In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended regular tree expressions (using negation or intersection operators), and we show how to compute a Brzozowski-like inductive tree automaton; the fixed point of this construction, when it exists, is the derivative tree automaton. Such a deterministic tree automaton can be used to solve the membership test efficiently: the whole structure is not necessarily computed, and the derivative computations can be performed in parallel. We also show how to solve the membership test using our (Bottom-Up) partial derivatives, without computing an automaton.

中文翻译：

---

树表达式的自底向上导数

在本文中，我们使用已知的商的归纳公式将（词）导数和偏导数的概念（分别）由 Brzozowski 和 Antimirov 扩展到树导数。我们定义了一个新的扩展正则树表达式系列（使用否定或交集运算符），并展示了如何计算类似 Brzozowski 的归纳树自动机；这种构造的不动点，当它存在时，就是派生树自动机。这种确定性树自动机可用于有效地解决隶属度测试：不一定要计算整个结构，并且可以并行执行导数计算。我们还展示了如何使用我们的（自下而上）偏导数来解决成员资格测试，而无需计算自动机。