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Distributionally-Hard Languages Lance Fortnow A. Pavan Alan L. Selman

Summary: Distributionally-Hard Languages
Lance Fortnow A. Pavan Alan L. Selman
Cai and Selman [CS99] defined a modification of Levin's notion of average poly-
nomial time and proved, for every P-bi-immune language L and every polynomial-time
computable distribution with infinite support, that L is not recognizable in poly-
nomial time on the -average. We call such languages distributionally-hard. Pavan
and Selman [PS00] 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. 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 average-case complexity and
he provided the central notions for its study. More recently, Cai and Selman [CS99] ob-
served that Levin's definition of Average-P has limitations when applied to distributional


Source: Aduri, Pavan - Department of Computer Science, Iowa State University


Collections: Computer Technologies and Information Sciences