On Restricting the Size of Oracles Compared with Restricting Access to Oracles

Abstract

A restricted relativization of NP, denoted ${\text{NP}}_B ( \cdot )$,was introduced in [SIAM J. Comput., 13 (1984), pp. 461– 487] in the study of positive relativizations on the ${\text{P}} = ?{\text{ NP}}$ question. The ${\text{NP}}_B ( \cdot )$ restriction allows nondeterministic polynomial-time oracle machines to query only polynomially many strings in its computation trees. In this paper we compare the language classes ${\text{NP}}_B (A)$, relative to arbitrary sets A, with the language classes $\text{NP}(S)$, relative to sparse sets S, showing that it is not always possible to obtain a class specified by ${\text{NP}}_B (A)$ as an $\text{NP}(S)$ class and vice versa. As a corollary to these results, we prove that there is a sparse set S such that for all tally sets T, $S \not\equiv_T^{\text{SN}} T$. This implies that the relationship established in [5] that for every sparse set S there is a tally set T such that $S \equiv _T^{\text{NP}} T$ cannot be improved to any of the strong nondeterministic polynomial-time degrees. Finally, we strengthen a result appearing in [4] by showing that nondeterministic oracle programs for sets $B \in \Sigma _k^{\text{P}} - \Delta _k^{\text{P}} $, for $k \geqq 2$, must search through exponentially many strings (infinitely often) that are actually in the oracle set when using oracle sets from $\Sigma _{k - 1}^{\text{P}} $. As a consequence, sparse sets at some $\Sigma^{\text{P}} $ level of the polynomial-time hierarchy cannot be used as orcales for sets properly at the next $\Sigma^{\text{P}} $ level.