# A relationship between difference hierarchies and relativized polynomial hierarchies

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## 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 Σ _{2} ^{ p } . We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to level*k*, then PH collapses to (P _{(k−1)} ^{NP} )^{NP}, the class of sets recognized in polynomial time with*k* − 1 nonadaptive queries to a set in NP^{NP} and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class*C* that has ≤ _{ m } ^{p} -complete sets and is closed under ≤ _{conj} ^{ p } -and ≤ _{ m } ^{NP} -reductions (alternatively, closed under ≤ _{disj} ^{ p } -and ≤ _{ m } ^{co-NP} -reductions), if the difference hierarchy over*C* collapses to level*k*, then PH^{ C } = (P _{(k−1)−tt} ^{NP} )^{ C }. Then we show that the exact counting class C_P is closed under ≤ _{disj} ^{ p } - and ≤ _{ m } ^{co-NP} -reductions. Consequently, if the difference hierarchy over C_P collapses to level*k*, then PH^{PP}(= PH^{C_P}) is equal to (P _{(k−1)−tt} ^{NP} )^{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} ^{NP} : the restricted relativization P _{ k−tt} ^{NP} ^{ C } and the full relativization (P _{ k−tt} ^{NP} )^{ C }. If*C* is NP-hard, then we show that the two relativizations are different unless PH^{ C } collapses.

## Keywords

Maximal Sequence Polynomial Hierarchy Full Relativization Complete Language Boolean Hierarchy## Preview

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