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Algorithmica

, Volume 81, Issue 1, pp 179–200 | Cite as

Sparse Selfreducible Sets and Nonuniform Lower Bounds

  • Harry Buhrman
  • Leen Torenvliet
  • Falk Unger
  • Nikolay VereshchaginEmail author
Article
  • 27 Downloads

Abstract

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in \({\mathrm{EXP^{NP}}}\), or even in \({\mathrm{EXP}}\) that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that \({\mathrm{EXP^{NP}}}\) does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that \({\mathrm{NEXP}}\) does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of \({\mathrm{EXP}}\) is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for \({\mathrm{NEXP}}\).

Keywords

Computational complexity Sparseness Selfreducibility 

Notes

Acknowledgements

We thank the anonymous referee for helpful suggestions. Funding was provided by Russian Foundation for Basic Research (Grant No. 16-01-00362), Russian Academic Excellence Project (Grant No. 5-100).

References

  1. 1.
    Agrawal, M., Arvind, V.: Quasi-linear truth-table reductions to p-selective sets. Theor. Comput. Sci. 158, 361–370 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balcázar, J., Díaz, J., Gabarró, J.: Structural Complexity I. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  3. 3.
    Buhrman, H., Fortnow, L., Thierauf, T.: Nonrelativizing separations. In: IEEE Conference on Computational Complexity. IEEE Computer Society Press, pp. 8–12 (1998)Google Scholar
  4. 4.
    Berman, L., Hartmanis, H.: On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6, 305–322 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beigel, R., Kummer, M., Stephan, F.: Approximable sets. Inf. Comput. 120(2), 304–314 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buhrman, H., Torenvliet, L.: P-selective self-reducible sets: a new characterization of P. J. Comput. Syst. Sci. 53(2), 210–217 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fortnow, L., Klivans, A.: NP with small advice. In: Proceedings of the 20th IEEE Conference on Computational Complexity. IEEE Computer Society Press, pp. 228–234 (2005)Google Scholar
  8. 8.
    Faliszewski, P., Ogihara, M.: Separating the notions of self- and autoreducibility. In: MFCS, pp. 308–315 (2005)Google Scholar
  9. 9.
    Hemaspaandra, L.A., Torenvliet, L.: Theory of Semi-Feasible Algorithms. Monographs in Theoretical Computer Science. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Ko, K.-I.: On self-reducibility and weak P-selectivity. J. Comput. Syst. Sci. 26, 209–211 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ko, K., Schöning, U.: On circuit-size and the low hierarchy in NP. SIAM J. Comput. 14(1), 41–51 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lozano, A., Torán, J.: Self-reducible sets of small density. J. Math. Systems Theory 24(1), 83–100 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Meyer, A.: Oral communication, cited in [4] (1977)Google Scholar
  14. 14.
    Mocas, S.: Separating exponential time classes from polynomial time classes. PhD thesis, Northeastern University (1993)Google Scholar
  15. 15.
    Ogihara, M.: Polynomial-time membership comparable sets. SIAM J. Comput. 24(5), 1168–1181 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ogiwara, M., Watanabe, O.: On polynomial time bounded truth-table reducibility of NP sets to sparse sets. SIAM J. Comput. 20, 471–483 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Papadimitriou, C.H.: Computational Complexity. Addison Wesley, Boston (1994)zbMATHGoogle Scholar
  18. 18.
    Selman, A.: P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Syst. Theory 13, 55–65 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Selman, A.: Analogues of semicursive sets and effective reducibilities to the study of NP complexity. Inf. Control 52(1), 36–51 (1982)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shaltiel, R., Umans, C.: Pseudorandomness for approximate counting and sampling. Technical report TR04-086, ECCC (2004)Google Scholar
  21. 21.
    Wagner, K.: Bounded query computations. In: Proceedings of 3rd Structure in Complexity in Conference. IEEE Computer Society Press, pp. 260–278 (1988)Google Scholar
  22. 22.
    Wilson, C.B.: Relativized circuit complexity. J. Comput. Syst. Sci. 31, 169–181 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CWI AmsterdamAmsterdamThe Netherlands
  2. 2.ILLCUniversiteit van AmsterdamAmsterdamThe Netherlands
  3. 3.Moscow State University, National Research University Higher School of EconomicsMoscowRussia

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