State Complexity of Chromatic Memory in Infinite-Duration Games

Abstract

A major open problem in the area of infinite-duration games is to characterize winning conditions that are determined in finite-memory strategies. Infiniteduration games are usually studied over edge-colored graphs, with winning conditions that are defined in terms of sequences of colors. In this paper, we investigate a restricted class of finite-memory strategies called chromatic finite-memory strategies. While general finite-memory strategies operate with sequences of edges of a game graph, chromatic finite-memory strategies observe only colors of these edges.Recent results in this area show that studying finite-memory determinacy is more tractable when we restrict ourselves to chromatic strategies. On the other hand, as was shown by Le Roux (CiE 2020), determinacy in general finite-memory strategies implies determinacy in chromatic finite-memory strategies. Unfortunately, this result is quite inefficient in terms of the state complexity: to replace a winning strategy with few states of general memory, we might need much more states of chromatic memory. The goal of the present paper is to find out the exact state complexity of this transformation.For every winning condition and for every game graph with n nodes we show the following: if this game graph has a winning strategy with q states of general memory, then it also has a winning strategy with (q + 1) n states of chromatic memory. We also show that this bound is almost tight. For every q and n, we construct a winning condition and a game graph with n + O(1) nodes, where one can win with q states of general memory, but not with q n − 1 states of chromatic memory.

Infinite Separation between General and Chromatic Memory

“…There is no contradiction with Theorem 1 because k chr depends not only on k gen , but also on A. A tight bound on k chr in terms of k gen and the number of nodes of A was obtained in [12].…”

“…To answer it, we have to go beyond Muller and even ω-regular conditions (because ChrMem(W ) is finite for them). In [12], we raised this question in the following form.…”

On the size of good-for-games Rabin automata and its link with the memory in Muller games

“…More recently, these two results have been generalized to finite-memory conditions [8,7]. The memory requirements have been proved to be different to the chromatic memory requirements in general [10], but conditions that are finite-memory determined are also chromatic-finite-memory determined [25].…”