# Distribution of variables in lambda-terms with restrictions on De Bruijn indices and De Bruijn levels

## TL;DR

This paper analyzes the distribution of variables in restricted lambda-terms with bounded De Bruijn indices and levels, showing asymptotic normality and deriving formulas for key constants, with insights into their structural profiles.

## Contribution

It provides the first asymptotic analysis of variable distribution in lambda-terms with bounded De Bruijn indices and levels, including explicit formulas for constants and structural profiles.

## Key findings

- Variables are asymptotically normally distributed in both subclasses.
- Mean and variance of variables grow linearly with the size of the lambda-terms.
- The study reveals a distinctive 'unary profile' across De Bruijn levels.

## Abstract

We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound of the nesting levels of abstractions, \textit{i.e.}, the number of De Bruijn levels, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable. We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with bounded De Bruijn index. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we investigate the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape.

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Source: https://tomesphere.com/paper/1903.05243