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

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.

Universal Graphs and Good for Games Automata: New Tools for Infinite Duration Games

“…The idea of separating automata 4 was introduced by Bojańczyk and Czerwiński [BC18] to reformulate the first quasipolynomial time algorithm [CJK+17]. Czerwiński, Daviaud, Fijalkow, Jurdziński, Lazić, and Parys [CDF+19] showed that the other two quasipolynomial time algorithms [JL17,Leh18] also can be understood as the construction of separating automata.…”

“…We are paying now a closer attention to the particular case of the parity condition. The technical developments that follow give an alternative proof of the equivalence results proved in [CDF+19] between strongly separating automata and universal trees.…”

“…In particular we use a technique of saturation of graphs which simplifies greatly the arguments. The two theorems together give an alternative simpler proof of the result in [CDF+19].…”

“…A concrete instanciation of good for small games automata is the notion of separating automata, which was introduced by Bojańczyk and Czerwiński [BC18] to reformulate the first quasipolynomial time algorithm of [CJK+17]. Later Czerwiński, Daviaud, Fijalkow, Jurdziński, Lazić, and Parys [CDF+19] showed that the other two quasipolynomial time algorithms also can be understood as the construction of separating automata, and proved a quasipolynomial lower bound on the size of separating automata.…”

Symbolic Parity Game Solvers that Yield Winning Strategies

“…Second, the class of progress measures algorithms [21], where each vertex in the game has a value which increases monotonically, based on the values of the direct successors of the vertex. These values are typically tuples of integers, and in recent work various authors have proposed sets of values that are quasi-polynomially bounded in size yet sufficient to solve parity games [7,9]. Third, many different strategy improvement algorithms [33] have been studied, where one player iteratively improves its strategies by playing against the best response of the opponent.…”