article

The arithmetical complexity of dimension and randomness

Authors:

John M. Hitchcock,

Sebastiaan A. TerwijnAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 8, Issue 2

Pages 13 - es

https://doi.org/10.1145/1227839.1227845

Published: 01 April 2007 Publication History

Abstract

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) ∈ [0,1] and a strong dimension Dim(A) ∈ [0,1].

Let DIM^α and DIM^α_str be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM⁰ is properly Π⁰₂, and that for all Δ⁰₂-computable α ∈ (0, 1], DIM^α is properly Π⁰₃.

To classify the strong dimension classes, we use a more powerful effective Borel hierarchy where a coenumerable predicate is used rather than an enumerable predicate in the definition of the Σ⁰₁ level. For all Δ⁰₂-computable α ∈ [0, 1), we show that DIM^α_str is properly in the Π⁰₃ level of this hierarchy. We show that DIM¹_str is properly in the Π⁰₂ level of this hierarchy.

We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π⁰₃.

References

[1]

Ambos-Spies, K., Merkle, W., Reimann, J., and Stephan, F. 2001. Hausdorff dimension in exponential time. In Proceedings of the 16th IEEE Conference on Computational Complexity. 210--217.

Digital Library

[2]

Athreya, K. B., Hitchcock, J. M., Lutz, J. H., and Mayordomo, E. 2004. Effective strong dimension in algorithmic information and computational complexity. In Proceedings of the 21st Annual Symposium on Theoretical Aspects of Computer Science. Springer Verlag. 632--643. To appear in SIAM J. Comput.

[3]

Dai, J. J., Lathrop, J. I., Lutz, J. H., and Mayordomo, E. 2004. Finite-State dimension. Theor. Comput. Sci. 310, 1 3, 1--33.

Digital Library

[4]

Fenner, S. A. 2002. Gales and supergales are equivalent for defining constructive Hausdorff dimension. Tech. Rep. cs.CC/0208044, Computing Research Repository.

[5]

Fortnow, L. and Lutz, J. H. 2005. Prediction and dimension. J. Comput. Syst. Sci. 70, 4, 570--589.

Digital Library

[6]

Hausdorff, F. 1919. Dimension und äußeres Maß. Mathematische Annalen 79, 157--179.

[7]

Hitchcock, J. M. 2002. MAX3SAT is exponentially hard to approximate if NP has positive dimension. Theor. Comput. Sci. 289, 1, 861--869.

Digital Library

[8]

Hitchcock, J. M. 2003a. Fractal dimension and logarithmic loss unpredictability. Theor. Comput. Sci. 304, 1 3, 431--441.

[9]

Hitchcock, J. M. 2003b. Gales suffice for constructive dimension. Inf. Process. Lett. 86, 1, 9--12.

Digital Library

[10]

Jockusch, C. G. 1980. Degrees of generic sets. In Recursion Theory: Its Generalizations and Applications. London Mathematical Society Lecture Notes Series, vol. 45. Cambridge University Press, New York. 110--139.

[11]

Kechris, A. S. 1994. Classical Descriptive Set Theory. Springer Verlag.

[12]

Ki, H. and Linton, T. 1994. Normal numbers and subsets of N with given densities. Fundamenta Mathematicae 144, 163--179.

[13]

Kreisel, G. 1950. Note on arithmetical models for consistent formulae of the predicate calculus. Fundam. Mathematicae 37, 265--285.

[14]

Li, M. and Vitányi, P. M. B. 1997. An Introduction to Kolmogorov Complexity and its Applications, 2nd ed Springer Verlag, Berlin.

Digital Library

[15]

Lutz, J. H. 2003a. Dimension in complexity classes. SIAM J. Comput. 32, 5, 1236--1259.

Digital Library

[16]

Lutz, J. H. 2003b. The dimensions of individual strings and sequences. Inf. Comput. 187, 1, 49--79.

Digital Library

[17]

Martin-Löf, P. 1966. The definition of random sequences. Inf. Control 9, 602--619.

[18]

Mayordomo, E. 2002. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inf. Process. Lett. 84, 1, 1--3.

Digital Library

[19]

Moschovakis, Y. N. 1980. Descriptive Set Theory. North-Holland, Amsterdam.

[20]

Odifreddi, P. 1989. Classical recursion theory. In Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland.

