Tohoku Mathematical Journal

Mass problems associated with effectively closed sets

Stephen G. Simpson

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The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov's non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let $\mathcal{E}_\mathrm{w}$ be this lattice. We show that $\mathcal{E}_\mathrm{w}$ provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in $\mathcal{E}_\mathrm{w}$ which are associated with such problems. In addition, we present some structural results concerning the lattice $\mathcal{E}_\mathrm{w}$. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how $\mathcal{E}_\mathrm{w}$ can be applied in symbolic dynamics, toward the classification of tiling problems and $\boldsymbol{Z}^d$-subshifts of finite type.

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Tohoku Math. J. (2), Volume 63, Number 4 (2011), 489-517.

First available in Project Euclid: 6 January 2012

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Zentralblatt MATH identifier

Primary: 03D30: Other degrees and reducibilities
Secondary: 03D20: Recursive functions and relations, subrecursive hierarchies 03D25: Recursively (computably) enumerable sets and degrees 03D28: Other Turing degree structures 03D32: Algorithmic randomness and dimension [See also 68Q30] 03D55: Hierarchies 03D80: Applications of computability and recursion theory 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Mass problems unsolvable problems degrees of unsolvability Muchnik degrees algorithmic randomness Kolmogorov complexity resourse-bounded computational complexity recursively enumerable degrees hyperarithmetical hierarchy intuitionism proof theory


Simpson, Stephen G. Mass problems associated with effectively closed sets. Tohoku Math. J. (2) 63 (2011), no. 4, 489--517. doi:10.2748/tmj/1325886278.

