nomial time and proved, for every Pbiimmune language L and every polynomialtime
computable distribution µ with infinite support, that L is not recognizable in poly
nomial time on the µaverage. We call such languages distributionallyhard. Pavan
and Selman [PS00] proved that there exist distributionallyhard sets that are not Pbi
immune if and only P contains Pprintableimmune sets. We extend this characterizion
to include assertions about several traditional questions about immunity, about finding
witnesses for NPmachines, and about existence of oneway functions. Similarly, we
address the question of whether NP contains sets that are distributionally hard. Sev
eral 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 con
verse fails. We use the techniques of Kolmogorov complexity to describe our oracles
and to simplify the technical arguments.
1 Introduction
Levin [Lev86] was the first to advocate the general study of averagecase complexity and
he provided the central notions for its study. More recently, Cai and Selman [CS99] ob
served that Levin's definition of AverageP has limitations when applied to distributional
