TY - JOUR

T1 - On the probabilistic closure of the loose unambiguous hierarchy

AU - Hirsch, Edward A.

AU - Sokolov, Dmitry

N1 - Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - 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.

AB - 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.

KW - Computational complexity

KW - Randomized algorithms

KW - Toda's theorem

KW - Unambiguous computations

UR - http://www.scopus.com/inward/record.url?scp=84929956067&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2015.04.010

DO - 10.1016/j.ipl.2015.04.010

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AN - SCOPUS:84929956067

SN - 0020-0190

VL - 115

SP - 725

EP - 730

JO - Information Processing Letters

JF - Information Processing Letters

IS - 9

ER -