Abstract
The guarded fragment (GF) was introduced in [ABN98] as a fragment of first order logic which combines a great expressive power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. While GF has been established as a particularly well-behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [HM99]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. Being a closure property w.r.t. definability, the Beth property is of independent interest, both theoretically and for typical potential applications of GF, e.g., in the context of description logics. The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics. From this we obtain that, more specifically, even every n-variable guarded fragment with up to n-ary relations has the Beth property.
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H. Andréka, J. van Benthem, and I. Németi. Modal logics and bounded fragments of predicate logic. Journal of Philosophical Logic, 27(3):217–274, 1998.
J. van Benthem. Modal foundations for predicate logic. In E. Orlowska, editor, Memorial Volume for Elena Rasiowa, Studies in fuziness and soft computing, pages 39–55. Physica-Verlag, Heidelberg, New York, 1999.
E. W. Beth. On Padoa’s method in the theory of definition. Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math., 15:330–339, 1953.
E. Grädel, P. Kolaitis, and M. Vardi. On the decision problem for two-variable first order logics. Bulletin of Symbolic Logic, 3:53–69, 1997.
E. Grädel and M. Otto. On logics with two variables. Theoretical Computer Science, 1999. to appear.
E. Grädel. On the restraining power of guards. Technical report, RWTH Aachen, Lehrgebiet Mathematische Grundlagen der Informatik, 1997. To appear in the Journal of Symbolic Logic.
E. Grädel. Guarded fragments of first-order logic: a perspective for new description logics? In Proc. of 1998 Int. Workshop on Description Logics DL’ 98, Trento, CEUR Electronic Workshop Proceedings, 1998. Extended abstract, available at http://sunsite.informatik.nrth-aachen.de/Publications/CEUR-WS/Vol-11.
E. Grädel and I Walukiewicz. Guarded fixed point logic. In Proc. 14th Symp. on Logic in Computer Science, LICS’99, pages 45–54, 1999.
P. Hájek. Generalized quantifiers and finite sets. Ser. Konfer. No 1 14, Prace Nauk. Inst. Mat. Politech. Wroclaw, 1977.
E. Hoogland and M. Marx. Interpolation in guarded fragments. Technical report, Institute for Logic, Language and Computation, University of Amsterdam, 1999.
I. Hodkinson. Finite variable logics. Bull. Europ. Assoc. Theor. Comp. Sci., 51:111–140, 1993. Addendum in vol. 52. Also available at http://www.doc.ic.ac.uk/~imh/papers/yuri.html.
Maarten Marx. Algebraic Relativization and Arrow Logic. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 1995.
M. Marx. Interpolation in (fibered) modal logic. In A. Haeberer, editor, Proc. of AMAST 1998, Amazonia-Manaus, Brazil, 4–8 January 1999, 1999.
M. Marx and Y. Venema. Multi-dimensional Modal Logic. Applied Logic Series. Kluwer Academic Publishers, 1997.
I. Németi. Cylindric relativised set algebras have strong amalgamation. Journal of Symbolic Logic, 50(3):689–700, 1985.
I. Sain. Beth’s and Craig’s properties via epimorphisms and amalgamation in algebraic logic. In C. Bergman, R. Maddux, and D. Pigozzi, editors, Algebraic logic and universal algebra in computer science, volume 24 of Lecture Notes in Computer Science, pages 209–226. Springer Verlag, Berlin, Heidelberg, New York, 1990.
K. Schild. A correspondence theory for terminological logics. In Proceedings of the 12th IJCAI, pages 466–471, 1991.
M. Vardi. Why is modal logic so robustly decidable? In Descriptive Complexity and Finite Models: Proceedings of a DIMACS Workshop, volume 31 of Series in Discrete Mathematics and Theoretical Computer Science, pages 149–184. American Mathematical Society, 1998.
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Hoogland, E., Marx, M., Otto, M. (1999). Beth Definability for the Guarded Fragment. In: Ganzinger, H., McAllester, D., Voronkov, A. (eds) Logic for Programming and Automated Reasoning. LPAR 1999. Lecture Notes in Computer Science(), vol 1705. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48242-3_17
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DOI: https://doi.org/10.1007/3-540-48242-3_17
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