Skip to main content

Some Questions in Computable Mathematics

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

In honor of Rod Downey’s 60th birthday, this paper discusses a few open problems connected in one way or another with him.

To Rod Downey on his 60th Birthday.

Partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation. I thank Russell Miller, Benoit Monin, and Ludovic Patey for useful comments.

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

Buying options

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

Learn about institutional subscriptions

References

  1. Andrews, U., Cai, M., Diamondstone, D., Jockusch, C., Lempp, S.: Asymptotic density, computable traceability, and 1-randomness. Fundam. Math. 234, 41–53 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Astor, E.P., Hirschfeldt, D.R., Jockusch Jr., C.G.: Dense computability, upper cones, and minimal pairs, in preparation

    Google Scholar 

  3. Bienvenu, L., Greenberg, N., Kučera, A., Nies, A., Turetsky, D.: Coherent randomness tests and computing the \(K\)-trivial sets. J. Eur. Math. Soc. 18, 773–812 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonnet, R.: Stratifications et extension des genres de chaînes dénombrables. C. R. Acad. Sci. Ser. A-B 269, A880–A882 (1969)

    MATH  Google Scholar 

  5. Brattka, V.: Maintainer, Bibliography on Weihrauch complexity, Computability and Complexity in Analysis Network. http://cca-net.de/publications/weibib.php

  6. Cholak, P., McCoy, C.: Effective prime uniqueness (to appear)

    Google Scholar 

  7. Cholak, P., Igusa, G.: Density-1-bounding and quasiminimality in the generic degrees. J. Symbolic Logic (to appear)

    Google Scholar 

  8. Cholak, P.A., Jockusch Jr., C.G., Slaman, T.A.: On the strength of Ramsey’s Theorem for pairs. J. Symbolic Logic 66, 1–55 (2001). (Corrigendum in J. Symbolic Logic 74, 1438–1439 (2009))

    Article  MathSciNet  MATH  Google Scholar 

  9. Chong, C.T., Slaman, T.A., Yang, Y.: The metamathematics of stable Ramsey’s Theorem for pairs. J. Am. Math. Soc. 27, 863–892 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chong, C.T., Slaman, T.A., Yang, Y.: The inductive strength of Ramsey’s Theorem for pairs (to appear)

    Google Scholar 

  11. Csima, B.F.: Applications of Computability Theory to Prime Models and Differential Geometry. Ph.D. Dissertation, The University of Chicago (2003)

    Google Scholar 

  12. Csima, B.F.: Degree spectra of prime models. J. Symbolic Logic 69, 430–442 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc. 368, 1321–1359 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Downey, R.: On presentations of algebraic structures. In: Sorbi, A. (ed.) Complexity, Logic, and Recursion Theory, pp. 157–205. Dekker, New York (1997)

    Google Scholar 

  15. Downey, R.G.: Computability theory and linear orderings. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B., Marek, V.W. (eds.) Handbook of Recursive Mathematics, vol. II. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 823–976 (1998)

    Google Scholar 

  16. Downey, R.: Computability, definability, and algebraic structures. In: Downey, R., Decheng, D., Ping, T.S., Hui, Q.Y., Yasugi, M. (eds.) Proceedings of the 7th and 8th Asian Logic Conferences, pp. 63–102. Singapore University Press and World Scientific, Singapore (2003)

    Chapter  Google Scholar 

  17. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, New York (2010)

