# Register Games

## TL;DR

This paper introduces register games, a novel approach that leads to a quasi-polynomial algorithm for parity games, a new complexity measure called register index, and improved automata translations, advancing understanding of parity game complexity.

## Contribution

It presents register games as a new method for solving parity games in quasi-polynomial time and introduces the register index as a new complexity measure, along with automata translation improvements.

## Key findings

- Quasi-polynomial algorithm for parity games.
- Polynomial algorithm for restricted classes of parity games.
- Improved automata translation with quasi-polynomial size increase.

## Abstract

The complexity of parity games is a long standing open problem that saw a major breakthrough in 2017 when two quasi-polynomial algorithms were published. This article presents a third, independent approach to solving parity games in quasi-polynomial time, based on the notion of register game, a parameterised variant of a parity game. The analysis of register games leads to a quasi-polynomial algorithm for parity games, a polynomial algorithm for restricted classes of parity games and a novel measure of complexity, the register index, which aims to capture the combined complexity of the priority assignement and the underlying game graph. We further present a translation of alternating parity word automata into alternating weak automata with only a quasi-polynomial increase in size, based on register games; this improves on the previous exponential translation. We also use register games to investigate the parity index hierarchy: while for words the index hierarchy of alternating parity automata collapses to the weak level, and for trees it is strict, for structures between trees and words, it collapses logarithmically, in the sense that any parity tree automaton of size n is equivalent, on these particular classes of structures, to an automaton with a number of priorities logarithmic in n.