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**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.) |

### Keywords:

completeness; reflexive object; Cartesian closed category of complete partial orders and Scott-continuous functions; \(\lambda\)-calculus; topological model; \(\lambda\)-theories
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\textit{F. Honsell} and \textit{S. Ronchi della Rocca}, J. Comput. Syst. Sci. 45, No. 1, 49--75 (1992; Zbl 0763.03012)

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### References:

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