Combinatorics of $\lambda$-terms: a natural approach

Abstract

We consider combinatorial aspects of λ-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for λ-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral λ-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of λ-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of λ-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all λ-terms.

How to generate random lambda terms?

“…Should there be fewer abstractions than n + 1, then n represents a free variable occurrence. And so, in the De Bruijn notation both λxyz.xz(yz) and λabc.ac(bc) admit the same representation λλλ.20 (10), see Figure 1.…”

Towards the average-case analysis of substitution resolution in $λ$-calculus

“…The following results exhibit the closed-form of generating functions corresponding to pure terms as well as the general class of λυ-terms and explicit substitutions. Proposition 6.2 (see [5]). Let L ∞ (z) denote the generating function corresponding to the set of λ-terms in υ-normal form (i.e.…”

Statistical Properties of Lambda Terms

“…Enumeration. In the current paper we follow [7,8,27,15] and investigate the statistical properties of random λ-terms in the de Bruijn representation. We assume a unary base encoding of indices, i.e.…”

“…Investigations into quantitative properties of λ-terms in the de Bruijn notation were continued by Bendkowski et al [7,8] who showed that, in contrast to the canonical representation of David et al, asymptotically almost all λ-terms are not strongly normalising. In other words, the proportion of terms for which all evaluation strategies terminate approaches zero as the term size tends to infinity.…”