Abstract
We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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Balbiani, P., V. Shehtman, and I. Shapirovsky, ‘Every world can see a Sahlqvist world’, in G. Governatori, I. Hodkinson, and Y. Venema, (eds.), Proc. Advances in Modal Logic, College Publications, 2006, pp. 69–85.
Benthem J.: ‘A note on modal formulas and relational properties’. Journal of Symbolic Logic 40(1), 55–58 (1975)
van Benthem J.: Modal logic and classical logic. Bibliopolis, Naples (1985)
van Benthem J.: ‘Minimal predicates, fixed-points, and definability’. J. Symbolic Logic 70, 696–712 (2005)
van Benthem J.: ‘Modal frame correspondences and fixed-points’. Studia Logica 83, 133–155 (2006)
Bezhanishvili N., I. Hodkinson, ‘Sahlqvist theorem for modal fixed point logic’, 2010. Submitted.
Blackburn, P., M. de Rijke, and Y. Venema, Modal logic, Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, UK, 2001.
ten Cate B., Marx M., Viana P.: ‘Hybrid logics with Sahlqvist axioms’. Logic J. IGPL 13, 293–300 (2005)
Conradie W., Goranko V., Vakarelov D.: ‘Algorithmic correspondence and completeness in modal logic V: recursive extensions of SQEMA’. J. Applied Logic 8, 319–333 (2010)
Doherty P., Łukaszewicz W., Szałas A.: ‘Computing circumscription revisited: A reduction algorithm’. J. Automated Reasoning 18, 297–336 (1997)
Ebbinghaus H-D., J. Flum, Finite model theory, 2nd edn., Perspectives in mathematical logic, Springer-Verlag, New York, 1999.
Esakia L.L.: ‘Topological Kripke models’. Soviet Math. Dokl. 15, 147–151 (1974)
Fontaine G., Modal fixpoint logic: some model theoretic questions, Ph.D. thesis, ILLC, Amsterdam, 2010. ILLC Dissertation Series DS-2010-09.
Gabbay, D. M., and H-J. Ohlbach, ‘Quantifier elimination in second-order predicate logic’, in B. Nebel, C. Rich, and W. Swartout, (eds.), Principles of Knowledge Representation and Reasoning (KR92), Morgan Kaufmann, 1992, pp. 425–435.
Givant S., Venema Y.: ‘The preservation of Sahlqvist equations in completions of Boolean algebras with operators’. Algebra Universalis 41, 47–84 (1999)
Goldblatt R., Hodkinson I.: ‘The McKinsey–Lemmon logic is barely canonical’. Australasian J. Logic 5, 1–19 (2007)
Goranko, V., and M. Otto, ‘Model theory of modal logic’, in P. Blackburn, J. van Benthem, and F. Wolter, (eds.), Handbook of Modal Logic, Elsevier, Amsterdam, 2006, pp. 249–329.
Goranko V., Vakarelov D.: ‘Elementary canonical formulae: extending Sahlqvist’s theorem’. Ann. Pure. Appl. Logic 141, 180–217 (2006)
Grädel E., ‘The decidability of guarded fixed point logic’, in J. Gerbrandy, M. Marx, M. de Rijke, and Y. Venema, (eds.), JFAK. Essays Dedicated to Johan van Benthem on the Occasion of his 50th Birthday, Vossiuspers, Amsterdam University Press, Amsterdam, 1999. CD-ROM, ISBN 90 5629 104 1.
Kracht, M., ‘How completeness and correspondence theory got married’, in M. de Rijke, (ed.), Diamonds and Defaults, Kluwer Academic Publishers, 1993, pp. 175–214.
Nonnengart, A., and A. Szałas, ‘A fixpoint approach to second-order quantifier elimination with applications to correspondence theory’, in E. Orłowska, (ed.), Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, vol. 24 of Studies in Fuzziness and Soft Computing, Physica-Verlag, 1999, pp. 307–328.
Sahlqvist, H., ‘Completeness and correspondence in the first and second order semantics for modal logic’, in S. Kanger, (ed.), Proc. 3rd Scandinavian logic symposium, Uppsala, 1973, North Holland, Amsterdam, 1975, pp. 110–143.
Sambin G., Vaccaro V.: ‘A new proof of Sahlqvist’s theorem on modal definability and completeness’. J. Symbolic Logic 54, 992–999 (1989)
Tarski A.: ‘A lattice-theoretical fixpoint theorem and its applications’. Pacific Journal of Mathematics 5, 285–309 (1955)
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In memoriam Leo Esakia
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van Benthem, J., Bezhanishvili, N. & Hodkinson, I. Sahlqvist Correspondence for Modal mu-calculus. Stud Logica 100, 31–60 (2012). https://doi.org/10.1007/s11225-012-9388-9
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DOI: https://doi.org/10.1007/s11225-012-9388-9