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# A relationship between difference hierarchies and relativized polynomial hierarchies

Author: Richard Beigel; Richard Chang; Mitsunori Ogiwara Ithaca, N.Y. : Cornell University, Dept. of Computer Science, [1991] Cornell University.; Department of Computer Science.; Technical report Print book : EnglishView all editions and formats Chang and Kadin have shown that if the difference hierarchy over NP collapses to level $k$, then the polynomial hierarchy (PH) is equal to the $k$th level of the difference hierarchy over $\Sigma_{2}[superscript]{p}$. We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level $k$, then PH = $\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{NP}$. We also extend the result to classes other than NP: For any class $C$ that has $\leq_{m}[superscript]{p}$-complete sets and is closed under $\leq_{conj}[superscript]{p}$- and $\leq_{m}[superscript]{NP}$-reductions, if the difference hierarchy over $C$ collapses to level $k$, then $PH[superscript]{C} =$\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{C}$. Then we show that the exact counting class$C_{=}P$is closed under$\leq_{disj}[superscript]{p}$- and$\leq_{m}[superscript]{co-NP}$-reductions. Consequently, if the difference hierarchy over$C_{=}P$collapses to level$k$then$PH[superscript]{PP}$is equal to$\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{PP}$. In contrast, the difference hierarchy over the closely related class PP is known to collapse.Finally, we consider two ways of relativizing the bounded query class$P_{k-tt}[superscript]{NP}$: the restricted relativization$P_{k-tt}[superscript]{NP[superscript]{C}}$, and the full relativization$\left(P_{k-tt}[superscript]{NP}\right)[superscrip t]{C}$. If$C$is NP-hard, then we show that the two relativizations are different unless$PH[superscript]{C}$collapses. Read more... (not yet rated) 0 with reviews - Be the first. ## Find a copy in the library Finding libraries that hold this item... ## Details Document Type: Book Richard Beigel; Richard Chang; Mitsunori Ogiwara Find more information about: Richard Beigel Richard Chang Mitsunori Ogiwara 25776302 "January 1991." 17 pages ; 28 cm. Cornell University.; Department of Computer Science.; Technical report Richard Beigel, Richard Chang, Mitsunori Ogiwara. ### Abstract: Chang and Kadin have shown that if the difference hierarchy over NP collapses to level$k$, then the polynomial hierarchy (PH) is equal to the$k$th level of the difference hierarchy over$\Sigma_{2}[superscript]{p}$. We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level$k$, then PH =$\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{NP}$. We also extend the result to classes other than NP: For any class$C$that has$\leq_{m}[superscript]{p}$-complete sets and is closed under$\leq_{conj}[superscript]{p}$- and$\leq_{m}[superscript]{NP}$-reductions, if the difference hierarchy over$C$collapses to level$k$, then$PH[superscript]{C} = $\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{C}$. Then we show that the exact counting class $C_{=}P$ is closed under $\leq_{disj}[superscript]{p}$- and $\leq_{m}[superscript]{co-NP}$-reductions. Consequently, if the difference hierarchy over $C_{=}P$ collapses to level $k$ then $PH[superscript]{PP}$ is equal to $\left(P_{(k-1)-tt}[superscript]{NP}\right)[supers cript]{PP}$. In contrast, the difference hierarchy over the closely related class PP is known to collapse.

Finally, we consider two ways of relativizing the bounded query class $P_{k-tt}[superscript]{NP}$: the restricted relativization $P_{k-tt}[superscript]{NP[superscript]{C}}$, and the full relativization $\left(P_{k-tt}[superscript]{NP}\right)[superscrip t]{C}$. If $C$ is NP-hard, then we show that the two relativizations are different unless $PH[superscript]{C}$ collapses.

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