Visibly Tree Automata with Memory and Constraints

TL;DR

This paper introduces Visibly Tree Automata with Memory (VTAM), a new class of tree automata with unbounded memory that enjoys good closure properties and decidability, extending previous models with enhanced features.

Contribution

The paper generalizes visibly pushdown automata to tree automata with memory, establishing closure, determinization, and decidability results for VTAM and its extensions.

Findings

VTAM can be determinized and have decidable emptiness, membership, inclusion, and universality.

Extensions with constraints preserve closure and decidability properties.

VTAM can express complex tree languages with relevant examples.

Abstract

Tree automata with one memory have been introduced in 2001. They generalize both pushdown (word) automata and the tree automata with constraints of equality between brothers of Bogaert and Tison. Though it has a decidable emptiness problem, the main weakness of this model is its lack of good closure properties. We propose a generalization of the visibly pushdown automata of Alur and Madhusudan to a family of tree recognizers which carry along their (bottom-up) computation an auxiliary unbounded memory with a tree structure (instead of a symbol stack). In other words, these recognizers, called Visibly Tree Automata with Memory (VTAM) define a subclass of tree automata with one memory enjoying Boolean closure properties. We show in particular that they can be determinized and the problems like emptiness, membership, inclusion and universality are decidable for VTAM. Moreover, we propose several extensions of VTAM whose transitions may be constrained by different kinds of tests between memories and also constraints a la Bogaert and Tison comparing brother subtrees in the tree in input. We show that some of these classes of constrained VTAM keep the good closure and decidability properties, and we demonstrate their expressiveness with relevant examples of tree languages.