Skip to main content
SpringerLink
Log in

Degrees of Unsolvability: A Tutorial

  • Conference paper
  • First Online:
Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

Included in the following conference series:

Abstract

Given a problem \(P\), one associates to \(P\) a degree of unsolvability, i.e., a quantity which measures the amount of algorithmic unsolvability which is inherent in \(P\). We focus on two degree structures: the semilattice of Turing degrees, \(\mathcal {D}_\mathrm {T}\), and its completion, \(\mathcal {D}_\mathrm {w}=\widehat{\mathcal {D}_\mathrm {T}}\), the lattice of Muchnik degrees. We emphasize specific, natural degrees and their relationship to reverse mathematics. We show how Muchnik degrees can be used to classify tiling problems and symbolic dynamical systems of finite type. We describe how the category of sheaves over \(\mathcal {D}_\mathrm {w}\) forms a model of intuitionistic mathematics, known as the Muchnik topos. This model is a rigorous implementation of Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a “calculus of problems”.

MSC2010: Primary 03D28; Secondary 03D80, 03D32, 03D35, 03D55, 03F55, 03G30, 18F20, 37B10.

S.G. Simpson—This paper is a preview of a three-hour tutorial to be given at CiE in Bucharest, June 29 to July 3, 2015. The author’s research is supported by the Eberly College of Science and by Simons Foundation Collaboration Grant 276282.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    By unsolvable we mean algorithmically unsolvable, i.e., not solvable by a Turing program.

  2. 2.

    We are not offering a rigorous definition of what is meant by “specific and natural.” However, it is well known that considerations of specificity and naturalness play an important role in mathematics. Without such considerations, it would be difficult or impossible to pursue the ideal of “exquisite taste” in mathematical research, as famously enunciated by von Neumann.

  3. 3.

    In this paper we take reals to be points in the Baire space \(\mathbb {N}^\mathbb {N}\), i.e., functions \(X:\mathbb {N}\rightarrow \mathbb {N}\) where \(\mathbb {N}=\{0,1,2,\ldots \}=\) the natural numbers.

  4. 4.

    More specifically, each of the mentioned problems amounts to the question of deciding whether or not a given string of symbols from a fixed finite alphabet belongs to a particular set of such strings. The problem is then identified with the characteristic function of the set of Gödel numbers of the strings which belong to the set.

  5. 5.

    A.k.a., computably enumerable [73].

  6. 6.

    The Sacks Splitting Theorem says that \(\mathcal {E}_\mathrm {T}\) satisfies \(\forall x\,(x>0\Rightarrow \exists u\,\exists v\,(u<x\) and \(v<x\) and \(\sup (u,v)=x))\).

  7. 7.

    The Sacks Density Theorem says that \(\mathcal {E}_\mathrm {T}\) satisfies \(\forall x\,\forall y\,(x<y\Rightarrow \exists z\,(x<z<y))\).

  8. 8.

    When speaking of decidable theories, we identify a theory with the characteristic function \(X\in \{0,1\}^\mathbb {N}\) of the set of Gödel numbers of theorems of the theory.

  9. 9.

    This concept is from Medvedev [39]. As in footnote 3 a real is a function \(X\in \mathbb {N}^\mathbb {N}\).

  10. 10.

    This is Muchnik’s notion of weak reducibility [41, Definition 2].

  11. 11.

    For a more precise statement, see [5, Theorem 5.8].

  12. 12.

    This follows from a theorem of Kleene [33, p. 398]. See also [29, 57].

  13. 13.

    Here \(E_1\simeq E_2\) means that \(E_1\) and \(E_2\) are both undefined or both defined and equal.

  14. 14.

    Let \(\varphi _n\), \(n\in \mathbb {N}\) be a fixed, standard, partial recursive enumeration of the partial recursive functions. A function \(Z\in \mathbb {N}^\mathbb {N}\) is said to be diagonally nonrecursive [3, 23, 26, 32, 65] if \(Z\cap \psi =\emptyset \) where \(\psi \) is the well known diagonal function, defined by \(\psi (n)\simeq \varphi _n(n)\). Letting \(\mathrm {DNR}=\{Z\in \mathbb {N}^\mathbb {N}\mid Z\) is diagonally nonrecursive\(\}\) and \(\mathrm {DNR}_\mathrm {REC}=\{Z\in \mathrm {DNR}\mid Z\) is recursively bounded\(\}\), we have \(\mathbf {d}=\deg _\mathrm {w}(\mathrm {DNR})\) and \(\mathbf {d}_\mathrm {REC}=\deg _\mathrm {w}(\mathrm {DNR}_\mathrm {REC})\).

