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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)
Evaluating a boolean conjunctive query q over a guarded first-order theory T is equivalent to checking whether (T \& not q) is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment(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)
Finite model theory deals with the model theory of finite structures. As a branch of model theory it is concerned with the analysis of structural properties in terms of logics. The attention to finite structures is not so much a restriction in scope as a shift in perspective. The main parts of classical model theory (the model theory related to first-order(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 Benthem'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)
It is a classical result of Mortimer that L 2 , rst-order logic with two variables, is decidable for satissability. We show that going beyond L 2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, viz. (i) transitive closure operations (ii) (restricted) monadic xed-point operations weak access to cardinalities,(More)
We study first-order logic with two variables FO/sup 2/ and establish a small substructure property. Similar to the small model property for FO/sup 2/ we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO/sup 2/ under(More)