The Finitely Generated Types of the λ-Calculus
We answer a question raised by Richard Statman (cf. ) concerning the simply typed λ-calculus (having o as only ground type): Is it possible to generate from a finite set of combinators all the closed terms of a given type ? (By combinators we mean closed λ-terms of any types).
Let us call complexity of a λ-term t the least number of distinct variables required for its writing up to α-equivalence. We prove here that a type T can be generated from a finite set of combinators iff there is a constant bounding the complexity of every closed normal λ-term of type T. The types of rank ⩽ 2 and the types A 1→(A2→…(A n→o)) such that for all i = 1, … n: A i = o, Ai = o→o or A i = (o→(o→…(o→o)))→o, are thus the only inhabited finitely generated types.
KeywordsFree Variable Distinct Variable Small Term Ground Type Sketch Proof
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