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This paper is a survey and systematic presentation of decidability and complexity issues for modal and non-modal two-variable logics. A classical result due to Mortimer says that the two-variable fragment of rst-order logic, denoted FO 2 , has the nite model property and is therefore decidable for satissability. One of the reasons for the signiicance of… (More)

The satissability problem for the two-variable fragment of rst-order logic is investigated over nite and innnite linearly ordered, respectively wellordered domains, as well as over nite and innnite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations. It is shown that FO 2 over ordered, respectively… (More)

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Ben-them's theorem which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation… (More)

Consider the class of all those properties of worlds in nite Kripke structures (or of states in inite transition systems), that are recognizable in polynomial time, and closed under bisimulation equivalence. It is shown that the class of these bisimulation-invariant Ptime queries has a natural logical characterization. It is captured by the straightforward… (More)

We consider \generic" (isomorphism-invariant) queries on rela-tional databases embedded in an innnite background structure. Assume a generic query is expressible by a rst-order formula over the embedded domain that may involve both the relations of the database and the relations and functions of the background structure. Then this query is already… (More)

Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rst-order logic, there is an interpolant in which each relation symbol appears only in those polarities in which it appears in both the antecedent and the succedent of the given implication. This note proves a similar interpolation result under… (More)

We prove that the satissability problem for C 2 is decidable. C 2 is rst-order logic with only two variables in the presence of arbitrary counting quantiiers 9 >m , m > 1. It considerably extends L 2 , plain rst-order with only two variables, which is known to be decidable by a result of Mortimer. Unlike L 2 , C 2 does not have the nite model property. As C… (More)

Guarded fixed-point logic μGF extends the guarded fragment by means of least and greatest fixed points, and thus plays the same role within the domain of guarded logics as the modal μ-calculus plays within the modal domain. We provide a semantic characterization of μGF within an appropriate fragment of second-order logic, in terms of invariance… (More)

Characterisation theorems for modal and guarded fragments of first-order logic are explored over finite transition systems. We show that the classical characterisations in terms of semantic invariance under the appropriate forms of bisimulation equivalence can be recovered in finite model theory. The new, more constructive proofs naturally extend to… (More)