点深度二的分离问题
Separation for dot-depth two
AI总结：
针对点深度层级第二层的分离问题，提出判定算法，并推广至基础有限的连接层级第一层，解决自动机理论中的开放问题。
摘要：
Brzozowski 和 Cohen 提出的点深度层级对有限词的无星语言进行分类。根据 McNaughton 和 Papert 的定理，这些语言也是一阶可定义语言。点深度在 Thomas 的工作之后变得突出，他证明了点深度层级与一阶逻辑的量词交替层级之间存在精确对应关系：点深度层级中的每一层都由所有可以用规定数量的量词块定义的语言组成。自动机理论中最著名的开放问题之一是确定每一层的成员资格问题是否可判定：即是否有可能判定一个输入正则语言是否属于该层？尽管进行了大量的研究工作，但成员资格问题本身仅在低层级上得到解决。最近的一个突破是用一个更一般的问题——分离问题——取代成员资格问题。给定两个输入语言，必须判定是否存在一个属于所研究层级的第三种语言，它包含第一种语言且与第二种语言不相交。其动机是：（1）虽然分离问题更难，但回报更大；（2）它提供了一个更方便的框架；（3）所有最近的成员资格算法都是归约到较低层级的分离问题。我们提出了一个针对点深度二的分离算法。虽然这是我们的最突出应用，但我们的结果更具一般性。我们考虑一个包含点深度层级的层级族：连接层级。它们通过一个通用的构造过程构建。首先选择一个初始类，即基础，它是层级中的最低层。进一步的层级通过应用通用操作来构建。我们的主要定理指出，对于任何基础为有限的连接层级，第一层的分离问题都是可判定的。在点深度的特殊情况下，利用先前已知的结果，这可以提升到第二层。
英文摘要：
The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.
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