### Abstract

It is shown that many natural counting classes are at least as computationally hard as PH (the polynomial-time hierarchy) in the following sense: for each K of the counting classes, every set in K(PH) is polynomial-time randomized many-one reducible to a set in K with two-sided exponentially small error probability. As a consequence, these counting classes are computationally harder than PH unless PH collapses to a finite level. Some other consequences are also shown.

Original language | English (US) |
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Title of host publication | Proc 6 Annu Struct Complexity Theor |

Publisher | Publ by IEEE |

Pages | 2-12 |

Number of pages | 11 |

ISBN (Print) | 0818622555 |

State | Published - Dec 1 1991 |

Externally published | Yes |

Event | Proceedings of the 6th Annual Structure in Complexity Theory Conference - Chicago, IL, USA Duration: Jun 30 1991 → Jul 3 1991 |

### Publication series

Name | Proc 6 Annu Struct Complexity Theor |
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### Other

Other | Proceedings of the 6th Annual Structure in Complexity Theory Conference |
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City | Chicago, IL, USA |

Period | 6/30/91 → 7/3/91 |

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

Toda, S., & Ogiwara, M. (1991). Counting classes are at least as hard as the polynomial-time hierarchy. In

*Proc 6 Annu Struct Complexity Theor*(pp. 2-12). (Proc 6 Annu Struct Complexity Theor). Publ by IEEE.