No abstract

Guarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; 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.

No abstract

We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent from the variables z if the variables x are kept constant. We contrast this with dependence logic D [7] based on the atomic formula =( x, y), actually equivalent to y ⊥ x y, saying that the variables y are totally determined by the variables x. We show that y ⊥ x z gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that y ⊥ x z can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using =( x, y) have.Of the numerous uses of the word "dependence" we focus on the concept of an attribute 1 depending on a number of other similar attributes when we observe the world. We call these attributes variables. We follow the approach of [7] and focus on the strongest form of dependence, namely functional dependence.

We identify the computational complexity of the satisfiability problem for FO2, the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO2 has the finite-model property, which means that if an FO2-sentence is satisiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO2-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO2-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO2 is NEXPTIME-complete.