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) |