Beth Deenability for the Guarded Fragment

The guarded fragment (GF) was introduced in ABN98] as a fragment of rst order logic which combines a great expressive power with nice modal behavior. It consists of relational rst order formulas whose quantiiers are relativized by atoms in a certain way. While GF has been established as a particularly well-behaved fragment of rst order logic in many respects, interpolation fails in restriction to GF, HM99]. In this paper we consider the Beth property of rst order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. Being a closure property w.r.t. deenability, 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 speciically, even every n-variable guarded fragment with up to n-ary relations has the Beth property.

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

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

[3]  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.

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

[5]  Finite variable logics , 1993, Bull. EATCS.

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

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

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

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

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

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

[12]  Maarten Marx,et al.  Interpolation in Guarded Fragments , 2000 .