Strong Reductions and Immunity for Exponential Time

Abstract

This paper investigates the relation between immunity and hardness in exponential time. The idea that these concepts are related originated in computability theory where it led to Post’s program. It has been continued successfully in complexity theory [10],[14],[20]. We study three notions of immunity for exponential time. An infinite set A is called - EXP-immune, if it does not contain an infinite subset in EXP; - EXP-hyperimmune, if for every infinite sparse set B ∈ EXP and every polynomial p there is an x ∈ B such that y ∈ B : p^-1(∣x∣) ≤∣y∣ ≤ p(∣x∣) is disjoint from A; - EXP-avoiding, if the intersection A∩B is finite for every sparse set B ∈ EXP.

EXP-avoiding sets are always EXP-hyperimmune and EXP-hyperimmune sets are always EXP-immune but not vice versa. We analyze with respect to which polynomial-time reducibilities these sets can be hard for EXP. EXP-immune sets cannot be conjunctively hard for EXP although they can be hard for EXP with respect to disjunctive and parity-reducibility. EXP-hyperimmunes sets cannot be hard for EXP with respect to any of these three reducibilities. There is a relativized world in which there is an EXP-avoiding set which is hard with respect to positive truth-table reducibility. Furthermore, in every relativized world there is some EXP-avoiding set which is Turing-hard for EXP.

This work was written while F. Stephan was visiting DePaul University. He was supported by the Deutsche Forschungsgemeinschaft (DFG) Heisenberg grant Ste 967/1-1 and DePaul University.