### 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) |
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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 |

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### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Information Systems
- Computer Science Applications
- Computational Theory and Mathematics

### Cite this

*Information and Computation*,

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