    Book  MATH  Google Scholar 

  18. Downey, R.G., Hirschfeldt, D.R., LaForte, G.: Randomness and reducibility. J. Comput. Syst. Sci. 68, 96–114 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Downey, R.G., Hirschfeldt, D.R., Lempp, S., Solomon, R.: A \(\Delta ^0_2\) set with no infinite low subset in either it or its complement. J. Symbolic Logic 66, 1371–1381 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Downey, R.G., Hirschfeldt, D.R., Lempp, S., Solomon, R.: Computability-theoretic and proof-theoretic aspects of partial and linear orderings. Isr. J. Math. 138, 271–290 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Downey, R., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31, 1169–1183 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Downey, R.G., Hirschfeldt, D.R., Nies, A., Terwijn, S.A.: Calibrating randomness. Bull. Symbolic Logic 12, 411–491 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Downey, R., Jockusch, C.G.: Every low Boolean algebra is isomorphic to a recursive one. Proc. Am. Math. Soc. 122, 871–880 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Downey, R., Jockusch, C., McNicholl, T.H., Schupp, P.: Asymptotic density and the Ershov Hierarchy. Math. Logic Q. 61, 189–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Downey, R.G., Jockusch Jr., C.G.: On self-embeddings of computable linear orderings. Ann. Pure Appl. Logic 138, 52–76 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Downey, R.G., Jockusch Jr., C.G., Schupp, P.E.: Asymptotic density and computably enumerable sets. J. Math. Logic 13, 43 (2013). 1350005

    Article  MathSciNet  MATH  Google Scholar 

  27. Downey, R.G., Kastermans, B., Lempp, S.: On computable self-embeddings of computable linear orderings. J. Symbolic Logic 74, 1352–1366 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Downey, R.G., Lempp, S.: The proof-theoretic strength of the Dushnik-Miller theorem for countable linear orders. In: Arslanov, M.M., Lempp, S. (eds.) Recursion Theory and Complexity. Series in Logic and Its Applications, vol. 2, pp. 55–57. De Gruyter, Berlin (1999)

    Google Scholar 

  29. Downey, R., Lempp, S., Wu, G.: On the complexity of the successivity relation in computable linear orders. J. Math. Logic 10, 83–99 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Downey, R., Remmel, J.B.: Questions in computable algebra and combinatorics. In: Cholak, P.A., Lempp, S., Lerman, M., Shore, R.A. (eds.) Computability Theory and its Applications (Boulder, CO, 1999). Contemporary Mathematics, vol. 257, pp. 95–125. American Mathematical Society, Providence (2000)

    Google Scholar 

  31. Dushnik, B., Miller, E.W.: Concerning similarity transformations of linearly ordered sets. Bull. Am. Math. Soc. 40, 322–326 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dzhafarov, D.D.: Strong reductions between combinatorial principles. J. Symbolic Logic (to appear)

    Google Scholar 

  33. Dzhafarov, D.D., Patey, L., Solomon, R., Westrick, L.B.: Ramsey’s Theorem for singletons and strong computable reducibility. Proc. Am. Math. Soc. (to appear)

    Google Scholar 

  34. Figueira, S., Hirschfeldt, D.R., Miller, J.S., Ng, K.M., Nies, A.: Counting the changes of random \(\Delta ^0_2\) sets. J. Logic Comput. 25, 1073–1089 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fokina, E.B., Harizanov, V., Melnikov, A.G.: Computable model theory. In: Downey, R. (ed.) Turing’s Legacy: Developments from Turing’s Ideas in Logic. Lecture Notes in Logic, vol. 42, pp. 124–194. Association for Symbolic Logic, La Jolla. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  36. Frolov, A., Harizanov, V., Kalimullin, I., Kudinov, O., Miller, R.: Degree spectra of high\(_n\) and nonlow\(_n\) degrees. J. Logic Comput. 22, 755–777 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gács, P.: Every sequence is reducible to a random one. Inf. Control 70, 186–192 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  38. Goncharov, S.S.: Problem of the number of non-self-equivalent constructivizations. Algebra Logic 19, 401–414 (1980)

    Article  MATH  Google Scholar 

  39. Goncharov, S.S.: Limit equivalent constructivizations. In: Mathematical Logic and the Theory of Algorithms Trudy Instituta Matematiki, vol. 2, pp. 4–12. “Nauka” Sibirskoe otdelenie, Novosibirsk (1982)

    Google Scholar 

  40. Goncharov, S.S.: Countable Boolean Algebras and Decidability. Siberian School of Algebra and Logic. Consultants Bureau, New York (1997)

    MATH  Google Scholar 

  41. Goncharov, S.S., Dzgoev, V.D.: Autostability of models. Algebra Logic 19, 28–37 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Goncharov, S.S., Nurtazin, A.T.: Constructive models of complete decidable theories. Algebra Logic 12, 67–77 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  43. Harizanov, V.S.: Pure computable model theory. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B., Marek, V.W. (eds.) Handbook of Recursive Mathematics. Studies in Logic and the Foundations of Mathematics, vol. 138, pp. 3–114. North-Holland, Amsterdam (1998)

    Google Scholar 

  44. Harrington, L.: Recursively presentable prime models. J. Symbolic Logic 39, 305–309 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  45. Harrison-Trainor, M., Melnikov, A., Miller, R., Montalbán, A.: Computable functors and effective interpretability. J. Symbolic Logic (to appear)

    Google Scholar 

  46. Hirschfeldt, D.R.: Computable trees, prime models, and relative decidability. Proc. Am. Math. Soc. 134, 1495–1498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Hirschfeldt, D.R.: Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles. Lecture Notes Series, vol. 28. World Scientific, Singapore (2014). Institute for Mathematical Sciences, National University of Singapore

    MATH  Google Scholar 

  48. Hirschfeldt, D.R., Jockusch Jr., C.G.: On notions of computability theoretic reduction between \(\Pi ^1_2\) principles. J. Math. Logic 16, 1650002 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hirschfeldt, D.R., Jockusch Jr., C.G., Kjos-Hanssen, B., Lempp, S., Slaman, T.A.: The strength of some combinatorial principles related to Ramsey’s Theorem for pairs. In: Chong, C., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.) Computational Prospects of Infinity, Part II: Presented Talks. Lecture Notes Series, vol. 15, pp. 143–161. World Scientific, Singapore (2008). Institute for Mathematical Sciences, National University of Singapore

    Google Scholar 

  50. Hirschfeldt, D.R., Jockusch Jr., C.G., Kuyper, R., Schupp, P.E.: Coarse reducibility and algorithmic randomness. J. Symbolic Logic (to appear)

    Google Scholar 

  51. Hirschfeldt, D.R., Jockusch Jr., C.G., McNicholl, T., Schupp, P.E.: Asymptotic density and the coarse computability bound. Computability 5, 13–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Hirschfeldt, D.R., Khoussainov, B., Shore, R.A., Slinko, A.M.: Degree spectra and computable dimensions in algebraic structures. Ann. Pure Appl. Logic 115, 71–113 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  53. Hirschfeldt, D.R., Kramer, K., Miller, R., Shlapentokh, A.: Categoricity properties for computable algebraic fields. Trans. Am. Math. Soc. 367, 3981–4017 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Hirschfeldt, D.R., Nies, A., Stephan, F.: Using random sets as oracles. J. Lond. Math. Soc. 75, 610–622 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s Theorem for pairs. J. Symbolic Logic 72, 171–206 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hirst, J.L.: Combinatorics in Subsystems of Second Order Arithmetic. Ph.D. Dissertation, The Pennsylvania State University (1987)

    Google Scholar 

  57. Igusa, G.: Nonexistence of minimal pairs for generic computation. J. Symbolic Logic 78, 511–522 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Igusa, G.: The generic degrees of density-\(1\) sets, and a characterization of the hyperarithmetic reals. J. Symbolic Logic 80, 1290–1314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Jockusch Jr., C.G.: Ramsey’s Theorem and recursion theory. J. Symbolic Logic 37, 268–280 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  60. Jockusch Jr., C.G., Schupp, P.E.: Generic computability, and asymptotic density. J. Lond. Math. Soc. 85, 472–490 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Jockusch, C.G., Soare, R.I.: Degrees of orderings not isomorphic to recursive linear orderings. Ann. Pure Appl. Logic 52, 39–64 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  62. Jullien, P.: Contribution à L’étude des Types D’ordre Dispersés. Ph.D. Dissertation, Université d’Aix-Marseille (1969)

    Google Scholar 

  63. Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory and random walks. J. Algebra 264, 665–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  64. Kastermans, B., Lempp, S.: Comparing notions of randomness. Theoret. Comput. Sci. 411, 602–616 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kalimullin, I., Khoussainov, B., Melnikov, A.: Limitwise monotonic sequences and degree spectra of structures. Proc. Am. Math. Soc. 141, 3275–3289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  67. Knight, J.F.: Degrees coded in jumps of orderings. J. Symbolic Logic 51, 1034–1042 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  68. Knight, J.F., Stob, M.: Computable Boolean algebras. J. Symbolic Logic 65, 1605–1623 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  69. Kučera, A.: Measure, \(\Pi ^0_1\)-classes and complete extensions of PA. In: Ebbinghaus, H.D., Müller, G.H., Sacks, G.E. (eds.) Recursion Theory Week. Lecture Notes in Mathematics, vol. 1141, pp. 245–259. Springer-Verlag, Berlin (1985)

    Chapter  Google Scholar 

  70. LaRoche, P.: Recursively represented Boolean algebras. Not. Am. Math. Soc. 24, A-552 (1977). (research announcement)

    Google Scholar 

  71. Lempp, S., McCoy, C., Miller, R., Solomon, R.: Computable categoricity for trees of finite height. J. Symbolic Logic 70, 151–215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  72. Lempp, S., McCoy, C., Miller, R., Solomon, R.: The computable dimension of trees of infinite height. J. Symbolic Logic 70, 111–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  73. Lewis, A.E.M., Barmpalias, G.: Random reals and Lipschitz continuity. Math. Struct. Comput. Sci. 16, 737–749 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  74. Lewis, A.E.M., Barmpalias, G.: Randomness and the linear degrees of computability. Ann. Pure Appl. Logic 145, 252–257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  75. Liu, J.: RT\(^2_2\) does not imply WKL\(_0\). J. Symbolic Logic 77, 609–620 (2012)

    Article  MathSciNet  Google Scholar 

  76. Liu, L.: Cone avoiding closed sets. Trans. Am. Math. Soc. 367, 1609–1630 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  77. Lynch, N.: Approximations to the halting problem. J. Comput. Syst. Sci. 9, 143–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  78. Melnikov, A.G.: Enumerations and completely decomposable torsion-free abelian groups. Theor. Comput. Syst. 45, 897–916 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  79. Merkle, W.: The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences. J. Symbolic Logic 68, 1362–1376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  80. Merkle, W., Mihailović, N.: On the construction of effectively random sets. J. Symbolic Logic 69, 862–878 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  81. Merkle, W., Miller, J.S., Nies, A., Reimann, J., Stephan, F.: Kolmogorov-Loveland randomness and stochasticity. Ann. Pure Appl. Logic 138, 183–210 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  82. Meyer, A.R.: An open problem on creative sets. Recursive Funct. Theor. Newsl. 4, 15–16 (1973)

    Google Scholar 

  83. Mileti, J.R., Partition Theorems and Computability Theory. Ph.D. Dissertation, University of Illinois at Urbana-Champaign (2004)

    Google Scholar 

  84. Millar, T.S.: Foundations of recursive model theory. Ann. Math. Logic 13, 45–72 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  85. Millar, T.S.: Omitting types, type spectrums, and decidability. J. Symbolic Logic 48, 171–181 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  86. Miller, J.S., Nies, A.: Randomness and computability: open questions. Bull. Symbolic Logic 12, 390–410 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  87. Miller, R.G.: Computability, Definability, Categoricity, and Automorphisms. Ph.D. Dissertation, The University of Chicago (2000)

    Google Scholar 

  88. Miller, R.: The \(\Delta ^0_2\)-spectrum of a linear order. J. Symbolic Logic 66, 470–486 (2001)

    Article  MathSciNet  Google Scholar 

  89. Miller, R.: \({\bf d}\)-computable categoricity for algebraic fields. J. Symbolic Logic 74, 1325–1351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  90. Miller, R., Poonen, B., Schoutens, H., Shlapentokh, A.: A computable functor from graphs to fields (to appear)

