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Self-reducible sets of small density

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Abstract

We study the complexity of sets that are at the same time self-reducible and sparse orm-reducible to sparse sets. We show that sets of this kind are low for the complexity classes Δ p2 , Θ p2 , NP, or P, depending on the type of self-reducibility used and on certain restrictions imposed on the query mechanism of the self-reducibility machines. The proof of some of these results is based on graph-theoretic properties that hold for the graphs induced by the self-reducibility structures.

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References

  1. E. Allender and L. Hemachandra: Lower bounds for the low hierarchy.Proc. 16th ICALP (1989), pp. 31–46.

  2. T. P. Baker, J. Gill, and R. M. Solovay: Relativizations of the P = ?NP question.SIAM J. Comput. 4 (1975), 431–442.

    Google Scholar 

  3. J. L. Balcázar: Self-reducibility.J. Comput. System Sci., to appear.

  4. J. L. Balcázar, R. V. Book, and U. Schöning: The polynomial-time hierarchy and sparse oracles.J. Assoc. Comput. Mach. 33 (1986), 603–617.

    Google Scholar 

  5. J. L. Balcázar, J. Díaz, and J. Gabarró:Structural Complexity, Vol. I. Springer-Verlag, Berlin (1988).

    Google Scholar 

  6. S. Buss and L. Hay: On truth-table reducibility to SAT and the difference hierarchy over NP.Proc. 3rd Structure in Complexity Conference (1988), pp. 224–233.

  7. S. Fortune: A note on sparse complete sets.SIAM J. Comput. 8 (1979), 431–433.

    Google Scholar 

  8. J. Goldsmith, D. Joseph, and P. Young: Self-reducible,p-selective, near-testable andp-cheatable sets: the effect of internal structure on the complexity of a set.Proc. 2nd Structure in Complexity Conference (1987), pp. 50–59.

  9. J. Kadin: PNP[logn] and sparse Turing-complete sets for NP.Proc. 2nd Structure in Complexity Conference (1987), pp. 33–41.

  10. K. Ko: On self-reducibility and weakp-selectivity.J. Comput. System Sci. 26 (1983), 209–221.

    Google Scholar 

  11. K. Ko and U. Schöning: On circuit-size complexity and the low hierarchy in NP.SIAM J. Comput. 14 (1985), 41–51.

    Google Scholar 

  12. L. Longpré and A. Selman: Hard promise problems.Proc. 7th ST ACS (1990), pp. 216–227.

  13. A. Lozano and J. Torán: Relativized and positive separations of Δ p2 and Θ p2 . Tech. Report L.S.I.-89-30, U. Politecnica de Catalunya (1989).

  14. S. Mahaney: Sparse complete sets for NP: solution of a conjecture by Berman and Hartmanis.J. Comput. System Sci. 25 (1982), 130–142.

    Google Scholar 

  15. A. Meyer and M. Paterson: With what frequency are apparently intractable problems difficult? Tech. Report TM-126, M.I.T. (1989).

  16. M. Ogiwara and O. Watanabe: On polynomial bounded truth-table reducibility of NP sets to sparse sets.Proc. 22nd STOC (1990), pp. 457–468.

  17. U. Schöning: A low and a high hierarchy within NP.J. Comput. System Sci. 27 (1983), 14–28.

    Google Scholar 

  18. U. Schöning:Complexity and Structure. Lecture Notes in Computer Science, Vol. 211, Springer-Verlag, Berlin (1985).

    Google Scholar 

  19. K. Wagner: Bounded query computations.Proc. 3rd Structure in Complexity Conference (1988), pp. 260–278.

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Authors and Affiliations

  1. Department L.S.I., U. Politècnica de Catalunya, Pau Gargallo 5, 08028, Barcelona, Spain

    Antoni Lozano & Jacobo Torán

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  1. Antoni Lozano
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  2. Jacobo Torán
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Additional information

This research was partially supported by ESPRIT-II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).

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Lozano, A., Torán, J. Self-reducible sets of small density. Math. Systems Theory 24, 83–100 (1991). https://doi.org/10.1007/BF02090392

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  • DOI: https://doi.org/10.1007/BF02090392

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