On Logics with Two Variables

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 this result is that many propositional modal logics can be embedded into FO 2. Logics that are of interest for knowledge representation, for the speciication and veriication of concurrent systems and for other areas of computer science are often deened (or can be viewed) as extensions of modal logics by features like counting constructs, path quantiiers, transitive closure operators, least and greatest xed points etc. Examples of such logics are computation tree logic CTL, the modal-calculus L , or popular description logics used in artiicial intelligence. Although the additional features are usually not rst-order constructs, the resulting logics can still be seen as two-variable logics that are embedded in suitable extensions of FO 2. Typically, the applications call for an analysis of the satissability and model checking problems of the logics employed. The decidability and complexity issues for modal and non-modal two-variables logics have been studied quite intensively in the last years. It has turned out that the satissability problems for two-variable logics with full rst-order quantiication are usually much harder (and indeed highly undecidable in many cases) than the satissability problems for corresponding modal logics. On the other side, the situation is diierent for model checking problems. The model checking problem of a modal logic has essentially the same complexity as the model checking problem of the corresponding two variable logic with full quantiication.