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 kth level of the difference hierarchy over Σ2p. 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 NPNP 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 ≤mp-complete sets and is closed under ≤conjp-and ≤mNP-reductions (alternatively, closed under ≤disjp-and ≤mco-NP-reductions), if the difference hierarchy over C collapses to level k, then PHC = (P(k-1)-ttNP)C. Then we show that the exact counting class C_P is closed under ≤disjp- and ≤mco-NP-reductions. Consequently, if the difference hierarchy over C_P collapses to level k, then PHPP(= PHC_P) is equal to (P(k-1)-ttNP)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 Pk-ttNP: the restricted relativization Pk-ttNPC and the full relativization (Pk-ttNP)C. If C is NP-hard, then we show that the two relativizations are different unless PHC collapses.
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics