Innnite -calculus and Types ?

Recent work on innnitary versions of the lambda calculus has shown that the innnite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus. Working in the framework of the intersection type disciplines, we devise a type assignment system such that two terms are equal in the innnite lambda calculus ii they can be assigned the same types in any basis. A novel feature of the system is the presence of a type constant to denote the set of all terms of order zero, and the possibility of applying a type to another type. We prove a completeness and an approximation theorem for our system. Our results can be considered as a rst step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms. However some non-continuity phenomena of the innnite lambda calculus make a full realization of this idea (namely the construction of a lter model) a quite diicult task.

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