The quotient of a formal language [Formula: see text] by another language [Formula: see text] is the set of all strings obtained by taking a string from [Formula: see text] that ends with a suffix of a string from [Formula: see text], and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly [Formula: see text], where [Formula: see text] and [Formula: see text] are the numbers of states in the automata recognizing [Formula: see text] and [Formula: see text], respectively.

中文翻译：

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输入驱动下推自动机商运算的状态复杂度

一种形式语言 [Formula: see text] 与另一种语言 [Formula: see text] 的商是通过从 [Formula: see text] 中获取一个以 [Formula: see text] 中的字符串的后缀结尾的字符串获得的所有字符串的集合：见文本]，并删除该后缀。任何语言对正则语言的商总是正则的，而上下文无关语言及其许多子族，如线性语言和确定性语言，在商运算下不是封闭的。本文在商运算下建立了输入驱动下推自动机（IDPDA），也称为可视下推自动机识别的语言族的闭包。给出了表示操作结果的自动机的构造，并且其相对于非确定性 IDPDA 的状态复杂度正好为 [公式：