Linear Context-Free Tree Languages and Inverse Homomorphisms

TL;DR

This paper investigates the closure properties of linear context-free tree languages under inverse linear tree homomorphisms, showing non-closure in general but closure for linear monadic cases, with implications for formal language theory.

Contribution

It proves that linear context-free tree languages are not closed under inverse linear tree homomorphisms, but linear monadic variants are closed, clarifying their theoretical boundaries.

Findings

Linear context-free tree languages are not closed under inverse linear tree homomorphisms.

Linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.

The proof involves encoding Dyck words and analyzing preimages under homomorphisms.

Abstract

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. The proof is by contradiction: we encode Dyck words into a context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.