Statistical Properties of Lambda Terms

Abstract

We present a quantitative, statistical analysis of random lambda terms in the de Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the de Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

A note on the asymptotic expressiveness of ZF and ZFC

“…1 The current definition is a simplified version of [1,Definition 5.5]. The original one involves systems of functional equations and permits additional vectors of so-called marking variables which can be used, for instance, to track the behaviour of certain interesting sub-patterns of random structures.…”

“…We base our analysis on a mixture of methods from analytic combinatorics and, more specifically, recent advances in its application in the quantitative analysis of λ-terms [1,3,5].…”