分离问题的一个转移定理
A Transfer Theorem for the Separation Problem
摘要：
我们研究了C类正则词语言的两个问题。C-成员问题需要一个算法来判断输入语言是否属于C。C-分离问题要求一种算法，给定两种正则语言作为输入，判断C中是否存在包含第一种语言的第三种语言，同时与第二种语言不相交。这些问题被认为是手段，以获得一个深刻的理解类C。这类类通常由逻辑形式定义。逻辑通常通过添加新的谓词建立在彼此之上。一个自然的结构是用后继关系来丰富逻辑。本文给出了两个转移结果的简单的自包含证明：证明了对于适当的逻辑定义类，成员关系，用后继关系丰富的类的分离问题简化为原始类的相同问题。我们的约简对有限词和无限词的语言都有效。证明大多是自包含的，只需要正则语言的基本背景。因此，本文给出了新的，简单的证明结果，被认为是困难的，如决定性的隶属度问题的水平1，3/2，2和5/2的点深度层次。
英文摘要：
We investigate two problems for a class C of regular word languages. The C-membership problem asks for an algorithm to decide whether an input language belongs to C. The C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. These problems are considered as means to obtain a deep understanding of the class C. It is usual for such classes to be defined by logical formalisms. Logics are often built on top of each other, by adding new predicates. A natural construction is to enrich a logic with the successor relation. In this paper, we obtain simple self-contained proofs of two transfer results: we show that for suitable logically defined classes, the membership, resp. the separation problem for a class enriched with the successor relation reduces to the same problem for the original class. Our reductions work both for languages of finite words and infinite words. The proofs are mostly self-contained, and only require a basic background on regular languages. This paper therefore gives new, simple proofs of results that were considered as difficult, such as the decid- ability of the membership problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy.
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