On the enumeration of closures and environments with an application to random generation

TL;DR

This paper studies the combinatorial properties of environments and closures in lambda-calculus, providing asymptotic counts and efficient random generators for plain and closed structures.

Contribution

It introduces asymptotic enumeration formulas and effective sampling methods for environments and closures, including closed variants, in lambda-calculus.

Findings

Asymptotic approximation of the number of plain environments and closures.

Construction of effective samplers for plain and closed environments and closures.

Discussion of asymptotic counting and random generation for closed structures.

Abstract

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures.