Separating complexity classes using structural properties

We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f(n)-dense set S ∈ P, A - S is still complete, where f(n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: For every ≤_m^p-complete set A for EXP and any subexponential dense sets S ∈ P, A - S is still Turing complete and under a reasonable hardness assumption even ≤_m^p-complete. For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S ∈ P, A - S is still Turing complete. There exists a 3-truth-table complete set A for EEXPSPACE, and a log-dense set S ∈ P such that A - S is not Turing complete. This implies that settling this issue for EEXP will either separate P from PSPACE or PUfrom EXP. We show that the robustness results for EXP and the delta levels of the exponential hierarchy do not relativize.