State Complexity of Chromatic Memory in Infinite-Duration Games

TL;DR

This paper investigates the state complexity of transforming general finite-memory strategies into chromatic finite-memory strategies in infinite-duration games, providing tight bounds on the number of states needed for such transformations.

Contribution

It establishes the exact state complexity bounds for converting general strategies into chromatic strategies, showing the transformation can require exponentially more states.

Findings

If a game has a winning strategy with q states, a chromatic strategy may need up to (q+1)^n states.

The derived bounds are nearly tight, with constructions showing the necessity of exponential state increases.

The results clarify the relationship between general and chromatic finite-memory strategies in terms of state complexity.

Abstract

A major open problem in the area of infinite-duration games is to characterize winning conditions that are determined in finite-memory strategies. Infinite-duration 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.