Very Sparse Leaf Languages

  • Lance Fortnow
  • Mitsunori Ogihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


Unger studied the balanced leaf languages defined via poly-logarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ\(^{p}_{\rm 2}\) and that Σ\(^{p}_{\rm 2}\)-complete sets are not polynomial-time bounded-truth-table reducible (respectively), polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to Δ\(^{p}_{\rm 2}\) (respectively, Σ\(^{p}_{\rm 4}\)).

This paper studies the complexity of the class of such balanced leaf languages, which will be denoted by VSLL. In particular, the following tight upper and lower bounds of VSLL are shown:

1. coNP ⊆ VSLL ⊆ coNP/poly (the former inclusion is already shown by Unger).

2. coNP/1 \(\not\subseteq\) VSLL unless PH = Θ\(^{p}_{\rm 2}\).

3. For all constant c>0, VSLL \(\not\subseteq\) coNP/n c .

4. P/(loglog(n) + O(1)) ⊆ VSLL.

5. For all h(n) = loglog(n) + ω(1), P\(/h \not\subseteq\) VSLL.


Turing Machine Complexity Class Computation Tree Computation Path Kolmogorov Complexity 
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  1. 1.
    Barrington, D.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences 38, 150–164 (1989)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bovet, D., Crescenzi, P., Silvestri, R.: A uniform approach to define complexity classes. Theoretical Computer Science 104(2), 263–283 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, J.: S2 p ⊆ ZPPNP. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 620–629. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  4. 4.
    Cai, J., Chakaravarthy, V., Hemaspaandra, L., Ogihara, M.: Competing provers yield improved Karp–Lipton collapse results. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 535–546. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Cai, J., Furst, M.: PSPACE survives constant-width bottlenecks. International Journal on Foundations of Computer Science 2(1), 67–76 (1991)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Canetti, R.: More on BPP and the polynomial-time hierarchy. Information Processing Letters 57(5), 237–241 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fortnow, L., Ogihara, M.: Very sparse leaf languages. Technical Report 899. Department of Computer Science, University of Rochester, Rochester, NY (June 2006)Google Scholar
  8. 8.
    Goldschlager, L., Parberry, I.: On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science 43, 43–58 (1986)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM 38(3), 690–728 (1991)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kadin, J.: PNP[logn] and sparse Turing-complete sets for NP. SIAM Journal on Computing 17(6), 1263–1282 (1988); Erratum 20(2), 404 (1991)Google Scholar
  11. 11.
    Karp, R., Lipton, R.: Some connections between nonuniform and uniform complexity classes. In: Proceedings of the 12th Symposium on Theory of Computing, pp. 302–309. ACM Press, New York (1980)Google Scholar
  12. 12.
    Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its application. Springer, New York (1993)Google Scholar
  13. 13.
    Ogiwara, M., Watanabe, O.: On polynomial time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal of Computing 20(3), 471–483 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Russell, A., Sundaram, R.: Symmetric alternation captures BPP. Computational Complexity 7(2), 152–162 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Unger, F.: On small hard leaf languages. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 781–792. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Yap, C.-K.: Consequences of non-uniform conditions on uniform classes. Theoretical Computer Science 26, 287–300 (1983)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Lance Fortnow
    • 1
  • Mitsunori Ogihara
    • 2
  1. 1.University of Chicago 
  2. 2.University of Rochester 

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