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Finite variable logics. (English) Zbl 0788.03051

Summary: In this survey article we discuss some aspects of finite variable logics. We translate some well-known fixed-point logics into the infinitary logic \(L^ \omega_{\infty \omega}\), discussing complexity issues. We give a game characterisation of \(L^ \omega_{\infty \omega}\), and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for \(L^ \omega_{\infty \omega}\). Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout.

MSC:

03C75 Other infinitary logic
03C13 Model theory of finite structures
03C40 Interpolation, preservation, definability
03B45 Modal logic (including the logic of norms)