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  • S. Aanderaa and D. E. Cohen, Modular machines I, II, in [wp2?], Stud. Logic Found. Math. 95 (1980), 1–18, 19–28.
  • S. I. Adian, W. W. Boone and G. Higman (eds.), Word problems II: The Oxford Book, Stud. Logic Found. Math. 95, North-Holland Publishing Co., Amsterdam, New York, 1980.
  • K. Ambos-Spies, B. Kjos-Hanssen, S. Lempp and T. A. Slaman, Comparing DNR and WWKL, J. Symbolic Logic 69 (2004), 1089–1104.
  • J. Barwise, H. J. Keisler and K. Kunen (eds.), The Kleene Symposium, Stud. Logic Found. Math. 101, North-Holland, Publishing Co., Amsterdam-New York, 1980.
  • R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66, American Mathematical Society, 1966.
  • S. Binns, A splitting theorem for the Medvedev and Muchnik lattices, MLQ Math. Log. Q. 49 (2003), 327–335.
  • S. Binns, Small $\Pi^0_1$ classes, Arch. Math. Logic 45 (2006), 393–410.
  • S. Binns, Hyperimmunity in $2^{\boldsymbol{N}}$, Notre Dame J. Formal Logic 48 (2007), 293–316.
  • S. Binns, $\Pi^0_1$ classes with complex elements, J. Symbolic Logic 73 (2008), 1341–1353.
  • S. Binns, B. Kjos-Hanssen, M. Lerman and D. R. Solomon, On a question of Dobrinen and Simpson concerning almost everywhere domination, J. Symbolic Logic 71 (2006), 119–136.
  • S. Binns and S. G. Simpson, Embeddings into the Medvedev and Muchnik lattices of $\Pi^0_1$ classes, Arch. Math. Logic 43 (2004), 399–414.
  • G. Birkhoff, Lattice theory, American Mathematical Society, 1940.
  • G. Birkhoff, Lattice theory, revised edition, Amer. Math. Soc. Colloq. Pub. 25, American Mathematical Society, New York, 1948.
  • G. Birkhoff, Lattice theory, third edition, Amer. Math. Soc. Colloq. Pub. 25, American Mathematical Society, Providence, R.I., 1967.
  • M. Boyle, Open problems in symbolic dynamics, Contemp. Math. 469, 69–118 in [cm-469?], Amer. Math. Soc., Providence, RI, 2008.
  • D. K. Brown, M. Giusto and S. G. Simpson, Vitali's theorem and WWKL, Arch. Math. Logic 41 (2002), 191–206.
  • K. Burns, D. Dolgopyat and Y. Pesin (eds.), Geometric and probabilistic structures in dynamics, Contemp. Math. 469, American Mathematical Society, Providence, RI, 2008.
  • J.-Y. Cai, S. B. Cooper and A. Li (eds.), Theory and applications of models of computation, Lecture Notes in Comput. Sci. 3959, Springer-Verlag, Berlin, 2006.
  • P. Cholak, R. Coles, R. Downey and E. Herrmann, Automorphisms of the lattice of $\Pi^0_1$ classes; perfect thin classes and ANC degrees, Trans. Amer. Math. Soc. 353 (2001), 4899–4924.
  • P. Cholak, N. Greenberg and J. S. Miller, Uniform almost everywhere domination, J. Symbolic Logic 71 (2006), 1057–1072.
  • C.-T. Chong, Q. Feng, T. A. Slaman, W. H. Woodin and Y. Yang (eds.), Computational Prospects of Infinity: Proceedings of the Logic Workshop at the Institute for Mathematical Sciences, June 20–-August 15, 2005, Part II: Presented Talks, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 15, World Scientific Publishing Co. Pre. Ltd., Hackensack, NJ, 2008.
  • J. A. Cole and S. G. Simpson, Mass problems and hyperarithmeticity, J. Math. Logic 7 (2008), 125–143.
  • B. A. Davey and H. A. Priestley, Introduction to lattices and order, Cambridge University Press, 1990.
  • M. Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233–269.
  • A. R. Day and J. S. Miller, Randomness for non-computable measures, to appear in Trans. Amer. Math. Soc.
  • J. C. E. Dekker (ed.), Recursive function theory, Proc. Symp. Pure Math., American Mathematical Society, Providence, R.I., 1962.
  • N. L. Dobrinen and S. G. Simpson, Almost everywhere domination, J. Symbolic Logic 69 (2004), 914–922.
  • R. G. Downey and D. Hirschfeldt, Algorithmic randomness and complexity, Theory Appl. Comput., Springer-Verlag, New York, 2010.
  • B. Durand and W. Thomas (eds.), STACS 2006: Proceedings of the Twenty-Third Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23–25, 2006, Lecture Notes in Computer Science 3884, Springer-Verlag, Berlin, 2006.
  • B. Durand, A. Romashchenko and A. Shen, Fixed-point tile sets and their applications, 2010, arXiv:0910.2415v5.pdf.
  • H.-D. Ebbinghaus, G. H. Müller and G. E. Sacks (eds.), Recursion theory week, Lecture Notes in Math. 1141, Springer-Verlag, 1985.
  • K. Falconer, Fractal geometry, 2nd edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003.
  • S. Feferman, C. Parsons and S. G. Simpson (eds.), Kurt Gödel: essays for his centennial, Lect. Notes in Log. 33, Assoc. Symbol. Logic, Cambridge University Press, Cambridge, 2010.
  • J.-E. Fenstad, I. T. Frolov and R. Hilpinen (eds.), Logic, methodology and philosophy of science VIII, Stud. Logic Found. Math. 126, North-Holland, Amsterdam, 1989.
  • FOM e-mail list,, September 1997 to the present.
  • R. O. Gandy, G. Kreisel and W. W. Tait, Set existence, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 577–582.
  • K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I, Monatsh. Math. Phys. 38 (1931), 173–198.
