Finite variable logics

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∞ω, discussing complexity issues. We give a game characterisation of L∞ω, 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∞ω. Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout. 1 Some extensions of first-order logic Quisani: Hello. Who are you? I am Yuri’s imaginary student, and I usually talk to him at this time. Author: I’m afraid he may be a bit late. I am a computer scientist from London, England; maybe I can help. I was reading your earlier conversation on 0–1 laws [Gu3]. Quisani: I remember it. We examined the 0–1 law for logics such as L∞ω, in which infinite conjunctions and disjunctions are allowed, as well as the first-order operations, but only finitely many variables can occur in any formula. Author: Right: the top ‘ω’ represents this restriction on variables. The bottom ‘∞’ means that subject to this restriction, the conjunction or disjunction of any set of formulas, even an uncountable one, is a formula. The bottom ‘ω’ is fixed and is not so important here, but for the record it means that only finite quantifier depth is permitted — unlike ∗Supported by SERC Advanced Fellowship B/ITF/266.