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