Notre Dame Journal of Formal Logic

Levels of Uniformity

Rutger Kuyper

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We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of nonuniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other.

Article information

Notre Dame J. Formal Logic, Volume 60, Number 1 (2019), 119-138.

Received: 1 November 2015
Accepted: 30 November 2016
First available in Project Euclid: 18 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D30: Other degrees and reducibilities 03D32: Algorithmic randomness and dimension [See also 68Q30]
Secondary: 03G10: Lattices and related structures [See also 06Bxx] 03B20: Subsystems of classical logic (including intuitionistic logic)

Medvedev degrees Muchnik degrees algorithmic randomness


Kuyper, Rutger. Levels of Uniformity. Notre Dame J. Formal Logic 60 (2019), no. 1, 119--138. doi:10.1215/00294527-2018-0024.

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  • [1] Downey, R. G., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.
  • [2] Dyment, E. Z., “Certain properties of the Medvedev lattice,” Matematicheskii Sbornik, (NS), vol. 101(143) (1976), pp. 360–379.
  • [3] Higuchi, K., and T. Kihara, “Inside the Muchnik degrees, I: Discontinuity, learnability and constructivism,” Annals of Pure and Applied Logic, vol. 165 (2014), pp. 1058–114.
  • [4] Higuchi, K., and T. Kihara, “Inside the Muchnik degrees, II: The degree structures induced by the arithmetical hierarchy of countably continuous functions,” Annals of Pure and Applied Logic, vol. 165 (2014), pp. 1201–41.
  • [5] Hinman, P. G., “A survey of Mučnik and Medvedev degrees,” Bulletin of Symbolic Logic, vol. 18 (2012), pp. 161–229.
  • [6] Hoyrup, M., and C. Rojas, “An application of Martin-Löf randomness to effective probability theory,” pp. 260–69 in Mathematical Theory and Computational Practice, edited by K. Ambos-Spies, B. Löwe, and W. Merkle, vol. 5635 of Lecture Notes in Computer Science, Springer, Berlin, 2009.
  • [7] Jockusch, C. G., Jr., “Degrees of functions with no fixed points,” pp. 191–201 in Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987), edited by J. E. Fenstad, I. T. Frolov, and R. Hilpinen, vol. 126 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1989.
  • [8] Kautz, S. M., “Degrees of random sets,” Ph.D. dissertation, Cornell University, Ithaca, New York, 1991.
  • [9] Kučera, A., “Measure, $\Pi^{0}_{1}$-classes and complete extensions of PA,” pp. 245–59 in Recursion Theory Week (Oberwolfach, 1984), edited by H.-D. Ebbinghaus, G. H. Müller, and G. E. Sacks, vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985.
  • [10] Kurtz, S. A., “Notions of weak genericity,” Journal of Symbolic Logic, vol. 48 (1983), pp. 764–70.
  • [11] Kuyper, R., “Natural factors of the Muchnik lattice capturing IPC,” Annals of Pure and Applied Logic, vol. 164 (2013), pp. 1025–36.
  • [12] Kuyper, R., “Natural factors of the Medvedev lattice capturing IPC,” Archive for Mathematical Logic, vol. 53 (2014), pp. 865–79.
  • [13] Medvedev, Y. T., “Degrees of difficulty of the mass problem,” Doklady Akademii Nauk SSSR (NS), vol. 104 (1955), pp. 501–4.
  • [14] Muchnik, A. A., “On strong and weak reducibility of algorithmic problems,” Sibirskii Matematicheskii Zurnal, vol. 4 (1963), pp. 1328–41.
  • [15] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009.
  • [16] Odifreddi, P. G., Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1989.
  • [17] Shore, R. A., and T. A. Slaman, “Defining the Turing jump,” Mathematical Research Letters, vol. 6 (1999), pp. 711–22.
  • [18] Skvortsova, E. Z., “Exact interpretation of the intuitionistic propositional calculus by means of an initial segment of the Medvedev lattice,” Sibirskii Matematicheskii Zhurnal, vol. 29 (1988), pp. 171–78; English translation in Siberian Mathematical Journal, vol. 29 (1988), pp. 133–39.
  • [19] Sorbi, A., “Some remarks on the algebraic structure of the Medvedev lattice,” Journal of Symbolic Logic, vol. 55 (1990), pp. 831–53.
  • [20] Sorbi, A., “The Medvedev lattice of degrees of difficulty,” pp. 289–312 in Computability, Enumerability, Unsolvability: Directions in Recursion Theory, edited by S. B. Cooper, T. A. Slaman, and S. S. Wainer, vol. 224 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1996.
  • [21] Sorbi, A., and S. A. Terwijn, “Intuitionistic logic and Muchnik degrees,” Algebra Universalis, vol. 67 (2012), pp. 175–88.