Decidability of all minimal models
We consider a simply typed λ-calculus with constants of ground types, and assume that for one ground type o, there are finitely many constants of type o. We call minimal model the quotient by observational equivalence of the set of all closed terms whose type is of terminal subformula o. We show that this model is decidable: all classes of any given type are recursively representable, and observational equivalence on closed terms is a decidable relation. In particular, this result solves the question raised by R.Statman on the decidability of this model in the case of a unique ground type and two constants.
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