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 ∈ DTIME(2 k·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 setB k ∈ EXP such thatB k is reducible to A andB k ∈ 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, 1995) but E-trivial.