    Google Scholar 

  91. Monin, B.: Asymptotic density and error-correcting codes (to appear)

    Google Scholar 

  92. Monin, B., Nies, A.: A unifying approach to the Gamma question. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, pp. 585–596. IEEE Computer Society (2015)

    Google Scholar 

  93. Monin, B., Patey, L.: \(\Pi ^0_1\) encodability and omniscient reductions (to appear)

    Google Scholar 

  94. Montalbán, A.: Equivalence between Fraïssé’s Conjecture and Jullien’s Theorem. Ann. Pure Appl. Logic 139, 1–42 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  95. Muchnik, A.A., Semenov, A.L., Uspensky, V.A.: Mathematical metaphysics of randomness. Theoret. Comput. Sci. 207, 263–317 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  96. Patey, L.: The weakness of being cohesive, thin or free in reverse mathematics. Isr. J. Math. (to appear)

    Google Scholar 

  97. Patey, L.: Open questions about Ramsey-type statements in reverse mathematics. Bull. Symbolic Logic 22, 151–169 (2016)

    Article  MathSciNet  Google Scholar 

  98. Remmel, J.B.: Recursively categorical linear orderings. Proc. Am. Math. Soc. 83, 387–391 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  99. Richter, L.J.: Degrees of structures. J. Symbolic Logic 46, 723–731 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  100. Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press Inc., New York-London (1982)

    MATH  Google Scholar 

  101. Schnorr, C.-P.: A unified approach to the definition of a random sequence. Math. Syst. Theor. 5, 246–258 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  102. Schnorr, C.-P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Springer, Berlin (1971)

    Google Scholar 

  103. Schweber, N.: Do all linear orders in this class have computable copies? (2014). mathoverflow.net/questions/161434

  104. Schweber, N.: Finding limit-nondecreasing sets for certain functions (2016). mathoverflow.net/questions/227766

  105. Seetapun, D., Slaman, T.A.: On the strength of Ramsey’s Theorem. Notre Dame J. Formal Logic 36, 570–582 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  106. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, 1st edn. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  107. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge and Association for Symbolic Logic, Poughkeepsie (2009)

    Book  MATH  Google Scholar 

  108. Slaman, T.A.: Relative to any nonrecursive set. Proc. Am. Math. Soc. 126, 2117–2122 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  109. Solovay, R.M.: Hyperarithmetically encodable sets. Trans. Am. Math. Soc. 239, 99–122 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  110. Soskov, I.N.: Degree spectra and co-spectra of structures, Annuaire de l’Université de Sofia “St. Kliment Ohrisdski", Faculté de Mathématiques et Informatique 96, 45–68(2004)

    Google Scholar 

  111. Specker, E.: Ramsey’s Theorem does not hold in recursive set theory. In: Gandy, R.O., Yates, C.E.M. (eds.) Logic Colloquium 1969. Studies in Logic and the Foundations of Mathematics, pp. 439–442. North-Holland, Amsterdam (1971)

    Google Scholar 

  112. Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundam. Math. 16, 386–389 (1930)

    MATH  Google Scholar 

  113. Terwijn, S.A.: Computability and Measure. Ph.D. Dissertation, University of Amsterdam (1998)

    Google Scholar 

  114. Thurber, J.: Degrees of Boolean Algebras. Ph.D. Dissertation, University of Notre Dame (1994)

    Google Scholar 

  115. Wang, Y.: Randomness and Complexity. Ph.D. Dissertation, University of Heidelberg (1996)

    Google Scholar 

  116. Wang, Y.: A separation of two randomness concepts. Inf. Process. Lett. 69, 115–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  117. Wehner, S.: Enumerations, countable structures, and Turing degrees. Proc. Am. Math. Soc. 126, 2131–2139 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  118. Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Technical report TR-92-050. International Computer Science Institute, Berkeley (1992)

    Google Scholar 

  119. Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. In: Informatik Berichte FernUniversität Hagen, vol. 130. Hagen (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Denis R. Hirschfeldt .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Hirschfeldt, D.R. (2017). Some Questions in Computable Mathematics. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50062-1_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics