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Constructive interpolation in hybrid logic

  • Patrick Blackburn (a1) and Maarten Marx (a2)


Craig's interpolation lemma (if φψ is valid, then φθ and θψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.



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[1]Areces, C., Blackburn, P., and Marx, M., Hybrid logics: Characterization, interpolation and complexity, this Journal, vol. 66 (2001), no. 3, pp. 9771010.
[2]Areces, C., Blackburn, P., and Marx, M., Repairing the interpolation lemma in quantified modal logic, Technical Report PP-2001-19, Institute for Language, Logic and Computation, 2001, To appear in Annals of Pure and Applied Logic.
[3]Blackburn, P., de Rijke, M., and Venema, Y., Modal logic, Cambridge University Press, 2001.
[4]Blackburn, P. and Marx, M., Tableaux for quantified hybrid logic, Automated reasoning with analytic tableaux and related methods, TABLEAUX 2002 (Egly, U. and Fermüller, C., editors), LNAI, vol. 2381, Springer Verlag, 2002, pp. 3852.
[5]de Rijke, M., The modal logic of inequality, this Journal, vol. 57 (1992), pp. 566584.
[6]Fine, K., Failures of the interpolation lemma in quantified modal logic, this Journal, vol. 44 (1979), no. 2, pp. 201206.
[7]Fitting, M., Proof methods for modal and intuitionistic logics, Synthese Library, vol. 169, Reidel, Dordrecht, 1983.
[8]Fitting, M., First order logic and automated theorem proving, second ed., Springer Verlag, 1996.
[9]Fitting, M., Interpolation for first order S5, this Journal, vol. 67 (2002), pp. 621634.
[10]Maksimova, L., Amalgamation and interpolation in normal modal logics, Studia Logica, vol. L (1991), no. 3/4, pp. 457471.
[11]Marx, M. and Venema, Y., Multi-dimensional modal logic, Applied Logic Series, Kluwer Academic Publishers, 1997.
[12]Rautenberg, W., Modal tableau calculi and interpolation, Journal of Philosophical Logic, vol. 12 (1983), pp. 403423.
[13]Smullyan, R., First order logic, Springer-Verlag, 1968.


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