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RAMSEY’S THEOREM FOR PAIRS AND K COLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

  • STEFANO BERARDI (a1) and SILVIA STEILA (a2)
Abstract

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments of k-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \ge 2$ , Ramsey’s Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for ${\rm{\Sigma }}_3^0$ formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some $i < k$ and some branch with infinitely many children of index i.

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[1] Akama, Y., Berardi, S., Hayashi, S., and Kohlenbach, U., An arithmetical hierarchy of the law of excluded middle and related principles , 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, 2004, pp. 192201.
[2] Argyros, S. A. and Todorcevic, S., Ramsey Methods in Analysis, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, Basel, 2005.
[3] Berardi, S. and Steila, S., Ramsey Theorem for pairs as a classical principle in intuitionistic arithmetic , 19th International Conference on Types for Proofs and Programs, TYPES 2013 (Matthes, R. and Schubert, A., editors), LIPIcs, vol. 26, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, 2014, pp. 6483.
[4] Berardi, S. and Steila, S., An intuitionistic version of Ramsey’s Theorem and its use in Program Termination . Annals of Pure and Applied Logic, vol. 166 (2015), no. 12, pp. 13821406.
[5] Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, NY, 1967.
[6] Brattka, V. and Rakotoniaina, T., On the uniform computational content of Ramsey’s Theorem, 2016, arXiv:1508.00471.
[7] Büchi, J. R., On a decision method in restricted second order arithmetic , Logic, Methodology and Philosophy of Science (Proceeding of the 1960 International Congress) (Nagel, E., Suppes, P., and Tarsk, A., editors), Stanford University Press, Stanford, 1962, pp. 111.
[8] Coquand, T., A direct proof of Ramsey’s Theorem , Author’s website, revised in 2011, 1994.
[9] 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.
[10] Gasarch, W. I., Proving programs terminate using well-founded orderings , Ramsey’s Theorem, and matrices. Advances in Computers, vol. 97 (2015), pp. 147200.
[11] Jockusch, C. G., Ramsey’s Theorem and recursion theory , this JOURNAL, vol. 37 (1972), pp. 268280.
[12] Kreuzer, A. and Kohlenbach, U., Ramsey’s Theorem for pairs and provably recursive functions . Notre Dame Journal of Formal Logic, vol. 50 (2009), no. 4, pp. 427444.
[13] Lee, C. S., Jones, N. D., and Ben-Amram, A. M., The size-change principle for program termination , Conference Record of POPL 2001 (Hankin, C. and Schmidt, D., editors), ACM, New York, NY, 2001, pp. 8192.
[14] Podelski, A. and Rybalchenko, A., Transition invariants , 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, 2004, pp. 3241.
[15] Ramsey, F. P., On a problem in formal logic . Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264286.
[16] Specker, E., Ramsey’s theorem does not hold in recursive set theory , Logic Colloquium ’69 (Proceedings of the Summer School and Colloquium, Manchester, 1969) (Gandy, R. O. and Yates, C. M. E., editors), North-Holland, Amsterdam, 1971, pp. 439442.
[17] Steila, S., An intuitionistic analysis of size-change termination , 20th International Conference on Types for Proofs and Programs, TYPES 2014 (Herbelin, H., Letouzey, P., and Sozeau, M., editors), LIPIcs, vol. 39, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, 2015, pp. 288307.
[18] Vytiniotis, D., Coquand, T., and Wahlstedt, D., Stop when you are almost-full - adventures in constructive termination , Interactive Theorem Proving, ITP 2012 (Beringer, L. and Felty, A. P., editors), Lecture Notes in Computer Science, vol. 7406, Springer, Berlin, Heidelberg, 2012, pp. 250265.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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