关于闭包与环境的枚举及其在随机生成中的应用
On the enumeration of closures and environments with an application to random generation
AI总结：
本文研究λ-演算中环境和闭包的渐近计数与随机生成，针对朴素与封闭两类结构，利用生成函数构造有效采样器，并讨论封闭类的相关问题。
摘要：
环境和闭包是λ-演算中求值的两个主要组成部分。闭包是由一个λ-项和一个环境组成的对，而环境是分配给自由变量的λ-项列表。在本文中，我们研究了λ-演算中求值的某些动态方面，考虑了环境和闭包的定量组合性质。我们聚焦于两类环境和闭包，即所谓的朴素类和封闭类，并考虑它们的渐近计数和有效随机生成问题。我们提供了大小为$n$的朴素环境和朴素闭包数量的渐近近似。利用相关的生成函数，我们为这两类组合结构构造了有效的采样器。最后，我们讨论了封闭环境和封闭闭包的渐近计数与随机生成的相关问题。
英文摘要：
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.
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