Immunity, Relativizations, and Nondeterminism

Abstract

It is known [8], [13] that there is a recursive set A such that ${\text{NP }}(A)$ contains a set that is ${\text{P }}(A)$ -immune; that is, there is an infinite set $L \in {\text{NP}}(A)$ such that no infinite subset of L is in ${\text{P}}(A)$. The first result generalizes this fact to situations where the running times of the machines specifying the “larger” class bound the size of the sets of strings queried in the computation trees of the machines specifying the “smaller” class. The second result is of a different type. For relativized complexity classes specified by bounds on the number of oracle queries and the number of nondeterministic steps allowed in computations, it is known [14] that one can describe a set A such that there is an infinite hierarchy of classes relative to A where each new class in the hierarchy is obtained by increasing the amount of nondeterminism. Here it is shown that the conditions allowing infinite hierarchies to exist also allow for each $i \geqq 0$, the existence of a set in the $(i + 1)$st class which has no infinite subset in the ith class.