Problem
Research questions and friction points this paper is trying to address.
Infinite Grid Games
Memory Capacity
Optimal Strategy
Characterising memory in infinite games
🤖 AI Summary
This work investigates the memory complexity of strategies in infinite games played on infinite graphs, aiming to characterize the minimal memory capacity required to realize optimal strategies. Methodologically, it establishes an exact correspondence between ε-memory and the size of antichains in well-founded monotone universal graphs, thereby unifying finite, countable, and unbounded memory regimes for the first time. It introduces a universal graph framework tailored for colored memory, integrating universal graph theory, order theory (well-foundedness and well-ordering), game semantics, and infinitary combinatorial analysis. The results provide a complete characterization of optimal strategies under arbitrary memory bounds—including infinite ones—derive general closure properties, and validate the framework’s efficacy and tightness via canonical game instances. The core innovation lies in a tight quantitative link between the optimality of objective functions and structural parameters of universal graphs.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Colored Memory
Infinite Games
Enhanced Muller Games Approach
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