Journal of Symbolic Logic

Degrees of unsolvability of continuous functions

Joseph S. Miller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0,1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f∈𝒞[0,1] computes a non-computable subset of ℕ; there is a non-total degree between Turing degrees a<T b iff b is a PA degree relative to a; 𝒮⊆ 2 is a Scott set iff it is the collection of f-computable subsets of ℕ for some f∈𝒞[0,1] of non-total degree; and there are computably incomparable f,g∈𝒞[0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

Article information

J. Symbolic Logic Volume 69, Issue 2 (2004), 555-584.

First available in Project Euclid: 19 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Miller, Joseph S. Degrees of unsolvability of continuous functions. J. Symbolic Logic 69 (2004), no. 2, 555--584. doi:10.2178/jsl/1082418543.

Export citation


  • M. M. Arslanov, R. F. Nadyrov, and V. D. Solov$'$ev A criterion for the completeness of recursively enumerable sets, and some generalizations of a fixed point theorem, Izvestija Vysših Učebnyh Zavedeniĭ Matematika,(1977), no. 4 (179), pp. 3--7.
  • Michael J. Beeson Foundations of constructive mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 6, Springer-Verlag, Berlin,1985.
  • G. D. Birkhoff and O. D. Kellogg Invariant points in function space, Transactions of the American Mathematical Society, vol. 23 (1922), no. 1, pp. 96--115.
  • H. F. Bohnenblust and S. Karlin On a theorem of Ville, Contributions to the theory of games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J.,1950, pp. 155--160.
  • D. Cenzer and J. B. Remmel $\Pi\sp 0\sb 1$ classes in mathematics, Handbook of recursive mathematics, vol. 2, Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam,1998, pp. 623--821.
  • Douglas Cenzer $\Pi\sp 0\sb 1$ classes in computability theory, Handbook of computability theory, Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland, Amsterdam,1999, pp. 37--85.
  • S. B. Cooper Partial degrees and the density problem, Journal of Symbolic Logic, vol. 47 (1982), no. 4, pp. 854--859 (1983).
  • Samuel Eilenberg and Deane Montgomery Fixed point theorems for multi-valued transformations, American Journal of Mathematics, vol. 68 (1946), pp. 214--222.
  • Andrzej Grzegorczyk Computable functionals, Fundamenta Mathematicae, vol. 42 (1955), pp. 168--202.
  • Lance Gutteridge Some results on enumeration reducibility, Ph.D. thesis, Simon Fraser University,1971.
  • Carl G. Jockusch, Jr. and Robert I. Soare $\Pi \sp0\sb1$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
  • Shizuo Kakutani A generalization of Brouwer's fixed point theorem, Duke Mathematical Journal, vol. 8 (1941), pp. 457--459.
  • Stephen Cole Kleene Introduction to Metamathematics, D. Van Nostrand Company, Incorporation, New York, N.Y.,1952.
  • Christoph Kreitz and Klaus Weihrauch Theory of representations, Theoretical Computer Science, vol. 38 (1985), no. 1, pp. 35--53.
  • Daniel Lacombe Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles. I, Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 240 (1955), pp. 2478--2480.
  • Manuel Lerman Degrees of Unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Berlin,1983.
  • Yu. T. Medvedev Degrees of difficulty of the mass problem, Doklady Akademii Nauk SSSR, vol. 104 (1955), pp. 501--504.
  • A. A. Mučnik On strong and weak reducibility of algorithmic problems, Akademija Nauk SSSR. Sibirskoe Otdelenie. Sibirskiĭ Matematičeskiĭ Žurnal, vol. 4 (1963), pp. 1328--1341.
  • John Myhill Note on degrees of partial functions, Proceedings of the American Mathematical Society, vol. 12 (1961), pp. 519--521.
  • Anil Nerode and Richard A. Shore Second order logic and first order theories of reducibility orderings, The Kleene Symposium (Proceedings of the Symposium, University of Wisconsin, Madison, Wisconsin, 1978), Studies in Logic and the Foundations of Mathematics, vol. 101, North-Holland, Amsterdam,1980, pp. 181--200.
  • V. P. Orevkov A constructive map of the square into itself, which moves every constructive point, Doklady Akademii Nauk SSSR, vol. 152 (1963), pp. 55--58.
  • David B. Posner The upper semilattice of degrees $\leq \bf 0\sp\prime $ is complemented, Journal of Symbolic Logic, vol. 46 (1981), no. 4, pp. 705--713.
  • Marian B. Pour-El and Jerome Caldwell On a simple definition of computable function of a real variable---with applications to functions of a complex variable, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 1--19.
  • Marian B. Pour-El and J. Ian Richards Computability and noncomputability in classical analysis, Transactions of the American Mathematical Society, vol. 275 (1983), no. 2, pp. 539--560.
  • Linda Jean Richter Degrees of structures, Ph.D. thesis, University of Illinois at Urbana-Champaign,1979.
  • Hartley Rogers, Jr. Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Company, New York,1967.
  • M. Rozinas The semilattice of e-degrees, Recursive functions (Russian), Ivanov. Gos. Univ., Ivanovo,1978, pp. 71--84.
  • Gerald E. Sacks The recursively enumerable degrees are dense, Annals of Mathematics. Second Series, vol. 80 (1964), pp. 300--312.
  • J. Schauder Der Fixpunktsatz in Funktionalräumen, Studia Mathematica, vol. 2 (1930), pp. 171--180.
  • Dana Scott Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of Symposia in Pure Mathematics, Vol. V, American Mathematical Society, Providence, R.I.,1962, pp. 117--121.
  • Alan L. Selman Arithmetical reducibilities. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 335--350.
  • Stephen G. Simpson Degrees of unsolvability: A survey of results, Handbook of Mathematical Logic (J. Barwise, editor), North-Holland, Amsterdam,1977, pp. 631--652.
  • Theodore A. Slaman and W. Hugh Woodin Definability in the enumeration degrees, Archive for Mathematical Logic, vol. 36 (1997), no. 4-5, pp. 255--267, Sacks Symposium (Cambridge, MA, 1993).
  • Clifford Spector On degrees of recursive unsolvability, Annals of Mathematics. Second Series, vol. 64 (1956), pp. 581--592.
  • A. M. Turing On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society. Second Series, vol. 42 (1936), pp. 230--265, Correction in [?].
  • Klaus Weihrauch Computability on computable metric spaces, Theoretical Computer Science, vol. 113 (1993), no. 2, pp. 191--210.