Weak Completeness Notions for Exponential Time

Lutz (SIAM J. Comput. 24(6), 1170-1189, 1995) 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 linear exponential time class E = DTIME(2(lin)), 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. In this paper we introduce two new weak hardness notions for E - E-nontriviality and strongly E-nontriviality. They generalize Lutz's weak hardness notion for E, but are much simpler conceptually. Namely, a set A is E-nontrivial if, for any k >= 1, there is a set B-k is an element of E which can be reduced to A (by a polynomial time many-one reduction) and which cannot be computed in time O(2(kn)), and a set A is strongly E-nontrivial if the set B-k can be chosen to be almost everywhereO(2(kn))-complex, i.e. if B-k can be chosen such that any algorithm that computes B-k runs for more than 2(k|x|) steps on all but finitely many inputs x.