On The Restraining Power of Guards

Guarded fragments of rst-order logic were recently introduced by Andr eka, van Benthem and N emeti; they consist of relational rst-order formulae whose quantiiers 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 quantiier pattern or the number of variables) of almost all other known decidable fragments of rst-order logic. Here, we investigate the computational complexity of these fragments. We prove that the satissability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of rst-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, satissability 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 nite model property of the guarded fragment. It is also shown that some natural, modest extensions of the guarded fragments are undecidable.