### 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) |
---|---|

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 - 2003 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

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

*2607*, 535-546.

**Competing provers yield improved Karp-Lipton collapse results.** / Cai, Jin Yi; Chakaravarthy, Venkatesan T.; Hemaspaandra, Lane A.; Ogihara, Mitsunori.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 2607, pp. 535-546.

}

TY - JOUR

T1 - Competing provers yield improved Karp-Lipton collapse results

AU - Cai, Jin Yi

AU - Chakaravarthy, Venkatesan T.

AU - Hemaspaandra, Lane A.

AU - Ogihara, Mitsunori

PY - 2003

Y1 - 2003

N2 - Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2 A = S2. Building on this, we strengthen the Kämper-AFK Theorem, namely, we prove that if NP ⊆ (NP ∩ coNP)/poly then the polynomial hierarchy collapses to S2NP∩coNP. We also strengthen Yap's Theorem, namely, we prove that if NP ⊆ coNP/poly then the polynomial hierarchy collapses to S2 NP. Under the same assumptions, the best previously known collapses were to ZPPNP and ZPPNPNP respectively ([20,6], building on [18,1,17,30]). It is known that S2 ⊆ ZPPNP [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.

AB - Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2 A = S2. Building on this, we strengthen the Kämper-AFK Theorem, namely, we prove that if NP ⊆ (NP ∩ coNP)/poly then the polynomial hierarchy collapses to S2NP∩coNP. We also strengthen Yap's Theorem, namely, we prove that if NP ⊆ coNP/poly then the polynomial hierarchy collapses to S2 NP. Under the same assumptions, the best previously known collapses were to ZPPNP and ZPPNPNP respectively ([20,6], building on [18,1,17,30]). It is known that S2 ⊆ ZPPNP [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.

UR - http://www.scopus.com/inward/record.url?scp=35248819735&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248819735&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:35248819735

VL - 2607

SP - 535

EP - 546

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -