迭代的层次：从 Elgot 到 Kleene
Shades of Iteration: from Elgot to Kleene
AI总结：
本文以 Elgot 单子与 Kleene 单子刻画两类迭代，并提出 while-monads，以连接并涵盖支持 while 循环但未必满足 Kleene 代数定律的模型。
摘要：
迭代的概念范围很广，从可以说是最一般的 Elgot 迭代，到非常具体的 Kleene 迭代。Bloom 和 Esik 以迭代理论的形式广泛探讨了 Elgot 迭代的根本性质，而 Kleene 迭代作为自动机理论、正则表达式和 Kleene 代数等（无类型）形式体系的组成部分而变得极为流行。在本文中，我们分别以 Elgot 单子和 Kleene 单子的形式，建立了 Elgot 迭代与 Kleene 迭代之间的形式联系。我们还引入了一类新的 while-monads；与 Kleene 单子类似，它们允许用代数术语进行相对简单的描述。与 Elgot 单子类似，while-monads 涵盖了大量有意义地支持 while 循环的模型，但这些模型可能不满足 Kleene 代数定律，甚至可能完全不支持 Kleene 迭代算子。
英文摘要：
Notions of iteration range from the arguably most general Elgot iteration to a very specific Kleene iteration. The fundamental nature of Elgot iteration has been extensively explored by Bloom and Esik in the form of iteration theories, while Kleene iteration became extremely popular as an integral part of (untyped) formalisms, such as automata theory, regular expressions and Kleene algebra. Here, we establish a formal connection between Elgot iteration and Kleene iteration in the form of Elgot monads and Kleene monads, respectively. We also introduce a novel class of while-monads, which like Kleene monads admit a relatively simple description in algebraic terms. Like Elgot monads, while-monads cover a large variety of models that meaningfully support while-loops, but may fail the Kleene algebra laws, or even fail to support a Kleen iteration operator altogether.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
