Half-Positional Objectives Recognized by Deterministic Büchi Automata

Bouyer, Patricia; Casares, Antonio; Randour, Mickaël; Vandenhove, Pierre

doi:10.48550/arxiv.2205.01365

Cited by 2 publications

(3 citation statements)

References 26 publications

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“…Going further, we provided two generic constructions for combining together universal graphs for pre x-independent objectives: nite lexicographical products in Section 5, and countable unions of objectives which exclude healing in Section 6. Despite being introduced only very recently in [27], well-monotone graphs have already been exploited by Bouyer, Casares, Randour and Vandenhove [2] to characterize positional objectives recognized by deterministic Büchi automata.…”

Section: Discussionmentioning

confidence: 99%

“…(Also it may be empty for some 𝑢 2 's which is not an issue either. )For each 𝑢 2 ∈ 𝑉 (𝑈 2 ), the pregraph 𝐺 𝑢2 1 satis es 𝑊 since 𝐺 does, and it even satis es 𝑊 1 since it has only edges with color in 𝐶 1 and 𝑊 ∩ 𝐶 𝜔 1 = 𝑊 1 . Thus there exists for each 𝑢 2 ∈ 𝑉 (𝑈 2 ) a morphism 𝜙 𝑢 2 1 from 𝐺 𝑢 2 1 to 𝑈 1 .…”

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confidence: 99%

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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: Discussionmentioning

confidence: 99%

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confidence: 99%

Characterizing Positionality in Games of Infinite Duration over Infinite Graphs

Ohlmann¹

2023

TheoretiCS

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“…As far as we know, Conjecture 1 is open in restriction to ω-regular conditions. In a recent preprint [3], Bouyer et al (among other results) prove Conjecture 1 for ω-regular conditions that can be recognized by deterministic Büchi automata.…”

Section: Conjecture 2 Every Bi-positional Prefix-independent Winning ...mentioning

confidence: 95%

Energy Games over Totally Ordered Groups

Kozachinskiy¹

2022

Preprint

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Kopczyński (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.

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Half-Positional Objectives Recognized by Deterministic Büchi Automata

Cited by 2 publications

References 26 publications

Characterizing Positionality in Games of Infinite Duration over Infinite Graphs

Characterizing Positionality in Games of Infinite Duration over Infinite Graphs

Energy Games over Totally Ordered Groups

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