  15. 15.

    For example, \(f(n)\) could be \(n/2\) or \(n/3\) or \(\sqrt{n}\) or \(\root 3 \of {n}\) or \(\log _2n\) or \(\log _3n\) or \(\log _2\log _2n\), etc., or \(f\) could be the inverse Ackermann function.

  16. 16.

    The inhabitants of this menagerie are downwardly closed sets of Turing degrees, but the complements of such sets are essentially the same thing as Muchnik degrees.

  17. 17.

    See also the English translation in [5, Appendix].

  18. 18.

    This topological space was considered by Muchnik [41, p. 1332] [5, p. 35].

References

  1. Aanderaa, S., Cohen, D.E.: Modular machines I, II. In: [2], pp. 1–18, 19–28 (1980)

    Google Scholar 

  2. Adian, S.I., Boone, W.W., Higman, G. (eds.): Word Problems II: The Oxford Book. Studies in Logic and the Foundations of Mathematics, X + 578 p., North-Holland (1980)

    Google Scholar 

  3. Ambos-Spies, K., Kjos-Hanssen, B., Lempp, S., Slaman, T.A.: Comparing DNR and WWKL. J. Symbolic Logic 69, 1089–1104 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barwise, J., Keisler, H.J., Kunen, K. (eds.): The Kleene Symposium. Studies in Logic and the Foundations of Mathematics, XX + 425 p., North-Holland (1980)

    Google Scholar 

  5. Basu, S.S., Simpson, S.G.: Mass problems and intuitionistic higher-order logic, 44 p., 12 August 2014. http://arxiv.org/abs/1408.2763

  6. Berger, R.: The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society, vol. 66, p. 72. American Mathematical Society, Providence (1966)

    Google Scholar 

  7. Bienvenu, L., Porter, C.P.: Deep \(\Pi ^0_1\) classes, 37 p., 4 June 2014. http://arxiv.org/abs/1403.0450v2

  8. Binns, S.: A splitting theorem for the medvedev and muchnik lattices. Math. Logic Q. 49(4), 327–335 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Binns, S., Shore, R.A., Simpson, S.G.: Mass problems and density, in preparation, 5 p., 1 March 2014

    Google Scholar 

  10. Binns, S., Simpson, S.G.: Embeddings into the medvedev and muchnik lattices of \(\Pi ^0_1\) classes. Arch. Math. Logic 43, 399–414 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chong, C.T., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.): Computational Prospects of Infinity. In: Proceedings of the Logic Workshop at the Institute for Mathematical Sciences, Part I: Tutorials in Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific, 20 June–15 August, 2005, no. 14, 264 p. (2008)

    Google Scholar 

  12. Church, A.: A note on the Entscheidungsproblem. J. Symbolic Logic 1, 40–41 (1936)

    Article  Google Scholar 

  13. Cole, J.A., Simpson, S.G.: Mass problems and hyperarithmeticity. J. Math. Logic 7(2), 125–143 (2008)

    Article  MathSciNet  Google Scholar 

  14. Davis, M.: Hilbert’s tenth problem is unsolvable. Am. Math. Mon. 80, 233–269 (1973)

    Article  MATH  Google Scholar 

  15. Dekker, J.C.E. (ed.): Recursive function theory. In: Proceedings of Symposia in Pure Mathematics, American Mathematical Society, VII + 247 p. (1962)

    Google Scholar 

  16. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability, XXVIII + 855 p. Springer, New York (2010)

    Google Scholar 

  17. Durand, B., Romashchenko, A., Shen, A.: Fixed-point tile sets and their applications. J. Comput. Syst. Sci. 78(3), 731–764 (2012). doi:10.1016/j.jcss.2011.11.001

    Article  MATH  MathSciNet  Google Scholar 

  18. Fenstad, J.E., Frolov, I.T., Hilpinen, R. (eds.): Logic, Methodology and Philosophy of Science VIII. No. 126 in Studies in Logic and the Foundations of Mathematics, XVII + 702 p., North-Holland (1989)

    Google Scholar 

  19. FOM e-mail list September 1997 to the present. http://www.cs.nyu.edu/mailman/listinfo/fom/

  20. Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.): Applications of Sheaves, Proceedings, Durham, 1977. No. 753 in Lecture Notes in Mathematics, XIV + 779 p., Springer (1979)

