Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds

  • Harry Buhrman
  • Leen Torenvliet
  • Falk Unger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3884)


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 EXPNP, or even in 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 Toran [10] that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also show that this result is optimal with respect to relativizing proofs, by exhibiting an oracle relative to which all of 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 NEXP.


Computational Complexity Sparseness Selfreducibility 


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

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Harry Buhrman
    • 1
    • 2
  • Leen Torenvliet
    • 2
  • Falk Unger
    • 1
  1. 1.CWI Amsterdam 
  2. 2.Universiteit van Amsterdam 

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