# Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs

arXiv: https://arxiv.org/abs/2110.01276 · HTML: https://mlsanity.com/paper/2110.01276/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2021.

## Built on by (most cited first, 2 of 2)

- [Mixing Any Cocktail with Limited Ingredients: On the Structure of Payoff Sets in Multi-Objective POMDPs and its Impact on Randomised Strategies](https://mlsanity.com/paper/2502.18296/) (2025)
- [Simplicity Lies in the Eye of the Beholder: A Strategic Perspective on Controllers in Reactive Synthesis](https://mlsanity.com/paper/2509.04129/) (2025)

## Abstract (arXiv)

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ω-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ω-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ω-regular. This provides a game-theoretic characterization of ω-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ω-regularity depends on the value of some parameters.