Abstract
The existence of setsnot being ≤ P tt -reducible to low sets is investigated for several complexity classes such as UP, NP, the polynomial-time hierarchy, PSPACE, and EXPTIME. The p-selective sets are mainly considered as a class of low sets. Such investigations were done in many earlier works, but almost all of these have dealt withpositive reductions in order to imply the strongest consequence such as P=NP under the assumption that all sets in NP are polynomial-time reducible to low sets. Currently, there seems to be some difficulty in obtaining the same strong results undernonpositive reducibilities. The purpose of this paper is to develop a useful technique to show for many complexity classes that if each set in the class is polynomial-time reducible to a p-selective set via anonpositive reduction, then the class is already contained in P. The following results are shown in this paper.
(1) If each set in UP is ≤ P tt -reducible to a p-selective set, then P=UP.
(2) If each set in NP is ≤ P tt -reducible to a p-selective set, then P=FewP and R=NP.
(3) If each set in Δ P2 is ≤ P tt -reducible to a p-selective set, then P=NP.
(4) If each set in PSPACE is ≤ P tt -reducible to a p-selective set, then P=PSPACE.
(5) There is a set in EXPTIME that is not ≤ P tt -reducible to any p-selective set.
It remains open whether P=NP follows from a weaker assumption that each set in NP is ≤ P tt -reducible to a p-selective set.
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Toda, S. On polynomial-time truth-table reducibility of intractable sets to P-selective sets. Math. Systems Theory 24, 69–82 (1991). https://doi.org/10.1007/BF02090391
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DOI: https://doi.org/10.1007/BF02090391