On symmetric differences of NP-hard sets with weakly-P-selective sets
The symmetric differences of N P—hard sets with weakly — P-selective sets are investigated in this paper. We show that if there exist a weakly-P-selective set A and a NP-≤ m p -hard set H such that H-A ∃ P btt (Sparse) and A-H ∃ Pm (Sparse) then P=N P So no NP-≤ m p -hard set has sparse symmetric difference with any weakly-P-selective set unless P=N P. In addition we show there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(Sparse).
KeywordsTuring Machine Complexity Class Symmetric Difference Selector Function Polynomial Time Hierarchy
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- [BGS88]J.Balcarzar, J.Diaz, and J.Gabarro: Structural Complexity I,II. Springer-Verlag, 1988 and 1990.Google Scholar
- [F91]B.Fu: On Lower Bounds of the Closeness between Complexity Classes. To appear in Math. Sys.Theory.Google Scholar
- [FL92]B.Fu and H.Li: On Closeness of NP-Hard sets to other Complexity Classes. Proc.7th Annual Conference in structural complexity theory, IEEE, 1992, 243–248.Google Scholar
- [K88]K.Ko: Distinguishing Bounded Reducibility by Sparse Sets. Proc.3th Conference on Structural Complexity Theory, IEEE, 1988, 181–191.Google Scholar
- [OW90]M.Ogiwara and O.Watanabe: On Polynomial-Time Bounded Truth-table Reducibility of NP sets to Sparse Sets. Proc. of the 22th ACM Symposium on Theory of Computing, 1990, 457–467.Google Scholar
- [Sc86]U.Schöning: Complete Sets and Closeness to Complexity Classes. Math.Sys. Theory, 13(1986), 55–65.Google Scholar
- [Se79]A.L.Selman: P-selective Sets, Tally Languages and the Behavior of Polynomial-Time Reducibility on NP. Math.Sys.Theory 13(1981), 326–332.Google Scholar
- [TFL91]S.Tang, B.Fu and T.Liu: Exponential Time and Subexponential Time Sets. Proc.6th Annual Conference in structural complexity theory, IEEE, 1991, 230–237.Google Scholar
- [T91]S.Toda: On Polynomial-Time Truth Table Reducibility of Intractable Sets to P-Selective Sets. Math.Sys.Theory 24(1991), 69–82.Google Scholar