[21]

Rogers, Jr, H. 1967. Theory of Recursive Functions and Effective Computability. McGraw - Hill, New York.

Digital Library

[22]

Schnorr, C. P. 1971. Zufälligkeit und wahrscheinlichkeit. Lecture Not. Math. 218.

[23]

Schnorr, C. P. 1977. A survey of the theory of random sequences. In Basic Problems in Methodology and Linguistics, R. E. Butts and J. Hintikka, Eds. D. Reidel. 193--211.

[24]

Staiger, L. 1993. Recursive automata on infinite words. In Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science. Springer Verlag. 629--639.

Digital Library

[25]

Staiger, L. 2000. On the power of reading the whole infinite input tape. In Finite Versus Infinite: Contributions to an Eternal Dilemma, C. S. Calude and G. Paun, Eds. Springer Verlag. 335--348.

[26]

Staiger, L. 2005. Constructive dimension equals Kolmogorov complexity. Inf. Process. Lett. 93, 3, 149--153.

Digital Library

[27]

Terwijn, S. A. 2004. Complexity and randomness. Rendiconti del Seminario Matematico di Torino 62, 1, 1--38.

[28]

van Lambalgen, M. 1987. Random sequences. Ph.D. thesis, Department of Mathematics, University of Amsterdam.

[29]

Ville, J. 1939. Étude Critique de la Notion de Collectif. Gauthier Villars, Paris.

[30]

Wadge, W. W. 1972. Degrees of complexity of subsets of the Baire space. Not. Amer. Math. Soc. 19, 714--715.

[31]

Wadge, W. W. 1983. Reducibility and determinateness on the Baire space. Ph.D. thesis, U. C. Berkeley.

[32]

Wang, Y. 1996. Randomness and complexity. Ph.D. thesis, Department of Mathematics, University of Heidelberg.

Cited By

Marcone AValenti M(2022)Effective aspects of Hausdorff and Fourier dimensionComputability10.3233/COM-21037211:3-4(299-333)Online publication date: 21-Dec-2022
https://doi.org/10.3233/COM-210372
Kjos-Hanssen BStephan FTeutsch J(2012)Arithmetic complexity via effective names for random sequencesACM Transactions on Computational Logic10.1145/2287718.228772413:3(1-18)Online publication date: 28-Aug-2012
https://dl.acm.org/doi/10.1145/2287718.2287724
Lutz JWeihrauch K(2008)Connectivity Properties of Dimension Level SetsElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2008.03.022202(295-304)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1016/j.entcs.2008.03.022
Show More Cited By

Index Terms

The arithmetical complexity of dimension and randomness
1. Theory of computation
  1. Logic
  2. Models of computation
    1. Computability

Recommendations

Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-Löf, computably, Schnorr, and Kurtz random ...
Effective Randomness for Computable Probability Measures

Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of ...
A divergence formula for randomness and dimension

If S is an infinite sequence over a finite alphabet @S and @b is a probability measure on @S, then the dimension of S with respect to @b, written dim^@b(S), is a constructive version of Billingsley dimension that coincides with the (constructive ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 8, Issue 2

April 2007

196 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/1227839

Issue’s Table of Contents

Copyright © 2007 ACM.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 April 2007

Published in TOCL Volume 8, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
281
Total Downloads

Downloads (Last 12 months)4
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Marcone AValenti M(2022)Effective aspects of Hausdorff and Fourier dimensionComputability10.3233/COM-21037211:3-4(299-333)Online publication date: 21-Dec-2022
https://doi.org/10.3233/COM-210372
Kjos-Hanssen BStephan FTeutsch J(2012)Arithmetic complexity via effective names for random sequencesACM Transactions on Computational Logic10.1145/2287718.228772413:3(1-18)Online publication date: 28-Aug-2012
https://dl.acm.org/doi/10.1145/2287718.2287724
Lutz JWeihrauch K(2008)Connectivity Properties of Dimension Level SetsElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2008.03.022202(295-304)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1016/j.entcs.2008.03.022
Lutz JWeihrauch K(2008)Connectivity properties of dimension level setsMathematical Logic Quarterly10.1002/malq.20071006054:5(483-491)Online publication date: 27-Aug-2008
https://dl.acm.org/doi/10.1002/malq.200710060

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents