Universal trees grow inside separating automata: Quasi-polynomial lower bounds for parity games

TL;DR

This paper establishes a quasi-polynomial lower bound on the size of separating automata for parity games, revealing fundamental limits that challenge existing approaches and advancing understanding of the problem's complexity.

Contribution

It introduces the concept of universal ordered trees and proves a quasi-polynomial lower bound on their size, impacting automata-based methods for solving parity games.

Findings

Quasi-polynomial lower bound on separating automata size

Universal ordered trees are embedded in automaton state spaces

Barrier to existing approaches for polynomial-time algorithms

Abstract

Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving parity games, a key technical component of which is constructing (explicitly or implicitly) an automaton that separates languages of words encoding plays that are (decisively) won by either of the two players. Our main technical result is a quasi-polynomial lower bound on the size of such separating automata that nearly matches the current best upper bounds. This forms a barrier that all existing approaches must overcome in the ongoing quest for a polynomial-time algorithm for solving parity games. The key and fundamental concept that we introduce and study is a universal ordered tree. The technical highlights are a quasi-polynomial lower bound on the size of universal ordered trees and a proof that every separating safety automaton has a universal tree hidden in its state space.