Abstract. We investigate the theories of the λY-calculus, i.e. simply typed λ-calculus with fixpoint combinators. Non-terminating λY-terms exhibit a rich behavior, and one can reflect in λY many results of untyped λ-calculus concerning theories. All theories can be characterized as contextual theories à la Morris, w.r.t. a suitable set of observables. We focus on theories arising from natural classes of observables, where Y can be approximated, albeit not always initially. In particular, we present the standard theory, induced by terminating terms, which features a canonical interpretation of Y as “minimal fixpoint”, and another theory, induced by pure λ-terms, which features a non-canonical interpretation of Y. The interest of these two theories is that the term model of the λY-calculus w.r.t. the first theory gives a fully complete model of the maximal theory of the simply typed λ-calculus, while the term model of the latter theory provides a fully complete model for the observational equivalence in unary PCF. Throughout the paper we raise open questions and conjectures
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