Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T09:19:22.288Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

Published online by Cambridge University Press:  12 March 2019

TAKAYUKI KIHARA
TAKAYUKI KIHARA*
Affiliation:
GRADUATE SCHOOL OF INFORMATICS NAGOYA UNIVERSITY, NAGOYA 464-8601, JAPANE-mail: kihara@i.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we study an analogue of tt-reducibility for points in computable metric spaces. We characterize the notion of the metric tt-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.

Keywords

Primary 03D30Secondary 03C6203D7854H05truth table degreeeffective dimensioncomputable metic space
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 726 - 749
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, U., Igusa, G., Miller, J. S., and Soskova, M. I., Characterizing the continuous degrees, submitted.Google Scholar
Banakh, T. and Bokalo, B., On scatteredly continuous maps between topological spaces. Topology and Its Applications, vol. 157 (2010), no. 1, pp. 108–122.Google Scholar
Bauer, A. and Yoshimura, K., Reductions in computability theory from a constructive point of view, a talk at The Logic Coloquium, July 14–19, 2014.Google Scholar
Bienvenu, L. and Merkle, W., Reconciling data compression and Kolmogorov complexity, Automata, Languages and Programming (Arge, L., Cachin, C., Jurdiński, T., and Tarlecki, A., editors), Lecture Notes in Computer Science, vol. 4596, Springer, Berlin, 2007, pp. 643–654.Google Scholar
Császár, Á. and Laczkovich, M., Discrete and equal convergence. Studia Scientiarum Mathematicarum Hungarica, vol. 10 (1975), no. 3–4, pp. 463–472.Google Scholar
Day, A. R. and Miller, J. S., Randomness for noncomputable measures. Transactions of the American Mathematical Society, vol. 365 (2013), no. 7, pp. 3575–3591.Google Scholar
Downey, R. G. and Hirschfeldt, D. R., Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.Google Scholar
Engelking, R., Theory of Dimensions Finite and Infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995.Google Scholar
Gregoriades, V., Kihara, T., and Ng, K. M., Turing degrees in Polish spaces and decomposability of Borel functions, preprint, arXiv:1410.1052.Google Scholar
Grubba, T., Schröder, M., and Weihrauch, K., Computable metrization. Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 381–395.Google Scholar
Hurewicz, W. and Wallman, H., Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, New Jersey, 1941.Google Scholar
Jayne, J. E. and Rogers, C. A., Borel isomorphisms at the first level. I. Mathematika, vol. 26 (1979), no. 1, pp. 125–156.Google Scholar
Jayne, J. E. and Rogers, C. A., Borel isomorphisms at the first level. II. Mathematika, vol. 26 (1979), no. 2, pp. 157–179.Google Scholar
Jayne, J. E. and Rogers, C. A., Piecewise closed functions. Mathematische Annalen, vol. 255 (1981), no. 4, pp. 499–518.Google Scholar
Jayne, J. E. and Rogers, C. A., First-level Borel functions and isomorphisms. Journal de Mathématiques Pures et Appliquées (9), vol. 61 (1982), no. 2, pp. 177–205.Google Scholar
Joyce, H., A relationship between packing and topological dimensions. Mathematika, vol. 45 (1998), no. 1, pp. 43–53.Google Scholar
Kenny, R., Effective zero-dimensionality for computable metric spaces. Logical Methods in Computer Science, vol. 11 (2015), pp. 1:11,25.Google Scholar
Kihara, T. and Miyabe, K., Uniform Kurtz randomness. Journal of Logic and Computation, vol. 24 (2014), no. 4, pp. 863–882.Google Scholar
Kihara, T. and Miyabe, K., Unified characterizations of lowness properties via Kolmogorov complexity. Archive for Mathematical Logic, vol. 54 (2015), no. 3–4, pp. 329–358.Google Scholar
Kihara, T., Ng, K. M., and Pauly, A., Enumeration degrees and nonmetrizable topology, preprint.Google Scholar
Kihara, T. and Pauly, A., Point degree spectra of represented spaces, submitted.Google Scholar
Lutz, J. H. and Lutz, N., Algorithmic information, plane Kakeya sets, and conditional dimension, 34th Symposium on Theoretical Aspects of Computer Science (Vollmer, H. and Vallée, B., editors), LIPICS - Leibniz International Proceedings in Informatics, vol. 66, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern, 2017, Art. No. 53.Google Scholar
Lutz, J. H. and Mayordomo, E., Dimensions of points in self-similar fractals. SIAM Journal on Computing, vol. 38 (2008), no. 3, pp. 1080–1112.Google Scholar
Lutz, N. and Stull, D. M., Bounding the dimension of points on a line, Theory and Applications of Models of Computation (Gopal, T. V., Jäger, G., and Steila, S., editors), Lecture Notes in Computer Science, vol. 10185, Springer, Cham, 2017, pp. 425–439.Google Scholar
Lutz, N. and Stull, D. M., Dimension spectra of lines, Unveiling Dynamics and Complexity, Lecture Notes in Computer Science, vol. 10307, Springer, Cham, 2017, pp. 304–314.Google Scholar
Luukkainen, J., Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures. Journal of the Korean Mathematical Society, vol. 35 (1998), no. 1, pp. 23–76.Google Scholar
McNicholl, T. H. and Rute, J., A uniform reducibility in computably presented Polish spaces, A talk at AMS Sectional Meeting AMS Special Session on “Effective Mathematics in Discrete and Continuous Worlds”, October 28–30, 2016.Google Scholar
Miller, J. S., Degrees of unsolvability of continuous functions, this Journal, vol. 69 (2004), no. 2, pp. 555–584.Google Scholar
Nagata, J.-i., Modern Dimension Theory, revised ed., Sigma Series in Pure Mathematics, vol. 2, Heldermann Verlag, Berlin, 1983.Google Scholar
Odifreddi, P., Classical Recursion Theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam, 1989.Google Scholar
O’Malley, R. J., Approximately differentiable functions: The r topology. Pacific Journal of Mathematics, vol. 72 (1977), no. 1, pp. 207–222.Google Scholar
Pol, R. and Zakrzewski, P., On Borel mappings and σ-ideals generated by closed sets. Advances in Mathematics, vol. 231 (2012), no. 2, pp. 651–663.Google Scholar
Shakhmatov, D. B., Baire isomorphisms at the first level and dimension. Topology and Its Applications, vol. 107 (2000), no. 1–2, pp. 153–159, 15th Anniversary of the Chair of General Topology and Geometry at Moscow State University.Google Scholar
Soare, R. I., Turing computability, Theory and Applications, Theory and Applications of Computability, Springer-Verlag, Berlin, 2016.Google Scholar
van Mill, J, The Infinite-Dimensional Topology of Function Spaces, North-Holland Mathematical Library, vol. 64, North-Holland, Amsterdam, 2001.Google Scholar
Weihrauch, K., Computable Analysis, Springer, Berlin, 2000.Google Scholar
Zapletal, J., Dimension theory and forcing. Topology and Its Applications, vol. 167 (2014), pp. 31–35.Google Scholar