  • O. Goldreich, Computational complexity: a conceptual perspective, Cambridge University Press, Cambridge, 2008.
  • D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437–479.
  • P. G. Hinman, Recursion-theoretic hierarchies, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, New York, 1978.
  • J. M. Hitchcock, J. H. Lutz and E. Mayordomo, The fractal geometry of complexity classes, SIGACT News 36 (2005), 24–38.
  • M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Ann. Math. 171 (2010), 2011–2038.
  • P. Hudelson, Mass problems and initial segment complexity, 2010, in preparation.
  • C. G. Jockusch, Jr., Degrees of functions with no fixed points, in [lmps8?] (1989), 191–201.
  • C. G. Jockusch, Jr. and R. A. Shore, Pseudo-jump operators I: the r.e. case, Trans. Amer. Math. Soc. 275 (1983), 599–609.
  • C. G. Jockusch, Jr. and R. A. Shore, Pseudo-jump operators II: transfinite iterations, hierarchies, and minimal covers, J. Symbolic Logic 49 (1984), 1205–1236.
  • C. G. Jockusch, Jr. and S. G. Simpson, A degree theoretic definition of the ramified analytical hierarchy, Ann. Math. Logic 10 (1976), 1–32.
  • C. G. Jockusch, Jr. and R. I. Soare, Degrees of members of $\Pi^0_1$ classes, Pacific J. Math. 40 (1972), 605–616.
  • C. G. Jockusch, Jr. and R. I. Soare, $\Pi^0_1$ classes and degrees of theories, Trans. Amer. Math. Soc. 173 (1972), 35–56.
  • A. S. Kechris, Classical descriptive set theory, Grad. Texts in Math. 156, Springer-Verlag, New York, 1995.
  • T. Kent and A. E. M. Lewis, On the degree spectrum of a $\Pi^0_1$ class, Trans. Amer. Math. Soc. 362 (2010), 5283–5319.
  • B. Kjos-Hanssen, Low-for-random reals and positive-measure domination, Proc. Amer. Math. Soc. 135 (2007), 3703–3709.
  • B. Kjos-Hanssen, W. Merkle and F. Stephan, Kolmogorov complexity and the recursion theorem, 149–161 in [stacs06?], Lecture Notes in Comput. Sci. 3884, Springer, Berlin, 2006.
  • B. Kjos-Hanssen, J. S. Miller and D. R. Solomon, Lowness notions, measure and domination, to appear in Proc. London Math. Soc.
  • B. Kjos-Hanssen and S. G. Simpson, Mass problems and Kolmogorov complexity, October 2006, in preparation.
  • S. C. Kleene and E. L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. Math. 59 (1954), 379–407.
  • A. Kolmogoroff, Zur Deutung der intuitionistischen Logik, Math. Z. 35 (1932), 58–65.
  • A. N. Kolmogorov, On the interpretation of intuitionistic logic, in [k-sel-1?] (1991), 151–158, 451–466. (Translation of [k-iil?] with commentary and additional references.)
  • M. Kumabe and A. E. M. Lewis, A fixed-point-free minimal degree, J. London Math. Soc. (2) 80 (2009), 785–797.
  • A. Kučera, Measure, $\Pi^0_1$ classes and complete extensions of PA, Recursion theory week (Oberwolfach, 1984), 245–259 in [rtw85?], Lecture Notes in Math. 1141, Springer, Berlin, 1985.
  • M. Lerman, A framework for priority arguments, Lecture Notes in Logic 34, Association for Symbolic Logic, Cambridge University Press, Cambridge, 2010.
  • M. Li and P. Vitányi, An introduction to Kolmogorov complexity and its applications, 2nd edition, Grad. Texts in Comput. Sci., Springer-Verlag, New York, 1997.
  • D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
  • S. MacLane and I. Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, Universitext, Springer-Verlag, New York, 1992.
  • R. Mansfield and G. Weitkamp, Recursive aspects of descriptive set theory, Oxford Logic Guides 11, Oxford University Press, New York, 1985.
  • P. Martin-Löf, The definition of random sequences, Information and Control 9 (1966), 602–619.
  • Y. T. Medvedev, Degrees of difficulty of mass problems, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 501–504.
  • J. S. Miller, Extracting information is hard: a Turing degree of non-integral effective Hausdorff dimension, Adv. Math. 226 (2011), 373–384.
  • Y. N. Moschovakis, Descriptive set theory, Stud. Logic Found. Math. 100, North-Holland Publishing Co., Amsterdam, New York, 1980.
  • S. Mozes, Tilings, substitution systems, and the dynamical systems generated by them, J. Analyse Math. 53 (1989), 139–186.
  • A. A. Muchnik, On strong and weak reducibilities of algorithmic problems, Sibirskii Matematicheskii Zhurnal 4 (1963), 1328–1341.
  • D. Myers, Nonrecursive tilings of the plane, II, J. Symbolic Logic 39 (1974), 286–294.
  • A. Nabutovsky, Einstein structures: existence versus uniqueness, Geom. Funct. Anal. 5 (1995), 76–91.
  • A. Nabutovsky and S. Weinberger, Betti numbers of finitely presented groups and very rapidly growing functions, Topology 46 (2007), 211–233.
  • A. Nies, Lowness properties and randomness, Adv. Math. 197 (2005), 274–305.
  • A. Nies, Computability and randomness, Oxford Logic Guides 51, Oxford University Press, Oxford, 2009.
  • A. Nies, R. A. Shore and T. A. Slaman, Interpretability and definability in the recursively enumerable degrees, Proc. London Math. Soc. (3) 77 (1998), 241–291.
  • D. B. Posner and R. W. Robinson, Degrees joining to $0'$, J. Symbolic Logic 46 (1981), 714–722.
