An open problem posed by Milner asks for a proof that a certain axiomatisation, which Milner showed is sound with respect to bisimilarity for regular expressions, is also complete. One of the main difficulties of the problem is the lack of a full Kleene theorem, since there are automata that can not be specified, up to bisimilarity, by an expression. Grabmayer and Fokkink (2020) characterise those automata that can be expressed by regular expressions without the constant 1, and use this characterisation to give a positive answer to Milner's question for this subset of expressions. In this paper, we analyse Grabmayer and Fokkink's proof of completeness from the perspective of universal coalgebra, and thereby give an abstract account of their proof method. We then compare this proof method to another approach to completeness proofs from coalgebraic language theory. This culminates in two abstract proof methods for completeness, what we call the local and global approaches, and a description of when one method can be used in place of the other.

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关于星型表达式和代数完备性定理

Milner 提出的一个公开问题要求证明某个公理化（Milner 证明在正则表达式的双相似性方面是合理的）也是完备的。该问题的主要困难之一是缺乏完整的 Kleene 定理，因为存在无法通过表达式指定的自动机，直至双相似性。Grabmayer 和 Fokkink (2020) 刻画了那些可以通过没有常量 1 的正则表达式表达的自动机，并使用这种刻画对 Milner 对这个表达式子集的问题给出肯定的回答。在本文中，我们从泛代数的角度分析了 Grabmayer 和 Fokkink 的完备性证明，从而对他们的证明方法进行了抽象说明。然后，我们将这种证明方法与来自代数语言理论的另一种完整性证明方法进行比较。这最终形成了两种抽象的完整性证明方法，我们称之为局部方法和全局方法，以及何时可以使用一种方法代替另一种方法的描述。