## An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus.(English)Zbl 0763.03012

Summary: It is well known that a reflexive object in the Cartesian closed category of complete partial orders and Scott-continuous functions is a model of $$\lambda$$-calculus (briefly a topological model). A topological model, through the interpretation function, induces a $$\lambda$$-theory, i.e., a congruence relation on $$\lambda$$-terms closed under $$\alpha$$- and $$\beta$$-reduction. It is natural to ask if all possible $$\lambda$$- theories are induced by a topological model, i.e., if topological models are complete w.r.t. $$\lambda$$-calculus. The authors prove an Approximation Theorem, which holds in all topological models. Using this theorem, they analyze some topological models and their induced $$\lambda$$-theories, and they exhibit a $$\lambda$$-theory which cannot be induced by a topological model. So they prove that topological models are not complete w.r.t. $$\lambda$$-calculus.

### MSC:

 03B40 Combinatory logic and lambda calculus 68Q55 Semantics in the theory of computing 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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### References:

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