# Pushing for weighted tree automata

## TL;DR

This paper introduces a weight normalization procedure called pushing for weighted tree automata over commutative semifields, enabling efficient minimization and equivalence testing, especially for bottom-up deterministic automata.

## Contribution

It presents a novel pushing normalization method for weighted tree automata that preserves recognized languages and improves equivalence testing efficiency.

## Key findings

- Normalization preserves recognized weighted tree languages.
- New equivalence test runs in near-linear time, improving over previous methods.
- Reduction to unweighted automata simplifies minimization and testing.

## Abstract

A weight normalization procedure, commonly called pushing, is introduced for weighted tree automata (wta) over commutative semifields. The normalization preserves the recognized weighted tree language even for nondeterministic wta, but it is most useful for bottom-up deterministic wta, where it can be used for minimization and equivalence testing. In both applications a careful selection of the weights to be redistributed followed by normalization allows a reduction of the general problem to the corresponding problem for bottom-up deterministic unweighted tree automata. This approach was already successfully used by Mohri and Eisner for the minimization of deterministic weighted string automata. Moreover, the new equivalence test for two wta $M$ and $M'$ runs in time $\mathcal O((\lvert M \rvert + \lvert M'\rvert) \cdot \log {(\lvert Q\rvert + \lvert Q'\rvert)})$, where $Q$ and $Q'$ are the states of $M$ and $M'$, respectively, which improves the previously best run-time $\mathcal O(\lvert M \rvert \cdot \lvert M'\rvert)$.