Abstract
We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is not EXP/1-immune. That is, we prove that some infinite P-selective set has no infinite EXP-time subset, but we also prove that every infinite P-selective set has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so fragile that it is pierced by a single bit of information.
The above claims follow from broader results that we obtain about the immunity of the P-selective sets. In particular, we prove that for every recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is not \({\it \Pi}^{p}_{2}\)/1-immune.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Berman, L.: On the Structure of Complete Sets. In: Proceedings of the 17th IEEE Symposium on Foundations of Computer Science, October 1976, pp. 76–80. IEEE Computer Society, Los Alamitos (1976)
Cai, J.: Probability one Separation of the Boolean Hierarchy. In: Brandenburg, F.J., Wirsing, M., Vidal-Naquet, G. (eds.) STACS 1987. LNCS, vol. 247, pp. 148–158. Springer, Heidelberg (1987)
Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The Boolean Hierarchy I: Structural Properties. SIAM Journal on Computing 17(6), 1232–1252 (1988)
Hemaspaandra, L., Hempel, H., Nickelsen, A.: Algebraic Properties for Selector Functions. SIAM Journal on Computing 33(6), 1309–1337 (2004)
Hemaspaandra, E., Hemaspaandra, L., Watanabe, O.: The Complexity of Kings. Technical Report TR-870, Department of Computer Science, University of Rochester, Rochester, NY (June 2005)
Hemaspaandra, L., Naik, A., Ogihara, M., Selman, A.: Computing Solutions Uniquely Collapses the Polynomial Hierarchy. SIAM Journal on Computing 25(4), 697–708 (1996)
Hemaspaandra, L., Ogihara, M.: The Complexity Theory Companion. Springer, Heidelberg (2002)
Hemaspaandra, L., Ogihara, M., Zaki, M., Zimand, M.: The Complexity of Finding Top-Toda-Equivalence-Class Members MINOR PANIC: hem-ogi-zak-zim: Missing year/issue/voume, and remove. Theory of Computing Systems (to appear); Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 90–99. Springer, Heidelberg (2004)
Hemaspaandra, L., Torenvliet, L.: Optimal Advice. Theoretical Computer Science 154(2), 367–377 (1996)
Hemaspaandra, L., Torenvliet, L.: Theory of Semi-Feasible Algorithms. Springer, Heidelberg (2003)
Karp, R., Lipton, R.: Some Connections between Nonuniform and Uniform Complexity Classes. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 302–309. ACM Press, New York (1980); An extended version has also appeared as: Turing Machines that Take Advice, L’Enseignement Mathématique. 2nd series, vol. 28, pp. 191–209 (1982)
Ko, K.: On Self-Reducibility and Weak P-Selectivity. Journal of Computer and System Sciences 26(2), 209–221 (1983)
Landau, H.: On Dominance Relations and the Structure of Animal Societies, III: The Condition for Score Structure. Bulletin of Mathematical Biophysics 15(2), 143–148 (1953)
Nickelsen, A., Tantau, T.: On Reachability in Graphs with Bounded Independence Number. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 554–563. Springer, Heidelberg (2002)
Nickelsen, A., Tantau, T.: Partial information classes. SIGACT News 34(1), 32–46 (2003)
Rogers Jr., H.: The Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
Schöning, U., Book, R.: Immunity, Relativization, and Nondeterminism. SIAM Journal on Computing 13, 329–337 (1984)
Selman, A.: P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP. Mathematical Systems Theory 13(1), 55–65 (1979)
Selman, A.: Some Observations on NP Real Numbers and P-Selective Sets. Journal of Computer and System Sciences 23(3), 326–332 (1981)
Selman, A.: Analogues of Semirecursive Sets and Effective Reducibilities to the Study of NP Complexity. Information and Control 52(1), 36–51 (1982)
Selman, A.: Reductions on NP and P-Selective Sets. Theoretical Computer Science 19(3), 287–304 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hemaspaandra, L.A., Torenvliet, L. (2006). P-Selectivity, Immunity, and the Power of One Bit. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds) SOFSEM 2006: Theory and Practice of Computer Science. SOFSEM 2006. Lecture Notes in Computer Science, vol 3831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11611257_30
Download citation
DOI: https://doi.org/10.1007/11611257_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31198-0
Online ISBN: 978-3-540-32217-7
eBook Packages: Computer ScienceComputer Science (R0)