On the probabilistic closure of the loose unambiguous hierarchy

Edward A. Hirsch, Dmitry Sokolov

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)725-730
Number of pages6
JournalInformation Processing Letters
Issue number9
StatePublished - 1 Sep 2015
Externally publishedYes

Scopus subject areas

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


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


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