Nondeterministic bounded query reducibilities

Abstract

A query-bounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic query bounded Turing machines. In particular we study how easily such machines can compute the function F A n ( x 1,..., x n ) from A, where A ⊆ ▪ and F A n ( x 1,..., x n )=< X A ( x 1),..., X A 〉. We show that each truth-table degree contain s a set A such that F A n can be nondeterministically computed from A by asking at most one question per nondeterministic branch; and that every set of the form Á also has this property. On the other hand, we show that if A is a 1-generic set, then F A n cannot be nondeterministically computed from A in less that n queries to A; and that each non-zero r.e. Turing degree contains an r.e. set A with the same property. If the machines involved can only make queries that are part of their input, then all sets such that F A n can be computed with one query per branch are weakly r.e.