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

Abstract

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