Weighted Tree Automata with Constraints

TL;DR

This paper extends the concept of tree automata with constraints to the weighted setting, demonstrating their ability to represent images of regular weighted tree languages under certain homomorphisms and exploring related decision problems.

Contribution

It introduces weighted tree automata with constraints, generalizing unweighted automata, and investigates their closure properties and decision problems.

Findings

Weighted tree automata with constraints can represent images of regular weighted tree languages.

Closure properties of weighted tree languages with constraints are established.

Decision problems for these automata are analyzed.

Abstract

The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Gim\'enez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints, can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.