Enumeration of generalized $BCI$ lambda-terms

TL;DR

This paper analyzes the asymptotic enumeration of specific classes of closed lambda-terms, deriving differential equations and recurrence relations to estimate their counts and facilitate computations.

Contribution

It introduces differential equations and recurrence relations for counting $BCI(p)$-terms, $BCK(p)$-terms, and closed lambda-terms, advancing enumeration methods in combinatory logic.

Findings

Derived asymptotic formulas for $BCI(p)$-terms

Established differential equations for generating functions

Provided efficient recurrence relations for counting sequences

Abstract

We investigate the asymptotic number of elements of size $n$ in a particular class of closed lambda-terms (so-called $BCI(p)$-terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of $BCK(p)$-terms and that of closed lambda-terms. Using elementary arguments we obtain upper and lowerestimates for the number of closed lambda-terms of size $n$. Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. $BCK(p)$-terms are discussed briefly.