覆盖问题
The Covering Problem
AI总结：
本文提出覆盖问题作为成员资格问题的替代，以更好地研究逻辑表达能力，并通过三个论点论证其合理性，同时提供相应数学框架与方法论。
摘要：
计算机科学中的一个重要努力是理解逻辑形式体系在离散结构（如单词）上的表达能力。自然，“理解”并非数学概念。因此，这项研究需要一个具体的目标来捕捉这种理解。在文献中，这一目标的标准选择是成员资格问题，其目的是寻找一个程序，以判定输入的正则语言是否能在所研究的逻辑中定义。这一方法由Schützenberger、McNaughton和Papert关于一阶逻辑的开创性工作确立为正确的方法，并自此一直沿用。然而，成员资格问题很困难：对于几个重要的片段，尽管经过数十年的研究，研究人员仍未成功。鉴于关于最著名的开放问题之一——即一阶逻辑的量词交替层级——的最新结果，一种解释可能是成员资格作为设置过于严格。这些新结果确实是通过考虑比成员资格更一般的问题而获得的，利用了丰富数学设置中增加的灵活性。这开辟了一条有前景的研究途径，并已致力于为自然片段识别和解决此类问题。然而到目前为止，这些问题都是特设的，大多数片段依赖于特定的一个问题。一个取代成员资格作为正确问题的独特新问题仍然缺失。本文的主要贡献是一个适合扮演这一角色的候选者：覆盖问题。我们用3个论点来论证这个问题。首先，它承认一个初等的集合论表述，类似于成员资格。其次，我们能够用这个问题重新解释或推广所有已知结果。第三，我们开发了一个数学框架和一种专门针对研究这个问题的方法论。
英文摘要：
An important endeavor in computer science is to understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. This investigation requires therefore a concrete objective to capture this understanding. In the literature, the standard choice for this objective is the membership problem, whose aim is to find a procedure deciding whether an input regular language can be defined in the logic under investigation. This approach was cemented as the right one by the seminal work of Schützenberger, McNaughton and Papert on first-order logic and has been in use since then. However, membership questions are hard: for several important fragments, researchers have failed in this endeavor despite decades of investigation. In view of recent results on one of the most famous open questions, namely the quantifier alternation hierarchy of first-order logic, an explanation may be that membership is too restrictive as a setting. These new results were indeed obtained by considering more general problems than membership, taking advantage of the increased flexibility of the enriched mathematical setting. This opens a promising research avenue and efforts have been devoted at identifying and solving such problems for natural fragments. Until now however, these problems have been ad hoc, most fragments relying on a specific one. A unique new problem replacing membership as the right one is still missing. The main contribution of this paper is a suitable candidate to play this role: the Covering Problem. We motivate this problem with 3 arguments. First, it admits an elementary set theoretic formulation, similar to membership. Second, we are able to reexplain or generalize all known results with this problem. Third, we develop a mathematical framework and a methodology tailored to the investigation of this problem.
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