Immunity (Extended Abstract)

There are a number of results about the separation of relativized complexity classes specified by deterministic or nondeterministic machines [1,4,11,12]. In the case of time-bounded machines, some individuals have taken the view that such results say little about the difference between determinism and nondeterminism in ordinary computation but rather illustrate the power of nondeterministic oracle machines to generate a large set of strings to be aueried. Recently, Xu, Doner, and Book [19] provided strong evidence for this thesis by establishing a general separating theorem in the context of "degrees of non. 9 i, determ• (that is, refined nondeterminism in the sense of Kintala [11,12]). The result of Xu, Doner, and Book is proved for classes specified by oracle machines where both the number of nondeterministic steps and also the number of oracle queries allowed in computations are bounded by functions of the length of the input. (See [4,6,7,18] for other results about classes specified by machines with these parameters bounded. ) In the present paper we consider "strong separation" theorems: to witness E1 ~ ~2' exhibit L @ ~2 such that L is infinite but no infinite subset of L is in ~i" The first context in which this is done is that of polynomial time-bounded oracle machines: for any A, a set L is P(A)-immune if L is infinite but has no infinite subset in P(A). Our first result shows the existence of a recursive set A such that NP(A) has a P(A)-immune set. (This was first established by Homer and Maass [9] .) Our proof does not depend on polynomial timebounds as such, but rather on polynomial bounds on the number of nondeterministic steps and the number of oracle queries in computations.