Properties of quasi-alphabetic tree bimorphisms

TL;DR

This paper investigates quasi-alphabetic tree bimorphisms, providing a canonical form, analyzing closure properties, and establishing their relation to context-free languages and other tree transformation classes.

Contribution

It introduces a canonical representation of quasi-alphabetic relations, explores their closure properties, and connects them to context-free languages and various tree transformation frameworks.

Findings

Closed under union

Not closed under intersection and complement

Equivalent to products of context-free string languages

Abstract

We study the class of quasi-alphabetic relations, i.e., tree transformations defined by tree bimorphisms with two quasi-alphabetic tree homomorphisms and a regular tree language. We present a canonical representation of these relations; as an immediate consequence, we get the closure under union. Also, we show that they are not closed under intersection and complement, and do not preserve most common operations on trees (branches, subtrees, v-product, v-quotient, f-top-catenation). Moreover, we prove that the translations defined by quasi-alphabetic tree bimorphism are exactly products of context-free string languages. We conclude by presenting the connections between quasi-alphabetic relations, alphabetic relations and classes of tree transformations defined by several types of top-down tree transducers. Furthermore, we get that quasi-alphabetic relations preserve the recognizable and algebraic tree languages.