This paper is concerned with games of infinite duration played over potentially infinite graphs. Recently, Ohlmann (LICS 2022) presented a characterisation of objectives admitting optimal positional strategies, by means of universal graphs: an objective is positional if and only if it admits well-ordered monotone universal graphs. We extend Ohlmann's characterisation to encompass (finite or infinite) memory upper bounds. We prove that objectives admitting optimal strategies with $\varepsilon$-memory less than $m$ (a memory that cannot be updated when reading an $\varepsilon$-edge) are exactly those which admit well-founded monotone universal graphs whose antichains have size bounded by $m$. We also give a characterisation of chromatic memory by means of appropriate universal structures. Our results apply to finite as well as infinite memory bounds (for instance, to objectives with finite but unbounded memory, or with countable memory strategies). We illustrate the applicability of our framework by carrying out a few case studies, we provide examples witnessing limitations of our approach, and we discuss general closure properties which follow from our results.

翻译：本文关注基于潜在无限图进行的无限时长博弈。近期，Ohlmann（LICS 2022）通过通用图理论提出了具有最优位置策略的目标表征：一个目标具有位置性当且仅当其承认良序单调通用图。我们扩展了Ohlmann的表征体系，将其覆盖至（有限或无限）记忆上界。我们证明：对于允许$\varepsilon$-记忆小于$m$（一种在读取$\varepsilon$-边时无法更新的记忆）的最优策略的目标，其充要条件为承认反链规模不超过$m$的良基单调通用图。我们还通过适当通用结构给出了色记忆的表征。我们的结论同时适用于有限与无限记忆上界（例如具有有限无界记忆或可数记忆策略的目标）。通过若干案例研究展示了框架的适用性，列举了方法局限性的实例，并讨论了由结论衍生的一般性封闭性质。