Nontriviality for Exponential Time w.r.t. Weak Reducibilities

Abstract

A set A is nontrivial for the linear exponential time class E = DTIME(2^lin) if A ∈ E and the sets from E which can be reduced to A are not from a single level DTIME(2^kn) of the linear exponential hierarchy. Similarly, a set A is nontrivial for the polynomial exponential time class EXP = DTIME(2^poly) if A ∈ EXP and the sets from EXP which can be reduced to A are not from a single level $\mathrm{DTIME}(2^{n^k})$ of the polynomial exponential hierarchy (see [1]). Here we compare the strength of the nontriviality notions with respect to the underlying reducibilities where we consider the polynomial-time variants of many-one, bounded truth-table, truth-table, and Turing reducibilities. Surprisingly, the results obtained for E and EXP differ. While the above reducibilities yield a proper hierarchy of nontriviality notions for E, nontriviality for EXP under many-one reducibility and truth-tab! le reducibility coincides.