[Paper Review] A Universal Attractor Decomposition Algorithm for Parity Games.

Marcin Jurdziński, Rémi Morvan|arXiv (Cornell University)|Jan 13, 2020

Formal Methods in Verification26 references4 citations

TL;DR

This paper introduces a universal attractor decomposition algorithm for solving parity games that generalizes McNaughton-Zielonka and recent quasi-polynomial algorithms. By leveraging embeddable ordered trees representing attractor decompositions of dominions, the algorithm achieves a symbolic space complexity of $O(\lg d)$, exponentially improving prior bounds and eliminating dependence on $n$, the number of vertices.

ABSTRACT

An attractor decomposition meta-algorithm for solving parity games is given that generalizes the classic McNaughton-Zielonka algorithm and its recent quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and Wojtczak (2019). The central concepts studied and exploited are attractor decompositions of dominia in parity games and the ordered trees that describe the inductive structure of attractor decompositions. The main technical results include the embeddable decomposition theorem and the dominion separation theorem that together help establish a precise structural condition for the correctness of the universal algorithm: it suffices that the two ordered trees given to the algorithm as inputs embed the trees of some attractor decompositions of the largest dominia for each of the two players, respectively. The universal algorithm yields McNaughton-Zielonka, Parys's, and Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal trees are given to it as inputs. The main technical results provide a unified proof of correctness and deep structural insights into those algorithms. A symbolic implementation of the universal algorithm is also given that improves the symbolic space complexity of solving parity games in quasi-polynomial time from $O(d \lg n)$---achieved by Chatterjee, Dvořak, Henzinger, and Svozil (2018)---down to $O(\lg d)$, where $n$ is the number of vertices and $d$ is the number of distinct priorities in a parity game. This not only exponentially improves the dependence on $d$, but it also entirely removes the dependence on $n$.

Motivation & Objective

• To unify existing quasi-polynomial algorithms for solving parity games under a single, general framework.
• To identify the precise structural conditions under which a universal algorithm correctly solves parity games.
• To reduce symbolic space complexity in solving parity games, especially in terms of dependency on the number of priorities $d$.
• To provide a symbolic implementation that improves space efficiency beyond previous quasi-polynomial methods.

Proposed method

• The algorithm uses attractor decompositions of dominions in parity games, structured via ordered trees that encode inductive decomposition processes.
• It introduces the embeddable decomposition theorem, ensuring correctness when input trees embed actual attractor decomposition trees of the largest dominions.
• The dominion separation theorem establishes structural conditions for decomposing games based on player-specific dominions.
• The universal algorithm is instantiated with specific universal trees to recover known algorithms, including McNaughton-Zielonka, Parys’s, and Lehtinen-Schewe-Wojtczak’s variants.
• A symbolic implementation is developed using tree embeddings and attractor-based operations, minimizing symbolic space usage.
• The symbolic space complexity is reduced to $O(\lg d)$, independent of $n$, by exploiting tree structure and attractor invariants.

Experimental results

Research questions

• RQ1 What structural conditions ensure the correctness of a universal attractor decomposition algorithm for parity games?
• RQ2 How can existing quasi-polynomial algorithms for parity games be unified under a single algorithmic framework?
• RQ3 Can symbolic space complexity in solving parity games be reduced to depend only on the number of distinct priorities $d$?
• RQ4 What role do ordered trees and tree embeddings play in characterizing attractor decompositions of dominions?
• RQ5 Is it possible to design a symbolic implementation that eliminates dependence on $n$ while maintaining quasi-polynomial time complexity?

Key findings

• The universal algorithm correctly solves parity games if and only if the input trees embed the attractor decomposition trees of the largest dominions for each player.
• The embeddable decomposition theorem and dominion separation theorem provide a precise structural foundation for the correctness of the universal algorithm.
• The symbolic space complexity of the universal algorithm is $O(\lg d)$, which is an exponential improvement over the previous $O(d \lg n)$ bound.
• The algorithm generalizes McNaughton-Zielonka, Parys’s, and Lehtinen-Schewe-Wojtczak’s algorithms as special cases through appropriate choice of input trees.
• The symbolic implementation achieves $O(\lg d)$ space complexity, completely removing dependence on the number of vertices $n$.
• The results provide a unified proof of correctness and deep structural insights into existing quasi-polynomial solvers for parity games.