Hausdorff reductions to sparse sets and to sets of high information content
We investigate the complexity of sets that have a rich internal structure and at the same time are reducible to sets of either low or very high information content. In particular, we show that every length-decreasing or word-decreasing self-reducible set that reduces to some sparse set via a non-monotone variant of the Hausdorff reducibility is low for Δ2p.
Measuring the information content of a set by the space-bounded Kolmogorov complexity of its characteristic sequence, we further investigate the (non-uniform) complexity of sets A in EXPSPACE/poly that reduce to some set having very high information content. Specifically, we show that if the reducibility used has a certain property, called “reliability,” then A in fact is reducible to a sparse set (under the same reducibility). As a consequence of our results, the existence of hard sets (under “reliable” reducibilities) of very high information content is unlikely for various complexity classes as for example NP, PP, and PSPACE.
KeywordsKolmogorov Complexity High Information Content Reliable Reduction Conjunctive Reducibility Polynomial Time Computable Function
Unable to display preview. Download preview PDF.
- 1.V. Arvind, Y. Han, L.A. Hemachandra, J. Köbler, A. Lozano, M. Mundhenk, M. Ogiwara, U. Schöning, R. Silvestri, and T. Thierauf. Reductions to sets of low information content. Proceedings of the 19th ICALP, Lecture Notes in Computer Science, #623:162–173, Springer Verlag, 1992.Google Scholar
- 2.V. Arvind, J. Köbler, and M. Mundhenk. On bounded truth-table, conjunctive, and randomized reductions to sparse sets. In Proceedings 12th Conference on the Foundations of Software Technology & Theoretical Computer Science, Lecture Notes in Computer Science, #652:140–151, Springer Verlag, 1992.Google Scholar
- 3.V. Arvind, J. Köbler, and M. Mundhenk. Lowness and the complexity of sparse and tally descriptions. In Proceedings Third International Symposium on Algorithms and Computation, Lecture Notes in Computer Science, #650:249–258, Springer Verlag, 1992.Google Scholar
- 4.V. Arvind, J. Köbler, and M. Mundhenk. Reliable reductions, high sets and low sets. Technical Report, Universität Ulm, Ulmer Informatik-Bericht 92-19, 1992.Google Scholar
- 5.J. Balcazár. Self-reducibility. Journal of Computer and System Sciences, 41:367–388, 1990.Google Scholar
- 6.J.L. Balcazár, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.Google Scholar
- 7.R. Book and J. Lutz. On languages with very high information content. Proceedings of the 7th Structure in Complexity Theory Conference, 255–259, IEEE Computer Society Press, 1992.Google Scholar
- 8.R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, 302–309, April 1980.Google Scholar
- 9.K. Ko. On self-reducibility and weak p-selectivity. Journal of Computer and System Sciences, 26:209–221, 1983.Google Scholar
- 10.R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities. Theoretical Computer Science, 1(2):103–124, 1975.Google Scholar
- 11.M. Li and P. Vitanyi. An introduction to Kolmogorov complexity and its application. Addison-Wesley, 1992.Google Scholar
- 12.A. Lozano and J. Torán. Self-reducible sets of small density. Mathematical Systems Theory, 24:83–100, 1991.Google Scholar
- 13.S. Mahaney. Sparse complete sets for NP: solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25(2):130–143, 1982.Google Scholar
- 14.M. Ogiwara and O. Watanabe. On polynomial-time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3):471–483, 1991.Google Scholar
- 15.D. Ranjan and P. Rohatgi. Randomized reductions to sparse sets. Proceedings of the 7th Structure in Complexity Theory Conference, IEEE Computer Society Press, 239–242, 1992.Google Scholar
- 16.K.W. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987.Google Scholar
- 17.K.W. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):83–846, 1990.Google Scholar
- 18.G. Wechsung and K.W. Wagner. On the boolean closure of NP. Manuscript. (Extended abstract by: G. Wechsung, On the boolean closure of NP, in Proc. International Conference on Fundamentals of Computation Theory, Lecture Notes in Computer Science, #199:485–493, Springer-Verlag, 1985.)Google Scholar