Weak completeness notions for exponential time

Abstract

Lutz [20] proposed the following generalization of hardness: While a problem A is hard for a complexity class C if all problems in C can be reduced to A, Lutz calls a problem weakly hard if a nonnegligible part of the problems in C can be reduced to A. For the exponential-time class E, Lutz formalized these ideas by introducing a resource-bounded (pseudo) measure on this class and by saying that a subclass of E is negligible if it has measure 0 in E. Here we introduce and investigate new weak hardness notions for E, called E-nontriviality and strong E-nontriviality, which generalize Lutz's weak hardness notion for E and which are conceptually much simpler than Lutz's concept. Moreover, E-nontriviality may be viewed as the most general consistent weak hardness notion for E. © 2010 Springer-Verlag Berlin Heidelberg.