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- W4238001574 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, $S notequiv_T^{text{SN}} T$. 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." @default.
- W4238001574 created "2022-05-12" @default.
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- W4238001574 date "1985-08-01" @default.
- W4238001574 modified "2023-10-05" @default.
- W4238001574 title "On Restricting the Size of Oracles Compared with Restricting Access to Oracles" @default.
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- W4238001574 doi "https://doi.org/10.1137/0214043" @default.
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