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：本文研究在可能无限的图上进行的无限时长博弈。最近，Ohlmann（LICS 2022）通过通用图提出了允许最优位置策略的目标的刻画：一个目标是位置性的，当且仅当它允许良序单调通用图。我们将Ohlmann的刻画扩展到涵盖（有限或无限）内存上界。我们证明，允许具有小于$m$的$\varepsilon$-内存（一种在读取$\varepsilon$-边时无法更新的内存）的最优策略的目标，正是那些允许其反链大小以$m$为界的良基单调通用图的目标。我们还通过适当的通用结构给出了色内存的刻画。我们的结果适用于有限以及无限的内存界（例如，适用于具有有限但无界内存或具有可数内存策略的目标）。我们通过进行几个案例研究来说明我们框架的适用性，提供证明我们方法局限性的例子，并讨论从我们结果中得出的一般闭包性质。