Random Fruits on the Zielonka Tree

TL;DR

This paper investigates how randomization in strategies can reduce memory requirements in stochastic Muller games, providing tight bounds for the amount of memory saved.

Contribution

It introduces the first tight bounds on memory reduction via randomization in stochastic Muller games, advancing understanding of controller synthesis in probabilistic settings.

Findings

Randomization can significantly reduce memory in stochastic Muller games.

Matching upper and lower bounds for memory savings are established.

Results improve the theoretical understanding of strategy complexity in stochastic games.

Abstract

Stochastic games are a natural model for the synthesis of controllers confronted to adversarial and/or random actions. In particular, $\omega$-regular games of infinite length can represent reactive systems which are not expected to reach a correct state, but rather to handle a continuous stream of events. One critical resource in such applications is the memory used by the controller. In this paper, we study the amount of memory that can be saved through the use of randomisation in strategies, and present matching upper and lower bounds for stochastic Muller games.