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STRONG REDUCTIONS BETWEEN COMBINATORIAL PRINCIPLES

  • DAMIR D. DZHAFAROV (a1)
Abstract
Abstract

This paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$ , and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$ , or strongly computably reducible to $SRT_2^2$ . The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

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[1] Brattka V., Bibliography on Weihrauch Complexity , http://cca-net.de/publications/weibib.php.
[2] Brattka V., Gherardi G., and Hölzl R., Probabilistic computability and choice . Information and Computation, vol. 242 (2015), pp. 249286.
[3] Brattka V. and Rakotoniaina T., On the Uniform Computational Content of Ramsey’s Theorem , submitted.
[4] Cholak P. A., Jockusch C. G., and Slaman T. A., On the strength of Ramsey’s theorem for pairs , this JOURNAL, vol. 66 (2001), no. 1, pp. 155.
[5] Chong C. T., Lempp S., and Yang Y., On the role of the collection principle for ${\rm{\Sigma }}_2^0$ -formulas in second-order reverse mathematics . Proceedings of the American Mathematical Society, vol. 138 (2010), no. 3, pp. 10931100.
[6] Chong C. T., Slaman T. A., and Yang Y., The metamathematics of stable Ramsey’s theorem for pairs . Journal of the American Mathematical Society, vol. 27 (2014), no. 3, pp. 863892.
[7] Dorais F. G., Dzhafarov D. D., Hirst J. L., Mileti J. R., and Shafer P., On uniform relationships between combinatorial problems . Transactions of the American Mathematical Society, vol. 368 (2016), no. 2, pp. 13211359.
[8] Downey R. G. and Hirschfeldt D. R., Algorithmic randomness and complexity, Theory and Applications of Computability, Springer, New York, 2010.
[9] Dzhafarov D. D., Cohesive avoidance and strong reductions . Proceedings of the American Mathematical Society, vol. 143 (2015), no. 2, pp. 869876.
[10] Dzhafarov D. D., The RM Zoo, 2015, http://rmzoo.uconn.edu.
[11] Dzhafarov D. D., Patey L., Solomon R., and Westrick L. B.. Ramsey’s theorem for singletons and strong computable reducibility, to appear.
[12] Hirschfeldt D. R., Slicing the Truth: On the Computable and Reverse Mathematics of Combinatorial Principles, Lecture Notes Series/Institute for Mathematical Sciences, National University of Singapore, World Scientific Publishing Company Incorporated, New York, 2014.
[13] Hirschfeldt D. R. and Jockusch C. G. Jr., On notions of computability theoretic reduction between ${\rm{\Pi }}_2^1$ principles, to appear.
[14] Hirschfeldt D. R., Jockusch C. G. Jr., Kjos-Hanssen B., Lempp S., and Slaman T. A., The strength of some combinatorial principles related to Ramsey’s theorem for pairs , Computational Prospects of Infinity. Part II. Presented Talks, Lecture Notes Series/Institute for Mathematical Sciences, National University of Singapore, vol. 15, World Scientific Publishing Company Incorporated, Hackensack, NJ, 2008, pp. 143161.
[15] Hirschfeldt D. R. and Shore R. A., Combinatorial principles weaker than Ramsey’s theorem for pairs, this JOURNAL, vol. 72 (2007), no. 1, pp. 171206.
[16] Jockusch C. G. Jr., Ramsey’s theorem and recursion theory, this JOURNAL, vol. 37 (1972), pp. 268280.
[17] Jockusch C. G., Degrees of generic sets, Recursion Theory: Its Generalisation and Applications (Proc. Logic Colloq., Univ. Leeds, Leeds, 1979), London Mathematical Society Lecture Note Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 110139.
[18] Mileti J. R., Partition Theorems and Computability Theory , Ph.D thesis, University of Illinois at Urbana-Champaign, 2004.
[19] Montalbán A., Open questions in reverse mathematics . Bulletin of Symbolic Logic, vol. 17 (2011), no. 3, pp. 431454.
[20] Patey L., The weakness of being cohesive, thin or free in reverse mathematics, submitted.
[21] Rakotoniaina T., The Computational Strength of Ramsey’s Theorem , Ph.D thesis, University of Cepe Town, 2015.
[22] Seetapun D. and Slaman T. A., On the strength of Ramsey’s theorem . Notre Dame Journal of Formal Logic, vol. 36 (1995), no. 4, pp. 570582.
[23] Shore R. A., Lecture notes on turing degrees , Computational Prospects of Infinity II: AII Graduate Summer School, Lecture Notes Series/Institute for Mathematical Sciences, National University of Singapore, World Scientific Publishing Company Incorporated, Hackensack, NJ, to appear.
[24] Simpson S. G., Degrees of unsolvability: A survey of results , Handbook of Mathematical Logic (Barwise J., editor), North-Holland, Amsterdam, 1977, pp. 631652.
[25] Simpson S. G., Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.
[26] Soare R. I., Computability theory and applications , Theory and Applications of Computability, Springer, New York, to appear.
[27] Weihrauch K., The Degrees of Discontinuity of Some Translators Between Representations of the Real Numbers , Informatik-Berichte 129, FernUniversität Hagen, 1992.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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