### 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 kth 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}^{NPC} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 293-310 |

Number of pages | 18 |

Journal | Mathematical Systems Theory |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics(all)
- Computational Theory and Mathematics

### Cite this

*Mathematical Systems Theory*,

*26*(3), 293-310. https://doi.org/10.1007/BF01371729

**A relationship between difference hierarchies and relativized polynomial hierarchies.** / Beigel, Richard; Chang, Richard; Ogihara, Mitsunori.

Research output: Contribution to journal › Article

*Mathematical Systems Theory*, vol. 26, no. 3, pp. 293-310. https://doi.org/10.1007/BF01371729

}

TY - JOUR

T1 - A relationship between difference hierarchies and relativized polynomial hierarchies

AU - Beigel, Richard

AU - Chang, Richard

AU - Ogihara, Mitsunori

PY - 1993/7

Y1 - 1993/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001656946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001656946&partnerID=8YFLogxK

U2 - 10.1007/BF01371729

DO - 10.1007/BF01371729

M3 - Article

AN - SCOPUS:0001656946

VL - 26

SP - 293

EP - 310

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 3

ER -