Solving Streett and Emerson-Lei Games with Universal Trees

2026-08-17

AI Summary

This study addresses the limitations of universal trees in Streett and Emerson-Lei games under bounded memory, where existing reduction methods fail to yield memory-optimal strategies. We elucidate the interaction between universal trees and Zielonka trees, proposing a direct solving algorithm that outperforms parity game reductions. This approach overcomes the restriction of universal trees to memoryless strategies by integrating symbolic algorithms, effectively replacing polynomial factors in complexity with the size of the universal tree. Consequently, our method significantly reduces time complexity while enabling the generation of memory-optimal strategies. These contributions establish a superior theoretical foundation and algorithmic framework for solving these classes of games, advancing beyond prior reduction-based techniques.

Abstract

Nearly a decade ago, Calude et al. showed that parity games can be solved in quasi-polynomial time. This result is now understood in terms of universal trees. By reduction to parity games, the quasi-polymonial result can benefit all omega-regular games. However, beyond such reductions, and with the exception of Rabin games, our understanding of the role of universal trees in direct solutions is still quite limited. In this work, we refute the common view that universal trees are relevant only for games that admit memoryless winning strategies. We contribute a full understanding of how universal trees interact with Zielonka trees for the solution of Streett and Emerson-Lei games. As a consequence, we show that winning regions and strategies in Streett games with $n$ vertices, $m$ edges, and $k$ pairs can be computed in time $O(mk\log(k)k!|U(n,k)|)$, where $U(n,k)$ is a universal tree for $n$ leaves and depth $k$. This improves upon the best previously known complexity result for Streett games, which relied on reduction to parity games and their quasi-polynomial solution. Furthermore, we show that winning regions and strategies for Emerson-Lei games with $n$ vertices, $m$ edges, and $c$ colors can be computed in time $O(mc\log(c)c!|U(n,c/2)|)$, again improving over reductions to parity games. Notably, our approach yields memory-optimal strategies, in contrast to those obtained via reductions to parity games. Finally, we show how universal trees can be used to bound the recursion tree of the Zielonka-McNaughton algorithm for Emerson-Lei games. This leads to a symbolic algorithm that replaces the factor $n^c$ in the time complexity of existing symbolic approaches with $|U(n,c)|$.

Problem

• Streett games
• Emerson-Lei games
• Universal trees
• Omega-regular games
• Memory-optimal strategies

Innovation

• Universal Trees
• Streett Games
• Emerson-Lei Games
• Zielonka Trees
• Symbolic Algorithm