### Abstract

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S_{2}^{A} = S_{2}. Building on this, we strengthen the Kämper-AFK Theorem, namely, we prove that if NP ⊆ (NP ∩ coNP)/poly then the polynomial hierarchy collapses to S_{2}NP∩coNP. We also strengthen Yap's Theorem, namely, we prove that if NP ⊆ coNP/poly then the polynomial hierarchy collapses to S_{2}^{NP}. Under the same assumptions, the best previously known collapses were to ZPP^{NP} and ZPP^{NPNP} respectively ([20,6], building on [18,1,17,30]). It is known that S_{2} ⊆ ZPP^{NP} [8]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kämper-AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results - ranging from the study of unique solutions to issues of approximation - our results implicitly strengthen all those results.

Original language | English (US) |
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Pages (from-to) | 535-546 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2607 |

State | Published - Dec 1 2003 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*2607*, 535-546.