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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andrews, U., Cai, M., Diamondstone, D., Jockusch, C., Lempp, S.: Asymptotic density, computable traceability, and 1-randomness. Fundam. Math. 234, 41–53 (2016)
Astor, E.P., Hirschfeldt, D.R., Jockusch Jr., C.G.: Dense computability, upper cones, and minimal pairs, in preparation
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)
Bonnet, R.: Stratifications et extension des genres de chaînes dénombrables. C. R. Acad. Sci. Ser. A-B 269, A880–A882 (1969)
Brattka, V.: Maintainer, Bibliography on Weihrauch complexity, Computability and Complexity in Analysis Network. http://cca-net.de/publications/weibib.php
Cholak, P., McCoy, C.: Effective prime uniqueness (to appear)
Cholak, P., Igusa, G.: Density-1-bounding and quasiminimality in the generic degrees. J. Symbolic Logic (to appear)
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))
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)
Chong, C.T., Slaman, T.A., Yang, Y.: The inductive strength of Ramsey’s Theorem for pairs (to appear)
Csima, B.F.: Applications of Computability Theory to Prime Models and Differential Geometry. Ph.D. Dissertation, The University of Chicago (2003)
Csima, B.F.: Degree spectra of prime models. J. Symbolic Logic 69, 430–442 (2004)
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)
Downey, R.: On presentations of algebraic structures. In: Sorbi, A. (ed.) Complexity, Logic, and Recursion Theory, pp. 157–205. Dekker, New York (1997)
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)
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)
Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, New York (2010)
Downey, R.G., Hirschfeldt, D.R., LaForte, G.: Randomness and reducibility. J. Comput. Syst. Sci. 68, 96–114 (2004)
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)
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)
Downey, R., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31, 1169–1183 (2002)
Downey, R.G., Hirschfeldt, D.R., Nies, A., Terwijn, S.A.: Calibrating randomness. Bull. Symbolic Logic 12, 411–491 (2006)
Downey, R., Jockusch, C.G.: Every low Boolean algebra is isomorphic to a recursive one. Proc. Am. Math. Soc. 122, 871–880 (1994)
Downey, R., Jockusch, C., McNicholl, T.H., Schupp, P.: Asymptotic density and the Ershov Hierarchy. Math. Logic Q. 61, 189–195 (2015)
Downey, R.G., Jockusch Jr., C.G.: On self-embeddings of computable linear orderings. Ann. Pure Appl. Logic 138, 52–76 (2006)
Downey, R.G., Jockusch Jr., C.G., Schupp, P.E.: Asymptotic density and computably enumerable sets. J. Math. Logic 13, 43 (2013). 1350005
Downey, R.G., Kastermans, B., Lempp, S.: On computable self-embeddings of computable linear orderings. J. Symbolic Logic 74, 1352–1366 (2009)
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)
Downey, R., Lempp, S., Wu, G.: On the complexity of the successivity relation in computable linear orders. J. Math. Logic 10, 83–99 (2010)
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)
Dushnik, B., Miller, E.W.: Concerning similarity transformations of linearly ordered sets. Bull. Am. Math. Soc. 40, 322–326 (1940)
Dzhafarov, D.D.: Strong reductions between combinatorial principles. J. Symbolic Logic (to appear)
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)
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)
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)
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)
Gács, P.: Every sequence is reducible to a random one. Inf. Control 70, 186–192 (1986)
Goncharov, S.S.: Problem of the number of non-self-equivalent constructivizations. Algebra Logic 19, 401–414 (1980)
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)
Goncharov, S.S.: Countable Boolean Algebras and Decidability. Siberian School of Algebra and Logic. Consultants Bureau, New York (1997)
Goncharov, S.S., Dzgoev, V.D.: Autostability of models. Algebra Logic 19, 28–37 (1980)
Goncharov, S.S., Nurtazin, A.T.: Constructive models of complete decidable theories. Algebra Logic 12, 67–77 (1973)
