Strong Reductions and Immunity for Exponential Time

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, and it has been continued successfully in complexity theory [9, 13, 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(|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 EXPalthough they can be disjunctively hard. EXP-hyperimmune sets cannot be conjunctively or disjunctively hard for EXP, but 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 Turinghard for EXP. 1 From Post’s Program to Complexity Theory Concepts of immunity have a long tradition in computability theory beginning with the famous paper of Post [11, 18] which introduced simple sets and showed that they are not hard in the sense that the halting problem cannot be many-one reduced to a simple set. In fact, no set without an infinite computable subset can be many-one hard for the halting problem. These sets are called immune and the present paper extends the study of resource bounded versions of This work was written while the second author was visiting DePaul University. He was supported by the Deutsche Forschungsgemeinschaft (DFG) Heisenberg grant Ste 967/1-1 and DePaul University.