Characterizing Positionality in Games of Infinite Duration over Infinite Graphs
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 st...
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| Format: | Article |
| Language: | English |
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TheoretiCS Foundation e.V.
2023-01-01
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| Series: | TheoretiCS |
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| Online Access: | https://theoretics.episciences.org/9724/pdf |
| _version_ | 1827363991729471488 |
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| author | Pierre Ohlmann |
| author_facet | Pierre Ohlmann |
| author_sort | Pierre Ohlmann |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T07:59:18Z |
| format | Article |
| id | doaj.art-106354f91ecb45f0b34927145f85034a |
| institution | Directory Open Access Journal |
| issn | 2751-4838 |
| language | English |
| last_indexed | 2024-03-08T07:59:18Z |
| publishDate | 2023-01-01 |
| publisher | TheoretiCS Foundation e.V. |
| record_format | Article |
| series | TheoretiCS |
| spelling | doaj.art-106354f91ecb45f0b34927145f85034a2024-02-02T12:32:47ZengTheoretiCS Foundation e.V.TheoretiCS2751-48382023-01-01Volume 210.46298/theoretics.23.39724Characterizing Positionality in Games of Infinite Duration over Infinite GraphsPierre OhlmannWe 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.https://theoretics.episciences.org/9724/pdfcomputer science - computer science and game theorycomputer science - logic in computer science |
| spellingShingle | Pierre Ohlmann Characterizing Positionality in Games of Infinite Duration over Infinite Graphs TheoretiCS computer science - computer science and game theory computer science - logic in computer science |
| title | Characterizing Positionality in Games of Infinite Duration over Infinite Graphs |
| title_full | Characterizing Positionality in Games of Infinite Duration over Infinite Graphs |
| title_fullStr | Characterizing Positionality in Games of Infinite Duration over Infinite Graphs |
| title_full_unstemmed | Characterizing Positionality in Games of Infinite Duration over Infinite Graphs |
| title_short | Characterizing Positionality in Games of Infinite Duration over Infinite Graphs |
| title_sort | characterizing positionality in games of infinite duration over infinite graphs |
| topic | computer science - computer science and game theory computer science - logic in computer science |
| url | https://theoretics.episciences.org/9724/pdf |
| work_keys_str_mv | AT pierreohlmann characterizingpositionalityingamesofinfinitedurationoverinfinitegraphs |