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)
Harrington, L.: Recursively presentable prime models. J. Symbolic Logic 39, 305–309 (1974)
Harrison-Trainor, M., Melnikov, A., Miller, R., Montalbán, A.: Computable functors and effective interpretability. J. Symbolic Logic (to appear)
Hirschfeldt, D.R.: Computable trees, prime models, and relative decidability. Proc. Am. Math. Soc. 134, 1495–1498 (2006)
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
Hirschfeldt, D.R., Jockusch Jr., C.G.: On notions of computability theoretic reduction between \(\Pi ^1_2\) principles. J. Math. Logic 16, 1650002 (2016)
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
Hirschfeldt, D.R., Jockusch Jr., C.G., Kuyper, R., Schupp, P.E.: Coarse reducibility and algorithmic randomness. J. Symbolic Logic (to appear)
Hirschfeldt, D.R., Jockusch Jr., C.G., McNicholl, T., Schupp, P.E.: Asymptotic density and the coarse computability bound. Computability 5, 13–27 (2016)
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)
Hirschfeldt, D.R., Kramer, K., Miller, R., Shlapentokh, A.: Categoricity properties for computable algebraic fields. Trans. Am. Math. Soc. 367, 3981–4017 (2015)
Hirschfeldt, D.R., Nies, A., Stephan, F.: Using random sets as oracles. J. Lond. Math. Soc. 75, 610–622 (2007)
Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s Theorem for pairs. J. Symbolic Logic 72, 171–206 (2007)
Hirst, J.L.: Combinatorics in Subsystems of Second Order Arithmetic. Ph.D. Dissertation, The Pennsylvania State University (1987)
Igusa, G.: Nonexistence of minimal pairs for generic computation. J. Symbolic Logic 78, 511–522 (2013)
Igusa, G.: The generic degrees of density-\(1\) sets, and a characterization of the hyperarithmetic reals. J. Symbolic Logic 80, 1290–1314 (2015)
Jockusch Jr., C.G.: Ramsey’s Theorem and recursion theory. J. Symbolic Logic 37, 268–280 (1972)
Jockusch Jr., C.G., Schupp, P.E.: Generic computability, and asymptotic density. J. Lond. Math. Soc. 85, 472–490 (2012)
Jockusch, C.G., Soare, R.I.: Degrees of orderings not isomorphic to recursive linear orderings. Ann. Pure Appl. Logic 52, 39–64 (1991)
Jullien, P.: Contribution à L’étude des Types D’ordre Dispersés. Ph.D. Dissertation, Université d’Aix-Marseille (1969)
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)
Kastermans, B., Lempp, S.: Comparing notions of randomness. Theoret. Comput. Sci. 411, 602–616 (2010)
Kalimullin, I., Khoussainov, B., Melnikov, A.: Limitwise monotonic sequences and degree spectra of structures. Proc. Am. Math. Soc. 141, 3275–3289 (2013)
Kjos-Hanssen, B., Merkle, W., Stephan, F.: Kolmogorov complexity and the recursion theorem. Trans. Am. Math. Soc. 363, 5465–5480 (2011)
Knight, J.F.: Degrees coded in jumps of orderings. J. Symbolic Logic 51, 1034–1042 (1986)
Knight, J.F., Stob, M.: Computable Boolean algebras. J. Symbolic Logic 65, 1605–1623 (2000)
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)
LaRoche, P.: Recursively represented Boolean algebras. Not. Am. Math. Soc. 24, A-552 (1977). (research announcement)
Lempp, S., McCoy, C., Miller, R., Solomon, R.: Computable categoricity for trees of finite height. J. Symbolic Logic 70, 151–215 (2005)
Lempp, S., McCoy, C., Miller, R., Solomon, R.: The computable dimension of trees of infinite height. J. Symbolic Logic 70, 111–141 (2005)
Lewis, A.E.M., Barmpalias, G.: Random reals and Lipschitz continuity. Math. Struct. Comput. Sci. 16, 737–749 (2006)
Lewis, A.E.M., Barmpalias, G.: Randomness and the linear degrees of computability. Ann. Pure Appl. Logic 145, 252–257 (2007)
Liu, J.: RT\(^2_2\) does not imply WKL\(_0\). J. Symbolic Logic 77, 609–620 (2012)
Liu, L.: Cone avoiding closed sets. Trans. Am. Math. Soc. 367, 1609–1630 (2015)
Lynch, N.: Approximations to the halting problem. J. Comput. Syst. Sci. 9, 143–150 (1974)
Melnikov, A.G.: Enumerations and completely decomposable torsion-free abelian groups. Theor. Comput. Syst. 45, 897–916 (2009)
Merkle, W.: The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences. J. Symbolic Logic 68, 1362–1376 (2003)
Merkle, W., Mihailović, N.: On the construction of effectively random sets. J. Symbolic Logic 69, 862–878 (2004)
Merkle, W., Miller, J.S., Nies, A., Reimann, J., Stephan, F.: Kolmogorov-Loveland randomness and stochasticity. Ann. Pure Appl. Logic 138, 183–210 (2006)
Meyer, A.R.: An open problem on creative sets. Recursive Funct. Theor. Newsl. 4, 15–16 (1973)
Mileti, J.R., Partition Theorems and Computability Theory. Ph.D. Dissertation, University of Illinois at Urbana-Champaign (2004)
Millar, T.S.: Foundations of recursive model theory. Ann. Math. Logic 13, 45–72 (1978)
Millar, T.S.: Omitting types, type spectrums, and decidability. J. Symbolic Logic 48, 171–181 (1983)
Miller, J.S., Nies, A.: Randomness and computability: open questions. Bull. Symbolic Logic 12, 390–410 (2006)
Miller, R.G.: Computability, Definability, Categoricity, and Automorphisms. Ph.D. Dissertation, The University of Chicago (2000)
Miller, R.: The \(\Delta ^0_2\)-spectrum of a linear order. J. Symbolic Logic 66, 470–486 (2001)
Miller, R.: \({\bf d}\)-computable categoricity for algebraic fields. J. Symbolic Logic 74, 1325–1351 (2009)
Miller, R., Poonen, B., Schoutens, H., Shlapentokh, A.: A computable functor from graphs to fields (to appear)
Monin, B.: Asymptotic density and error-correcting codes (to appear)
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)
Monin, B., Patey, L.: \(\Pi ^0_1\) encodability and omniscient reductions (to appear)
Montalbán, A.: Equivalence between Fraïssé’s Conjecture and Jullien’s Theorem. Ann. Pure Appl. Logic 139, 1–42 (2006)
Muchnik, A.A., Semenov, A.L., Uspensky, V.A.: Mathematical metaphysics of randomness. Theoret. Comput. Sci. 207, 263–317 (1998)
Patey, L.: The weakness of being cohesive, thin or free in reverse mathematics. Isr. J. Math. (to appear)
Patey, L.: Open questions about Ramsey-type statements in reverse mathematics. Bull. Symbolic Logic 22, 151–169 (2016)
Remmel, J.B.: Recursively categorical linear orderings. Proc. Am. Math. Soc. 83, 387–391 (1981)
Richter, L.J.: Degrees of structures. J. Symbolic Logic 46, 723–731 (1981)
Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press Inc., New York-London (1982)
Schnorr, C.-P.: A unified approach to the definition of a random sequence. Math. Syst. Theor. 5, 246–258 (1971)
Schnorr, C.-P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Springer, Berlin (1971)
Schweber, N.: Do all linear orders in this class have computable copies? (2014). mathoverflow.net/questions/161434
Schweber, N.: Finding limit-nondecreasing sets for certain functions (2016). mathoverflow.net/questions/227766
Seetapun, D., Slaman, T.A.: On the strength of Ramsey’s Theorem. Notre Dame J. Formal Logic 36, 570–582 (1995)
Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, 1st edn. Springer, Berlin (1999)
Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge and Association for Symbolic Logic, Poughkeepsie (2009)
Slaman, T.A.: Relative to any nonrecursive set. Proc. Am. Math. Soc. 126, 2117–2122 (1998)
Solovay, R.M.: Hyperarithmetically encodable sets. Trans. Am. Math. Soc. 239, 99–122 (1978)
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)
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)
Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundam. Math. 16, 386–389 (1930)
Terwijn, S.A.: Computability and Measure. Ph.D. Dissertation, University of Amsterdam (1998)
Thurber, J.: Degrees of Boolean Algebras. Ph.D. Dissertation, University of Notre Dame (1994)
Wang, Y.: Randomness and Complexity. Ph.D. Dissertation, University of Heidelberg (1996)
Wang, Y.: A separation of two randomness concepts. Inf. Process. Lett. 69, 115–118 (1999)
Wehner, S.: Enumerations, countable structures, and Turing degrees. Proc. Am. Math. Soc. 126, 2131–2139 (1998)
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)
Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. In: Informatik Berichte FernUniversität Hagen, vol. 130. Hagen (1992)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights 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)