Abstract
In this chapter, we introduce finite variable logics: a unifying tool for studying fixed point logics. These logics use infinitary connectives already seen in Chap. 8, but here we impose a different restriction: each formula can use only finitely many variables. We show that fixed point logics LFP, IFP, and PFP can be embedded in such a finite variable logic. Furthermore, the finite variable logic is easier to study: it can be characterized by games, and this gives us bounds on the expressive power of fixed point logics; in particular, we show that without a linear ordering, they fail to capture complexity classes. We then study definability and ordering of types in finite variable logics, and use these techniques to relate separating complexity classes to separating sonic fixed point logics over unordered structures.
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