    Google Scholar 

  21. Fourman, M.P., Scott, D.S.: Sheaves and logic. In: [20], pp. 302–401 (1979)

    Google Scholar 

  22. Gandy, R.O., Kreisel, G., Tait, W.W.: Set existence. In: Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 8, pp. 577–582 (1960)

    Google Scholar 

  23. Greenberg, N., Miller, J.S.: Diagonally non-recursive functions and effective Hausdorff dimension. Bull. Lond. Math. Soc. 43(4), 636–654 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hudelson, W.M.P.: Mass problems and initial segment complexity. J. Symbolic Logic 79(1), 20–44 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  25. Jech, T.: Set Theory, XI + 621 p., Academic Press (1978)

    Google Scholar 

  26. Jockusch Jr. C.G.: Degrees of functions with no fixed points. In: [18], pp. 191–201 (1989)

    Google Scholar 

  27. Jockusch Jr. C.G., Simpson, S.G.: A degree-theoretic definition of the ramified analytical hierarchy. Ann. Math. Logic 10, 1–32 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jockusch Jr. C.G., Soare, R.I.: Degrees of members of \(\Pi ^0_1\) classes. Pac. J. Math. 40, 605–616 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  29. Jockusch Jr. C.G., Soare, R.I.: \(\Pi ^0_1\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 35–56 (1972)

    MathSciNet  Google Scholar 

  30. Khan, M., Kjos-Hanssen, B., Miller, J.S.: The Computability Menagerie (2015). http://www.math.wisc.edu/~jmiller/

  31. Khan, M., Miller, J.S.: Forcing with bushy trees, 18 p., 30 March 2015. http://arxiv.org/abs/1503.08870v1

  32. Kjos-Hanssen, B., Merkle, W., Stephan, F.: Kolmogorov complexity and the recursion theorem. Trans. Am. Math. Soc. 363, 5465–5480 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kleene, S.C.: Introduction to Metamathematics, X + 550 p., Van Nostrand (1952)

    Google Scholar 

  34. Kleene, S.C., Post, E.L.: The upper semi-lattice of degrees of recursive unsolvability. Ann. Math. 59, 379–407 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kolmogoroff, A.: Zur Deutung der intuitionistischen Logik. Math. Z. 35, 58–65 (1932)

    Article  MathSciNet  Google Scholar 

  36. Kolmogorov, A.N.: On the interpretation of intuitionistic logic. In: [76], translation of [35] with commentary and additional references, pp. 151–158 and 451–466 (1991)

    Google Scholar 

  37. Lerman, M.: Degrees of Unsolvability. Perspectives in Mathematical Logic, XIII + 307 p., Springer, Berlin (1983)

    Google Scholar 

  38. Lerman, M.: A Framework for Priority Arguments. Lecture Notes in Logic, Association for Symbolic Logic, XVI + 176 p., Cambridge University Press (2010)

    Google Scholar 

  39. Medvedev, Y.T.: Degrees of difficulty of mass problems. Dokl. Akad. Nauk SSSR 104, 501–504 (1955). in Russian

    MATH  MathSciNet  Google Scholar 

  40. Miller, J.S.: Extracting information is hard: a turing degree of non-integral effective Hausdorff dimension. Adv. Math. 226, 373–384 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Muchnik, A.A.: On strong and weak reducibilities of algorithmic problems. Sib. Mat. Zh. 4, 1328–1341 (1963). in Russian

    MATH  Google Scholar 

  42. Nabutovsky, A.: Einstein structures: existence versus uniqueness. Geom. Funct. Anal. 5, 76–91 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. Nabutovsky, A., Weinberger, S.: Betti numbers of finitely presented groups and very rapidly growing functions. Topology 46, 211–233 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Nies, A.: Computability and Randomness, XV + 433 p., Oxford University Press (2009)

    Google Scholar 

  45. Nies, A., Shore, R.A., Slaman, T.A.: Interpretability and definability in the recursively enumerable degrees. Proc. Lon. Math. Soc. 77, 241–291 (1998)

    Article  MathSciNet  Google Scholar 

  46. Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284–316 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  47. Rabin, M.O.: Recursive unsolvability of group theoretic problems. Ann. Math. 67, 172–194 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  48. Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symbolic Logic 33, 514–520 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  49. Robinson, R.M.: Undecidability and nonperiodicity of tilings of the plane. Inventiones Math. 12, 177–209 (1971)

