Energy Games over Totally Ordered Groups

Kozachinskiy, Alexander

doi:10.48550/arxiv.2205.04508

Cited by 1 publication

(2 citation statements)

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“…Positionality in Infinite Games over nite game graphs was recently disproved by Kozachinskiy [25], however the question remains open for in nite game graphs (which are the focus of the present paper).…”

Section: / 51mentioning

confidence: 87%

“…The most tantalizing open question remains Kopczy ński conjecture: are unions of pre x-independent positional objectives positional? This was recently answered in the negative in the case of nite game graphs by Kozachinskiy [25]; however the question remains open in the setting of in nite game graphs considered in this paper. We believe that well-monotone graphs provide a nice tool to attack this question: on one hand, one can look for constructions combining well-monotone graphs to preserve unions (this is achieved in Section 6 assuming healing is excluded), and on the other hand, graphs that are "hard to embed" can provide counterexamples (see above for instance).…”

Section: Open Problemsmentioning

confidence: 92%

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Characterizing Positionality in Games of Infinite Duration over Infinite Graphs

Ohlmann¹

2023

TheoretiCS

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We study turn-based quantitative games of infinite duration opposing two antagonistic players and played over graphs. This model is widely accepted as providing the adequate framework for formalizing the synthesis question for reactive systems. This important application motivates the question of strategy complexity: which valuations (or payoff functions) admit optimal positional strategies (without memory)? Valuations for which both players have optimal positional strategies have been characterized by Gimbert and Zielonka for finite graphs and by Colcombet and Niwi\'nski for infinite graphs. However, for reactive synthesis, existence of optimal positional strategies for the opponent (which models an antagonistic environment) is irrelevant. Despite this fact, not much is known about valuations for which the protagonist admits optimal positional strategies, regardless of the opponent. In this work, we characterize valuations which admit such strategies over infinite game graphs. Our characterization uses the vocabulary of universal graphs, which has also proved useful in understanding recent breakthrough results regarding the complexity of parity games. More precisely, we show that a valuation admitting universal graphs which are monotone and well-ordered is positional over all game graphs, and -- more surprisingly -- that the converse is also true for valuations admitting neutral colors. We prove the applicability and elegance of the framework by unifying a number of known positionality results, proving new ones, and establishing closure under lexicographical products. Finally, we discuss a class of prefix-independent positional objectives which is closed under countable unions.

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Section: / 51mentioning

confidence: 87%

Section: Open Problemsmentioning

confidence: 92%