Abstract
This paper deals with balanced leaf language complexity classes, introduced independently in [1] and [14]. We propose the seed concept for leaf languages, which allows us to give “short” representations for leaf words. We then use seeds to show that leaf languages A with NP ⊆ BLeaf P(A) cannot be polylog-sparse (i.e. census A ∈ O(logO(1))), unless PH collapses.
We also generalize balanced ≤\(^{P,{bit}}_{m}\)-reductions, which were introduced in [6], to other bit-reductions, for example (balanced) truth-table- and Turing-bit-reductions. Then, similarly to above, we prove that NP and Σ\(^{P}_{\rm 2}\) cannot have polylog-sparse hard sets under those balanced truth-table- and Turing-bit-reductions, if the polynomial-time hierarchy is infinite.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bovet, D.P., Crescenzi, P., Silvestri, R.: A Uniform Approach to Define Complexity Classes. Theoretical Computer Science 104, 263–283 (1992)
Cai, J.-Y.: \(S^P_2 \subseteq ZPP^{NP}\). In: FOCS 2001, pp. 620–629 (2001)
Canetti, R.: More on BPP and the polynomial-time hierarchy. Information Processing Letters 57, 237–241 (1996)
Cai, J.-Y., Ogihara, M.: Sparse Sets versus Complexity Classes. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp. 53–80 (1997)
Homer, S., Longpré, L.: On reductions of NP sets to sparse sets. Journal of Computer and System Sciences 48(2), 324–336 (1994)
Hertrampf, U., Lautemann, C., Schwentick, T., Vollmer, H., Wagner, K.: On the Power of Polynomial Time Bit-Reductions. In: Proc. 8th Structure in Complexity Theory Conference, pp. 200–207 (1993)
Hertrampf, U., Vollmer, H., Wagner, K.: On Balanced vs. Unbalanced Computation Trees. Mathematical Systems Theory 29, 411–421 (1996)
Jenner, B., McKenzie, P., Therien, D.: Logspace and Logtime Leaf Languages. Information and Computation 141, 21–33 (1996)
Kadin, J.: P NP[logn] and sparse Turing-complete sets for NP. Journal of Computer and System Sciences 39, 282–298 (1989)
Mahaney, S.: Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences 25(2), 130–143 (1982)
Ogiwara, M., Watanabe, O.: On polynomial time bounded truth-table reducibility of NP sets to sparse sets. SIAM JC 20, 471–483 (1991)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Russell, A., Sundaram, R.: Symmetric alternation captures BPP. Computational Complexity 7(2), 152–162 (1998)
Vereshchagin, N.: Relativizable and Nonrelativizable Theorems in the Polynomial Theory of Algorithms. Russian Acad. Sci. Izv. Math. 42, 261–298 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Unger, F. (2005). On Small Hard Leaf Languages. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_67
Download citation
DOI: https://doi.org/10.1007/11549345_67
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28702-5
Online ISBN: 978-3-540-31867-5
eBook Packages: Computer ScienceComputer Science (R0)