Models of λ-Calculus and the Weak MSO Logic∗

We study the Weak MSO logic in relationship to infinitary λ-calculus. We show that for every formula φ of Weak MSO there exists a finitary model of infinitary λ-calculus recognizing the set of infinitary λ-terms whose Böhm tree satisfies φ. The model is effective, in the sense that for every λY -term we can effectively compute its value in the model. In particular, given a finite λY -term, one can decide whether the resulting Böhm tree satisfies a given formula of Weak MSO, which is a special case of the result of Ong [16], which concerns unrestricted MSO. The existence of effective models for Weak MSO and MSO was proved earlier by Salvati and Walukiewicz [19, 20] but our proof uses a different method, as it does not involve automata, but works directly with logics. 1998 ACM Subject Classification F.1.1 Models of Computation