Quantitative Aspects of Linear and Affine Closed Lambda Terms

Abstract

Affine λ-terms are λ-terms in which each bound variable occurs at most once, and linear λ-terms are λ-terms in which each bound variable occurs once and only once. In this article, we count the number of affine closed λ-terms of size n , linear closed λ-terms of size n , affine closed β-normal forms of size n , and linear closed β-normal forms of size n , for several measures of the size of λ-terms. From these formulas, we show how we can derive programs for generating all the terms of size n for each class. The foundation of all of this is a specific data structure, made of contexts in which one counts all the holes at each level of abstractions by λ’s.