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Interpolation properties for provability logics GL and GLP

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Abstract

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP.

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References

  1. S. N. Artemov and L. D. Beklemishev, “Provability Logic,” in Handbook of Philosophical Logic, Ed. by D. Gabbay and F. Guenthner, 2nd ed. (Springer, Dordrecht, 2005), Vol. 13, pp. 189–360.

    Google Scholar 

  2. L. D. Beklemishev, “Kripke Semantics for Provability Logic GLP,” Ann. Pure Appl. Logic 161, 756–774 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  3. L. D. Beklemishev, “On the Craig Interpolation and the Fixed Point Property for GLP,” in Proofs, Categories and Computations: Essays in Honor of G. Mints, Ed. by S. Feferman and W. Sieg (College Publ., London, 2010), Tributes 13, pp. 49–60.

    Google Scholar 

  4. L. D. Beklemishev, “A Simplified Proof of Arithmetical Completeness Theorem for Provability Logic GLP,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 274, 32–40 (2011) [Proc. Steklov Inst. Math. 274, 25–33 (2011)].

    Google Scholar 

  5. G. Boolos, The Unprovability of Consistency: An Essay in Modal Logic (Cambridge Univ. Press, Cambridge, 1979).

    MATH  Google Scholar 

  6. G. D’Agostino, “Uniform Interpolation, Bisimulation Quantifiers, and Fixed Points,” in Logic, Language, and Computation: Proc. 6th Int. Tbilisi Symp. TbiLLC 2005, Batumi, Georgia, Sept. 2005, Ed. by B. D. ten Cate and H. W. Zeevat (Springer, Berlin, 2007), Lect. Notes Comput. Sci. 4363, pp. 96–116.

    Google Scholar 

  7. G. D’Agostino and M. Hollenberg, “Logical Questions concerning the µ-Calculus: Interpolation, Lyndon and Łós-Tarski,” J. Symb. Log. 65(1), 310–332 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Ghilardi and M. Zawadowski, “A Sheaf Representation and Duality for Finitely Presented Heyting Algebras,” J. Symb. Log. 60(3), 911–939 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. Goranko and M. Otto, “Model Theory of Modal Logic,” in Handbook of Modal Logic, Ed. by P. Blackburn, J. van Benthem, and F. Wolter (Elsevier, Amsterdam, 2007), pp. 249–329.

    Chapter  Google Scholar 

  10. K. N. Ignatiev, “On Strong Provability Predicates and the Associated Modal Logics,” J. Symb. Log. 58(1), 249–290 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. K. Japaridze, “The Modal Logical Means of Investigation of Provability,” Candidate (Philos.) Dissertation (Moscow State Univ., Moscow, 1986).

    Google Scholar 

  12. A. M. Pitts, “On an Interpretation of Second Order Quantification in First Order Intuitionistic Propositional Logic,” J. Symb. Log. 57(1), 33–52 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Yu. Shavrukov, Subalgebras of Diagonalizable Algebras of Theories Containing Arithmetic (Inst. Mat., Pol. Acad. Sci., Warszawa, 1993), Diss. Math. 323.

    Google Scholar 

  14. C. Smoryński, “Beth’s Theorem and Self-referential Sentences,” in Logic Colloquium’ 77: Proc. Colloq., Wrocław, 1977, Ed. by A. Macintyre, L. Pacholski, and J. Paris (North-Holland, Amsterdam, 1978), pp. 253–261.

    Google Scholar 

  15. J. van Benthem, “Modal Frame Correspondences and Fixed-Points,” Stud. Log. 83(1–3), 133–155 (2006).

    Article  MATH  Google Scholar 

  16. A. Visser, “Uniform Interpolation and Layered Bisimulation,” in Gödel’ 96. Logical Foundations of Mathematics, Computer Science and Physics-Kurt Gödel’s Legacy: Proc. Conf., Brno, Czech Rep., Aug. 1996, Ed. by P. Hájek (Springer, Berlin, 1996), Lect. Notes Log. 6, pp. 139–164.

    Google Scholar 

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Vol. 274, pp. 329–342.

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Shamkanov, D.S. Interpolation properties for provability logics GL and GLP. Proc. Steklov Inst. Math. 274, 303–316 (2011). https://doi.org/10.1134/S0081543811060198

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