Solving the Weighted HOM-Problem With the Help of Unambiguity

TL;DR

This paper proves the decidability of the weighted HOM-problem over general semirings by introducing restrictions on the homomorphism and automaton ambiguity, extending known unweighted results.

Contribution

It extends the decidability of the HOM-problem to weighted cases over general semirings using new restrictions and ambiguity notions.

Findings

Decidability of the weighted HOM-problem for zero-sum free semirings.

Reduction of the weighted problem to the unweighted case under certain conditions.

Introduction of tetris-free homomorphisms and ambiguity notions for weighted automata.

Abstract

The HOM-problem, which asks whether the image of a regular tree language under a tree homomorphism is again regular, is known to be decidable by [Godoy, Gim\'enez, Ramos, \`Alvarez: The HOM problem is decidable. STOC (2010)]. Research on the weighted version of this problem, however, is still in its infancy since it requires customized investigations. In this paper we address the weighted HOM-problem and strive to keep the underlying semiring as general as possible. In return, we restrict the input: We require the tree homomorphism h to be tetris-free, a condition weaker than injectivity, and for the given weighted tree automaton, we propose an ambiguity notion with respect to h. These assumptions suffice to ensure decidability of the thus restricted HOM-problem for all zero-sum free semirings by allowing us to reduce it to the (decidable) unweighted case.