## Abstract

Unambiguous hierarchies [1-3] are defined similarly to the polynomial hierarchy; however, all witnesses must be unique. These hierarchies have subtle differences in the mode of using oracles. We consider a "loose" unambiguous hierarchy prUH• with relaxed definition of oracle access to promise problems. Namely, we allow to make queries that miss the promise set; however, the oracle answer in this case can be arbitrary (a similar definition of oracle access has been used in [4]). In this short note we prove that the first part of Toda's theorem PH⊂BP.⊕P⊂^{PPP} can be strengthened to PH=BP.prUH•, that is, the closure of our hierarchy under Schöning's BP operator equals the polynomial hierarchy. It is easily seen that BP.prUH•⊂BP.⊕P. The proof follows the same lines as Toda's proof, so the main contribution of the present note is a new definition that allows to characterize PH as a probabilistic closure of unambiguous computations.

Original language | English |
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Pages (from-to) | 725-730 |

Number of pages | 6 |

Journal | Information Processing Letters |

Volume | 115 |

Issue number | 9 |

DOIs | |

State | Published - 1 Sep 2015 |

Externally published | Yes |

## Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications

## Keywords

- Computational complexity
- Randomized algorithms
- Toda's theorem
- Unambiguous computations