Title On Using Oracles That Compute Values

This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are coinputable nondeterministically in $1$) $olynomial$ time. $(_{-}-$ oncerning deterniinistic polynomial-time reducibilities, it is shown that 1. A multivalued $pa\iota\cdot tial$ function is polynomial-time computable with $k$ adaptive quet ies to NPMV if $\omega\iota do11$ ]} if it $i_{\mathfrak{d}}$ poly nomial-tinre $Com_{1)utable}$ via $2^{k}-1$ nonadaptive quet ies to NPMV. 2. A characteristic function is polynomial-time computable with $k$ adaptive queries to NPMV if and only if it is polynomial-time computable with $k$ adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, $k$ adaptive (nonadaptive) queries to NPMV is different than $k+1$ adaptive (nonadaptive) queries to NPMV for every $k$ . Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for ] functions does not $collapt,\sim\neg e$ unless the Boolean hierarchy collapses, but, $su1^{\cdot}p_{1}\cdot i_{S1I1}gly$ , the levels of the difference and bounded query hierarchies do not interleave (as is th $e$ case $fo1^{\cdot}$ sets) unless the polynomial hierarchy collapses. ’Dept. of Computer Science, Portland, ME 04103. Research $partiaJl\backslash ’\backslash ^{-}npported$ by the National Science Foundation under grant no. CCR-9209833. $\uparrow$ Dept. of Computer Science, Boston, MA 01003. Research $1$) $al\cdot(iaJl\backslash \backslash ^{\backslash }tlppo1^{\cdot}ted$ by the National Science Foundation under grant no. CCR-9103055 $t_{Dept}$ . of $C_{oI11}p$ ul.er Science, $Chofu-arrow hi$ , Tokyo 18-, Japan. $Researc\cdot 1\downarrow pal(ia1]_{)^{r}}\llcorner\backslash ^{\backslash }\backslash \iota pl)orted$ bv the National Science Foundation under grant no. CCR-9002292. $\S_{Dept}$ . of Computer Science, Buffalo, NY 14260. $Re\llcorner\backslash eal\cdot ch$ ]) $\partial 1^{\cdot}ti_{d}11\backslash \cdot\backslash ^{-}n$ pported bv the National Science $F^{t}oundation$ under grant no. $CC^{t}li9t$ ) $0_{-}^{\underline{\prime}\rangle}9_{-}$ ). 数理解析研究所講究録 第 833巻 1993年 31-41