    Article  MATH  Google Scholar 

  50. Rogers Jr. H.: Theory of Recursive Functions and Effective Computability, XIX + 482 p.. MIT Press, Cambridge (1967)

    Google Scholar 

  51. Sacks, G.E.: Degrees of Unsolvability. No. 55 in Annals of Mathematics Studies, IX + 174 p., Princeton University Press, London (1963)

    Google Scholar 

  52. Sacks, G.E.: The recursively enumerable degrees are dense. Ann. Math. 80, 300–312 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  53. Sacks, G.E.: Higher Recursion Theory. Perspectives in Mathematical Logic, XV + 344 p., Springer (1990)

    Google Scholar 

  54. Scott, D.S.: Algebras of sets binumerable in complete extensions of arithmetic. In: [15], pp. 117–121 (1962)

    Google Scholar 

  55. Scott, D.S., Tennenbaum, S.: On the degrees of complete extensions of arithmetic (abstract). Not. Am. Math. Soc. 7, 242–243 (1960)

    Google Scholar 

  56. Shafer, P.: Coding true arithmetic in the Medvedev and Muchnik degrees. J. Symbolic Logic 76(1), 267–288 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  57. Shoenfield, J.R.: Degrees of models. J. Symbolic Logic 25, 233–237 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  58. Shoenfield, J.R.: Degrees of Unsolvability. No. 2 in North-Holland Mathematics Studies, VIII + 111 p., North-Holland (1971)

    Google Scholar 

  59. Shore, R.A.: The Turing degrees: an introduction, IV + 77 p. (2012). http://www.math.cornell.edu/~shore/

  60. Simpson, S.G.: The hierarchy based on the jump operator. In: [4], pp. 267–276 (1980)

    Google Scholar 

  61. Simpson, S.G.: FOM: natural r.e. degrees; Pi01 classes. FOM e-mail list [19], 13 August 1999

    Google Scholar 

  62. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, pp. XIV + 445, Second Edition, Springer (1999), Perspectives in Logic, Association for Symbolic Logic, XVI+ 444 p., Cambridge University Press (2009)

    Google Scholar 

  63. Simpson, S.G.: FOM: natural r.e. degrees. FOM e-mail list [19], 27 February 2005

    Google Scholar 

  64. Simpson, S.G.: Mass problems and randomness. Bull. Symbolic Logic 11, 1–27 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  65. Simpson, S.G.: An extension of the recursively enumerable turing degrees. J. Lond. Math. Soc. 75(2), 287–297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  66. Simpson, S.G.: Mass problems and almost everywhere domination. Math. Logic Quarterly 53, 483–492 (2007)

    Article  MATH  Google Scholar 

  67. Simpson, S.G.: Mass problems and intuitionism. Notre Dame J. Formal Logic 49, 127–136 (2008)

    Article  MATH  Google Scholar 

  68. Simpson, S.G.: Mass problems and measure-theoretic regularity. Bull. Symbolic Logic 15, 385–409 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  69. Simpson, S.G.: Mass problems associated with effectively closed sets. Tohoku Math. J. 63(4), 489–517 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  70. Simpson, S.G.: Medvedev degrees of 2-dimensional subshifts of finite type. Ergodic Theor. Dyn. Syst. 34(2), 665–674 (2014). doi:10.1017/etds.2012.152

    Article  Google Scholar 

  71. Slaman, T.A.: Global properties of the turing degrees and the turing jump. In: [11], pp. 83–101 (2008)

    Google Scholar 

  72. Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, XVIII + 437 p., Springer (1987)

    Google Scholar 

  73. Soare, R.I.: Computability and recursion. Bull. Symbolic Logic 2, 284–321 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  74. Sorbi, A., Terwijn, S.A.: Intuitionistic logic and Muchnik degrees. Algebra Univers. 67, 175–188 (2012). doi:10.1007/s00012-012-0176-1

    Article  MATH  MathSciNet  Google Scholar 

  75. Tarski, A., Mostowski, A., Robinson, R.M.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics, IX + 98 p., North-Holland (1953)

    Google Scholar 

  76. Tikhomirov, V.M. (ed.): Selected Works of A.N. Kolmogorov, vol. I, Mathematics and Mechanics. Mathematics and its Applications, Soviet Series, XIX + 551 p., Kluwer Academic Publishers (1991)

    Google Scholar 

  77. Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 42, 230–265 (1936)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, State College, PA, USA

    Stephen G. Simpson

Authors
  1. Stephen G. Simpson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen G. Simpson .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Simpson, S.G. (2015). Degrees of Unsolvability: A Tutorial. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20028-6_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20027-9

  • Online ISBN: 978-3-319-20028-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation