We are interested in regular expressions that represent word relations in an alphabet-invariant way—for example, the set of all word pairs $(u,v)$ where v is a prefix of u independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. Labelled graphs (transducers and automata) with alphabet-invariant and user-defined labels were considered in a recent paper. In this paper we study derivatives of regular expressions over labels (atomic objects) in some set B. These labels can be any strings as long as the strings represent subsets of a certain monoid. We show that the number of partial derivatives of any type B regular expression is linearly bounded, and that one can define partial derivative labelled graphs, whose transition labels can be elements of another label set X as long as X and B refer to the same monoid. We also show how to use derivatives directly to decide whether a given word is in the language of a regular expression over set specs. Set specs and pairing specs are label sets allowing one to express languages and relations over large alphabets in a natural and concise way such that many algorithms work directly on these labels without the need to expand these labels to linear or quadratic size expressions.

我们对以字母不变的方式表示单词关系的正则表达式感兴趣，例如，所有单词对的集合 $（ü，v）$其中v是u的前缀与字母无关。当前形式语言对象的软件系统没有定义此类对象的机制。最近的论文中考虑了带有字母不变和用户定义标签的标签图（传感器和自动机）。在本文中，我们研究了某些集合B中标签（原子对象）上的正则表达式的导数。这些标签可以是任何字符串，只要这些字符串代表某个类半体动物的子集即可。我们表明，任何B型正则表达式的偏导数的数量都是线性有界的，并且可以定义带有偏导数的图，只要X和B指代相同的类半圆，它们的过渡标记可以是另一个标记集X的元素。 。我们还将展示如何直接使用派生词来确定给定单词是否符合集合规范中的正则表达式语言。集合规范和配对规范是标签集，允许人们自然而简洁地表达语言和大字母之间的关系，这样许多算法就可以直接在这些标签上工作，而无需将这些标签扩展为线性或二次大小表达式。