Statistical properties of lambda terms

TL;DR

This paper provides a detailed statistical analysis of random lambda terms using generating functions, revealing insights into their structure and complexity, including average-case behavior for finding redexes.

Contribution

It introduces a novel analytic method for handling combinatorial parameters in infinite algebraic systems of generating functions for lambda terms.

Findings

Average number of redexes is constant in random lambda terms

Distribution of free variables and de Bruijn indices characterized

Efficient random generation methods for lambda terms with specified parameters

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.