Beth Definability for the Guarded Fragment

The gueirded 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 relationeil first order formulas whose qucintiiiers are relativized by atoms in a certain way. While GF hsis 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 appUcations 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.

[1]  István Németi,et al.  Cylindric-relativised set algebras have strong amalgamation , 1985, Journal of Symbolic Logic.

[2]  Erich Grädel,et al.  On the Restraining Power of Guards , 1999, Journal of Symbolic Logic.

[3]  Moshe Y. Vardi.  Why is Modal Logic So Robustly Decidable? , 1996, Descriptive Complexity and Finite Models.

[4]  Igor Walukiewicz,et al.  Guarded fixed point logic , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[5]  Klaus Schild,et al.  A Correspondence Theory for Terminological Logics: Preliminary Report , 1991, IJCAI.

[6]  maarten marx.  Algebraic Relativization and Arrow Logic , 1995 .

[7]  Roger D. Maddux,et al.  Algebraic Logic and Universal Algebra in Computer Science , 1990, Lecture Notes in Computer Science.

[8]  Evert W. Beth,et al.  On Padoa’s Method in the Theory of Definition , 1953 .

[9]  Phokion G. Kolaitis,et al.  On the Decision Problem for Two-Variable First-Order Logic , 1997, Bulletin of Symbolic Logic.

[10]  Martin Otto,et al.  On Logics with Two Variables , 1999, Theor. Comput. Sci..

[11]  Maarten Marx,et al.  Interpolation in Modal Logic , 1999, AMAST.

[12]  M. J. Marx.  Interpolation in (fibered) modal logic. , 1999 .

[13]  Johan van Benthem,et al.  Modal Foundations for Predicate Logic , 1997, Log. J. IGPL.

[14]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[15]  Maarten Marx,et al.  Multi-dimensional modal logic , 1997, Applied logic series.

[16]  Ildikó Sain,et al.  Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic , 1988, Algebraic Logic and Universal Algebra in Computer Science.