Infinite Separation between General and Chromatic Memory

Abstract

In this note, we answer a question from [Alexander Kozachinskiy. State Complexity of Chromatic Memory in Infinite-Duration Games, arXiv:2201.09297]. Namely, we construct a winning condition W over a finite set of colors such that, first, every finite arena has a strategy with 2 states of general memory which is optimal w.r.t. W , and second, there exists no k such that every finite arena has a strategy with k states of chromatic memory which is optimal w.r.t. W .