  • M. B. Pour-El and S. Kripke, Deduction-preserving “recursive isomorphisms” between theories, Fund. Math. 61 (1967), 141–163.
  • M. O. Rabin, Recursive unsolvability of group theoretic problems, Ann. Math. 67 (1958), 172–194.
  • D. Richardson, Some undecidable problems involving elementary functions of a real variable, J. Symbolic Logic 33 (1968), 514–520.
  • R. M. Robinson, Undecidability and nonperiodicity of tilings of the plane, Invent. Math. 12 (1971), 177–209.
  • H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967.
  • G. E. Sacks, The recursively enumerable degrees are dense, Ann. Math. 80 (1964), 300–312.
  • G. E. Sacks, Degrees of unsolvability, 2nd edition, Ann. Math. Studies 55, Princeton University Press, Princeton, 1966.
  • G. E. Sacks, Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.
  • K. Schütte, Proof theory, Grundlehren der Mathematischen Wissenschaften 225, Springer-Verlag, Berlin-New York, 1977.
  • D. S. Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proc. Sympos. Pure Math. 5, 117–121 in [rec-fcn-thy?], American Mathematical Society, Providence, R.I., 1962.
  • D. S. Scott and S. Tennenbaum, On the degrees of complete extensions of arithmetic (abstract), Notices Amer. Math. Soc. 7 (1960), 242–243.
  • R. A. Shore and T. A. Slaman, Defining the Turing jump, Math. Res. Lett. 6 (1999), 711–722.
  • S. G. Simpson (ed.), Reverse mathematics 2001, Lecture Notes in Logic 21, Association for Symbolic Logic, La Jolla, CA; A. K. Peters, Ltd., Wellesley, MA, 2005.
  • S. G. Simpson, The hierarchy based on the jump operator, in [kleene-symp?] (1980), 203–212.
  • S. G. Simpson, FOM: natural r.e. degrees; Pi01 classes, FOM e-mail list [fom?], 13 August, 1999.
  • S. G. Simpson, FOM: priority arguments; Kleene-r.e. degrees; Pi01 classes, FOM e-mail list [fom?], 16 August, 1999.
  • S. G. Simpson, Subsystems of second order arithmetic, Perspect. in Math. Logic, Springer-Verlag, Berlin, 1999; Second Edition, Perspect. Logic, Association for Symbolic Logic, Cambridge University Press, Cambridge, 2009.
  • S. G. Simpson, Mass problems and randomness, Bull. Symbolic Logic 11 (2005), 1–27.
  • S. G. Simpson, $\Pi^0_1$ sets and models of $\mathsf{WKL}_0$, Reverse mathematics 2001, 352–378 in [rm2001?], Lect. Notes Log. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005.
  • S. G. Simpson, Almost everywhere domination and superhighness, MLQ Math. Log. Q. 53 (2007), 462–482.
  • S. G. Simpson, An extension of the recursively enumerable Turing degrees, J. London Math. Soc. 75 (2007), 287–297.
  • S. G. Simpson, Mass problems and almost everywhere domination, MLQ Math. Log. Q. 53 (2007), 483–492.
  • S. G. Simpson, Mass problems and intuitionism, Notre Dame J. Form. Log. 49 (2008), 127–136.
  • S. G. Simpson, Some fundamental issues concerning degrees of unsolvability, in [cpoi?] (2008), 313–332.
  • S. G. Simpson, Mass problems and measure-theoretic regularity, Bull. Symbolic Logic 15 (2009), 385–409.
  • S. G. Simpson, The Gödel hierarchy and reverse mathematics, Kurt Gödel: essays for his centennial, 109–127 in [godel-fps?], Lect. Notes Log. 33, Assoc. Symbol. Logic, La Jolla, CA, 2010.
  • S. G. Simpson, Medvedev degrees of 2-dimensional subshifts of finite type, to appear in Ergodic Theory and Dynamical Systems.
  • S. G. Simpson, Symbolic dynamics: entropy $=$ dimension $=$ complexity, 2010, in preparation.
  • R. I. Soare, Recursively enumerable sets and degrees, Perspect. Math. Logic, Springer-Verlag, Berlin, 1987.
  • A. Sorbi and S. A. Terwijn, Intuitionistic logic and Muchnik degrees, preprint, 2010, arXiv:1003.4489v1.
  • J. R. Steel, Descending sequences of degrees, J. Symbolic Logic 40 (1975), 59–61.
  • A. Stukachev, On mass problems of presentability, Theory and applications of models of computation, 772–782 in [lncs3959?], Lecture Notes in Comput. Sci. 3959, Springer, Berlin, 2006.
  • G. Takeuti, Proof theory, 2nd edition, Stud. Logic Found. Math. 81. North-Holland Publishing Co., Amsterdam, 1987.
  • A. Tarski, A. Mostowski and R. M. Robinson, Undecidable theories, Stud. Logic Found. Math., North-Holland Publishing Co., Amsterdam, 1953.
  • V. M. Tikhomirov (ed.), Selected works of A. N. Kolmogorov, Volume I, mathematics and mechanics, Math. Appl. (Soviet Series) 25, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • A. S. Troelstra and D. van Dalen, Constructivism in mathematics, an Introduction, Vol. I, II, Stud. Log. Found. Math. 121, 123, North-Holland Publishing Co., Amsterdam, 1988.
  • A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42 (1936), 230–265.
  • S. S. Wainer, A classification of the ordinal recursive functions, Arch. Math. Logik und Grundlagenforsch. 13 (1970), 136–153.
  • H. Wang, Proving theorems by pattern recognition, II, Bell System Tech. J. 40 (1961), 1–42.
  • R. Weber, Invariance in $\mathcal{E}^*$ and $\mathcal{E}_\Pi$, Trans. Amer. Math. Soc. 358 (2006), 3023–3059.
  • X. Yu and S. G. Simpson, Measure theory and weak König's lemma, Arch. Math. Logic 30 (1990), 171–180.