On the Restraining Power of Guards

• Published in: The Journal of symbolic logic Vol. 64; no. 4; pp. 1719 - 1742
• Main Author: Gradel, Erich
• Format: Journal Article
• Language: English
• Published: New York, USA Cambridge University Press 01-12-1999; The Association for Symbolic Logic, Inc; Association for Symbolic Logic
• Subjects: Decidability; Determinism; Finite model property; Mathematical logic; Modal logic; Plant roots; Predicate logic; Predicates; Satisfiability; Tiling
• ISSN: 0022-4812; 1943-5886
• DOI: 10.2307/2586808

Guarded fragments of first-order logic were recently introduced by Andreka, van Benthem and Nemeti; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic. Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is EXPTIME-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment. It is also shown that some natural, modest extensions of the guarded fragments are undecidable.