### Abstract

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2A=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 S2NP. Under the same assumptions, the best previously known collapses were to ZPP^{NP} and ZPPNPNP, respectively ([SIAM Journal on Computing 28 (1) (1998) 311; Journal of Computer and System Sciences 52 (3) (1996) 421], building on [Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302-309; Journal of Computer and System Sciences 39 (1989) 21; Theoretical Computer Science 85 (2) (1991) 305; Theoretical Computer Science 26 (3) (1983) 287]). It is known that S_{2} ⊆ ZPP^{NP} [Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Silver Spring, MD, 2001, pp. 620-629]. That result and its relativized version show that our new collapses indeed improve the previously known results. 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-and so our results implicitly strengthen those results.

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

Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Information and Computation |

Volume | 198 |

Issue number | 1 |

DOIs | |

State | Published - Apr 10 2005 |

Externally published | Yes |

### Fingerprint

### Keywords

- Competing provers
- Kämper-AFK theorem
- Karp-Lipton theorem
- Lowness
- Nonuniform complexity
- Structural complexity
- Symmetric alternation
- Yap's theorem

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Information and Computation*,

*198*(1), 1-23. https://doi.org/10.1016/j.ic.2005.01.002

**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

*Information and Computation*, vol. 198, no. 1, pp. 1-23. https://doi.org/10.1016/j.ic.2005.01.002

}

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 - 2005/4/10

Y1 - 2005/4/10

N2 - Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2A=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 S2NP. Under the same assumptions, the best previously known collapses were to ZPPNP and ZPPNPNP, respectively ([SIAM Journal on Computing 28 (1) (1998) 311; Journal of Computer and System Sciences 52 (3) (1996) 421], building on [Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302-309; Journal of Computer and System Sciences 39 (1989) 21; Theoretical Computer Science 85 (2) (1991) 305; Theoretical Computer Science 26 (3) (1983) 287]). It is known that S2 ⊆ ZPPNP [Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Silver Spring, MD, 2001, pp. 620-629]. That result and its relativized version show that our new collapses indeed improve the previously known results. 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-and so our results implicitly strengthen those results.

AB - Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2A=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 S2NP. Under the same assumptions, the best previously known collapses were to ZPPNP and ZPPNPNP, respectively ([SIAM Journal on Computing 28 (1) (1998) 311; Journal of Computer and System Sciences 52 (3) (1996) 421], building on [Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302-309; Journal of Computer and System Sciences 39 (1989) 21; Theoretical Computer Science 85 (2) (1991) 305; Theoretical Computer Science 26 (3) (1983) 287]). It is known that S2 ⊆ ZPPNP [Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Silver Spring, MD, 2001, pp. 620-629]. That result and its relativized version show that our new collapses indeed improve the previously known results. 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-and so our results implicitly strengthen those results.

KW - Competing provers

KW - Kämper-AFK theorem

KW - Karp-Lipton theorem

KW - Lowness

KW - Nonuniform complexity

KW - Structural complexity

KW - Symmetric alternation

KW - Yap's theorem

UR - http://www.scopus.com/inward/record.url?scp=15844376276&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15844376276&partnerID=8YFLogxK

U2 - 10.1016/j.ic.2005.01.002

DO - 10.1016/j.ic.2005.01.002

M3 - Article

AN - SCOPUS:15844376276

VL - 198

SP - 1

EP - 23

JO - Information and Computation

JF - Information and Computation

SN - 0890-5401

IS - 1

ER -