Abstract
Scaled dimension has been introduced by Hitchcock et al. (2003) in order to quantitatively distinguish among classes such as SIZE(2αn) and SIZE(\( 2^{n^{\alpha}}\)) that have trivial dimension and measure in ESPACE.
This paper gives an exact characterization of effective scaled dimension in terms of resource-bounded Kolmogorov complexity. We can now view each result on the scaled dimension of a class of languages as upper and lower bounds on the Kolmogorov complexity of the languages in the class.
We prove a Small Span Theorem for Turing reductions that implies the class of ≤ P/poly T-hard sets for ESPACE has (–3)rd-pspace dimension 0.
As a consequence we have a nontrivial upper bound on the Kolmogorov complexity of all hard sets for ESPACE for this very general nonuniform reduction, ≤ P/poly T. This is, to our knowledge, the first such bound. We also show that this upper bound does not hold for most decidable languages, so \(\leq^{\rm P/poly}_{\rm T}\)-hard languages are unusually simple.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This research was supported in part by Spanish Government MEC project TIC 2002-04019-C03-03.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambos-Spies, K., Merkle, W., Reimann, J., Stephan, F.: Hausdorff dimension in exponential time. In: Proceedings of the 16th IEEE Conference on Computational Complexity, pp. 210–217 (2001)
Ambos-Spies, K., Neis, H.-C., Terwijn, S.A.: Genericity and measure for exponential time. Theoretical Computer Science 168(1), 3–19 (1996)
Athreya, K.B., Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Effective strong dimension in algorithmic information and computational complexity. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 632–643. Springer, Heidelberg (2004)
Balcázar, J.L., Book, R.V.: Sets with small generalized Kolmogorov complexity. Acta Informatica 23, 679–688 (1986)
Buhrman, H., van Melkebeek, D.: Hard sets are hard to find. Journal of Computer and System Sciences 59(2), 327–345 (1999)
Hartmanis, J.: Generalized Kolmogorov complexity and the structure of feasible computations. In: Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, pp. 439–445 (1983)
Hitchcock, J.M.: Effective Fractal Dimension: Foundations and Applications. PhD thesis, Iowa State University (2003)
Hitchcock, J.M.: Small spans in scaled dimension. In: Proceedings of the 19th IEEE Conference on Computational Complexity (2004) (to appear)
Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Scaled dimension and nonuniform complexity. Journal of Computer and System Sciences (to appear), Preliminary version appeared in Proceedings of the 30th International Colloquium on Automata, Languages, and Programming, pp. 278–290 (2003)
Hitchcock, J.M., Vinodchandran, N.V.: Dimension, entropy rates, and compression. In: Proceedings of the 19th IEEE Conference on Computational Complexity (2004) (to appear)
Hitchcock, J.M., Mayordomo, E.: Base invariance of feasible dimension. Manuscript (2003)
Huynh, D.T.: Resource-bounded Kolmogorov complexity of hard languages. In: Selman, A.L. (ed.) Structure in Complexity Theory. LNCS, vol. 223, pp. 184–195. Springer, Heidelberg (1986)
Juedes, D.W., Lutz, J.H.: The complexity and distribution of hard problems. SIAM Journal on Computing 24(2), 279–295 (1995)
Juedes, D.W., Lutz, J.H.: Completeness and weak completeness under polynomial-size circuits. Information and Computation 125(1), 13–31 (1996)
Kolmogorov, A.N.: Three approaches to the quantitative definition of ‘information’. Problems of Information Transmission 1, 1–7 (1965)
Levin, L.A.: Randomness conservation inequalities; information and independence in mathematical theories. Information and Control 61, 15–37 (1984)
Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Berlin (1997)
Lindner, W.: On the polynomial time bounded measure of one-truth-table degrees and p-selectivity (1993), Diplomarbeit, Technische Universität Berlin
Longpré, L.: Resource Bounded Kolmogorov Complexity, a Link Between Computational Complexity and Information Theory. PhD thesis, Cornell University (1986) Technical Report TR-86-776
Lutz, J.H.: Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44(2), 220–258 (1992)
Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003)
Sipser, M.: A complexity-theoretic approach to randomness. In: Proceedings of the 15th ACM Symposium on Theory of Computing, pp. 330–335 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hitchcock, J.M., López-Valdés, M., Mayordomo, E. (2004). Scaled Dimension and the Kolmogorov Complexity of Turing-Hard Sets. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28629-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-28629-5_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22823-3
Online ISBN: 978-3-540-28629-5
eBook Packages: Springer Book Archive