PP-2000-11: Interpolation in Guarded Fragments

PP-2000-11: Hoogland, Eva and Marx, Maarten (2000) Interpolation in Guarded Fragments. [Report]

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Abstract

The guarded fragment (GF) was introduced by Andreka, van Benthem and Nemeti as a fine-structure 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. Slightly generalizing the admissible relativizations yields the packed fragment (PF). In this paper we chart the behavior of these fragments with regard to interpolation. While GF and PF have been established as particularly well-behaved fragments of first order logic in many respects, it will be shown that the interpolation property of first order logic fails in restriction to GF and PF. However, each of these fragments turns out to have an alternative interpolation property that closely resembles the interpolation property usually studied in modal logic. These results are strong enough to entail the Beth definability theorem for GF and PF. Even better, every n-variable guarded or packed fragment with up to n-ary relations has the Beth property. Finally it will be demonstrated that any n-variable guarded fragment which contains at most k-ary relations has interpolation if and only if k<3 or n<3. In particular, the 2-variable guarded fragment has interpolation. The packed fragments have interpolation if and only if n<3 or k<2. Keyword(s): Guarded fragment, packed fragment, interpolation property, Beth definability property. Note: The part of this paper concerning Beth's definability property for the guarded fragment has appeared as: E. Hoogland, M. Marx, and M. Otto. Beth Definability for the Guarded Fragment. In: H. Ganzinger, D. McAllester, and A. Voronkov editors, Logic for Programming and Automated Reasoning, 6th International Conference LPAR99, Tbilisi, Georgia, volume 1705 of LNAI, pages 273--285. Springer, 1999.

Item Type: Report
Report Nr: PP-2000-11
Series Name: Prepublication (PP) Series
Year: 2000
Date Deposited: 12 Oct 2016 14:36
Last Modified: 12 Oct 2016 14:36
URI: https://eprints.illc.uva.nl/id/eprint/38

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