# Alternating Weak Automata from Universal Trees

## TL;DR

This paper presents an improved translation method from alternating parity automata to alternating weak automata, significantly reducing state blow-up and potentially enabling faster algorithms for solving parity games.

## Contribution

It introduces a quasi-polynomial translation based on universal trees, improving upon previous methods and impacting the complexity of solving parity games.

## Key findings

- State blow-up is quasi-polynomial, polynomial with logarithmic priorities.
- The translation improves upon previous exponential and quasi-polynomial methods.
- Potential for faster algorithms in solving parity games if the translation is efficiently constructive.

## Abstract

An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and only polynomial if the asymptotic number of priorities is logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if---like all presently known such translations---it is efficiently constructive) lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art (Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdzi\'nski and Lazi\'c, 2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough.