Comparing Nontriviality for E and EXP

Abstract

A set A is nontrivial for the linear-exponential-time class E=DTIME(2^lin) if for any k≥1 there is a set B _k ∈E such that B _k is (p-m-)reducible to A and $B_{k} \not\in \mathrm{DTIME}(2^{k\cdot n})$. I.e., intuitively, A is nontrivial for E if there are arbitrarily complex sets in E which can be reduced to A. Similarly, a set A is nontrivial for the polynomial-exponential-time class EXP=DTIME(2^poly) if for any k≥1 there is a set $\hat{B}_{k} \in \mathrm {EXP}$ such that $\hat{B}_{k}$ is reducible to A and $\hat{B}_{k} \not\in \mathrm{DTIME}(2^{n^{k}})$. We show that these notions are independent, namely, there are sets A ₁ and A ₂ in E such that A ₁ is nontrivial for E but trivial for EXP and A ₂ is nontrivial for EXP but trivial for E. In fact, the latter can be strengthened to show that there is a set A∈E which is weakly EXP-hard in the sense of Lutz (SIAM J. Comput. 24:1170–1189, 11) but E-trivial.