# Weighted Regular Tree Grammars with Storage

## TL;DR

This paper introduces weighted regular tree grammars with storage, combining concepts from regular tree grammars, storage, and weighted automata, and explores their properties, logical characterizations, and simplifications.

## Contribution

It presents a novel framework for weighted regular tree grammars with storage, including their characterization, support generation, and logical description.

## Key findings

- Support of weighted tree languages can be generated by unweighted grammars.
- Weighted tree languages are characterized by a composition of basic concepts.
- Results on elimination of chain rules and finite storage types.

## Abstract

We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a weighted tree language, which is a mapping from the set of trees to the multioperator monoid. We prove that, for multioperator monoids canonically associated to particular strong bi-monoids, the support of the generated weighted tree languages can be generated by (unweighted) regular tree grammars with storage. We characterize the class of all generated weighted tree languages by the composition of three basic concepts. Moreover, we prove results on the elimination of chain rules and of finite storage types, and we characterize weighted regular tree grammars with storage by a new weighted MSO-logic.