正则语言的剖析能力
The Dissecting Power of Regular Languages
摘要：
近年来对正则语言和上下文无关语言的结构性质的研究极大地促进了我们对这些语言复杂行为的基本认识。我们继续研究如何正则语言的行为时，他们需要削减许多无限的语言。一个特殊的兴趣在于这样一种情况，即正则语言需要将给定的无限语言“解剖”成两个无限大小的子集。每一种上下文无关语言都被精心挑选的正则语言分解（或者说它是REG可分解的）。在一个更大的图片，我们表明，不断增长的语言和半线性语言是REG解剖。在一定的自然条件下，半线性语言的补语和有限交也成为REG-可剖分的。受限于有界语言，许多上下文无关语言的交集，更令人惊讶的是，有界上下文无关语言上的整个布尔层次结构都是REG-可分割的。作为一个直接的应用程序的REG-dissectability，我们展示了另一个结构的属性，其中一个适当的有界上下文无关的语言可以“分离与无限的利润”两个给定的嵌套无限有界上下文无关的语言。
英文摘要：
A recent study on structural properties of regular and context-free languages has greatly promoted our basic understandings of the complex behaviors of those languages. We continue the study to examine how regular languages behave when they need to cut numerous infinite languages. A particular interest rests on a situation in which a regular language needs to "dissect" a given infinite language into two subsets of infinite size. Every context-free language is dissected by carefully chosen regular languages (or it is REG-dissectible). In a larger picture, we show that constantly-growing languages and semi-linear languages are REG-dissectible. Under certain natural conditions, complements and finite intersections of semi-linear languages also become REG-dissectible. Restricted to bounded languages, the intersections of finitely many context-free languages and, more surprisingly, the entire Boolean hierarchy over bounded context-free languages are REG-dissectible. As an immediate application of the REG-dissectibility, we show another structural property, in which an appropriate bounded context-free language can "separate with infinite margins" two given nested infinite bounded context-free languages.
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