One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

中文翻译：

---

关于置换文法的隶属问题——NP-完全性的直接证明

与形式语言相关的最重要的一类问题是隶属问题（也称为单词问题），即确定给定的输入单词是否属于指定的语言，例如生成语法。对于上下文无关语言，该问题可以通过各种众所周知的解析算法有效地解决。但是，有几种重要的语言不是上下文无关的。上下文相关语言类的成员问题是 PSPACE 完全的，因此，人们认为它通常不能以有效的方式解决。上述两个类之间存在各种语言类，它们具有各种复杂性的成员问题。这些类之一，即置换语言类，是由置换文法生成的，即用置换规则扩展的上下文无关文法，其中置换规则允许在句子形式中交换两个连续非终结符的位置。本文研究了置换语言的隶属问题。提供了一个证明来证明这个问题是 NP 完全的。