On the Balancedness of Tree-to-word Transducers

TL;DR

This paper investigates the properties of tree-to-word transducers, focusing on the balancedness of their output languages, and provides polynomial-time algorithms for related decision problems in formal language theory.

Contribution

It introduces polynomial-time algorithms to decide balancedness of output languages of non-linear tree transducers and analyzes well-formedness in context-free languages.

Findings

Well-formedness of context-free languages is decidable in polynomial time.

Longest common reduced suffix can be computed in polynomial time.

Decidability of balancedness for non-linear tree transducer outputs.

Abstract

A language over an alphabet $B = A \cup \overline{A}$ of opening ($A$) and closing ($\overline{A}$) brackets, is balanced if it is a subset of the Dyck language $D_B$ over $B$, and it is well-formed if all words are prefixes of words in $D_B$. We show that well-formedness of a context-free language is decidable in polynomial time, and that the longest common reduced suffix can be computed in polynomial time. With this at a hand we decide for the class 2-TWs of non-linear tree transducers with output alphabet $B^*$ whether or not the output language is balanced.