Bottom-Up Derivatives of Tree Expressions

TL;DR

This paper generalizes derivatives to tree expressions, enabling efficient membership testing through automaton construction or partial derivatives without full automaton computation.

Contribution

It introduces a novel approach to tree derivatives, extending regular expression derivatives to trees with negation and intersection, and provides methods for efficient membership testing.

Findings

Automaton construction for tree derivatives is feasible and efficient.

Partial derivatives enable membership testing without automaton construction.

The approach generalizes derivatives to complex tree expressions with negation and intersection.

Abstract

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.