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Memoryless determinacy of parity and mean payoff games: a simple proof

Henrik Björklund, Sven Sandberg, Sergei Vorobyov
2004 Theoretical Computer Science  
We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payo games.  ...  Second, we show that memoryless determinacy straightforwardly generalizes to inÿnite duration versions of parity and mean payo games.  ...  Memoryless determinacy of ÿnite duration parity games In this section we prove the main theorem of the paper, which implies memoryless determinacy of the inÿnite duration parity games (Section 6).  ... 
doi:10.1016/s0304-3975(03)00427-4 fatcat:riy5abyph5cnvoyydxqq27ga6y

On the positional determinacy of edge-labeled games

Thomas Colcombet, Damian Niwiński
2006 Theoretical Computer Science  
It is well known that games with the parity winning condition admit positional determinacy : the winner has always a positional (memoryless) strategy.  ...  That is, a winning condition over arbitrary set of colors admits positional determinacy in all games if and only if it can be reduced to a parity condition with some finite number of priorities.  ...  Acknowledgements We thank the anonymous referee for his or her careful reading, for providing us with many helpful comments, and in particular for suggesting a more transparent proof of Lemma 5.  ... 
doi:10.1016/j.tcs.2005.10.046 fatcat:adj3ymx2cnbizaqwxvcro72ghm

Half-Positional Determinacy of Infinite Games [chapter]

Eryk Kopczyński
2006 Lecture Notes in Computer Science  
We establish some closure properties of such conditions, and discover some common reasons behind several known and new positional determinacy results.  ...  We study infinite games where one of the players always has a positional (memory-less) winning strategy, while the other player may use a history-dependent strategy.  ...  This theorem gives yet another proof of finite positional determinacy of parity games, and also half-positional determinacy of unions of families of parity conditions (where each parity condition may use  ... 
doi:10.1007/11787006_29 fatcat:unmp5t6dpjgtbeqkl73pz7jaxa

How unprovable is Rabin's decidability theorem? [article]

Leszek Aleksander Kołodziejczyk, Henryk Michalewski
2015 arXiv   pre-print
the infinite binary tree, positional determinacy of parity games and determinacy of Bool(Σ^0_2) Gale-Stewart games are all equivalent.  ...  ACA_0 to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of Σ^0_2 sets.  ...  Let σ H be a positional winning strategy for one of the players, say ∃. We now turn to the positional determinacy of parity games.  ... 
arXiv:1508.06780v1 fatcat:3c55nyrbvrb4vhq6ectcwjizh4

Postinal Determinacy of Games with Infinitely Many Priorities [article]

Erich Graedel, Igor Walukiewicz
2012 arXiv   pre-print
Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs.  ...  A game is positionally determined if, from each position, one of the two players has a positional winning strategy.  ...  Determinacy of Muller Games. Theorem 5.12 classifies the Muller conditions that imply positional determinacy on all game graphs.  ... 
arXiv:cs/0610034v2 fatcat:inenij25hvcpvcil23wou76pzq

Positional Determinacy of Games with Infinitely Many Priorities

Erich Grädel, Igor Walukiewicz, Luke Ong
2006 Logical Methods in Computer Science  
Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs.  ...  A game is positionally determined if, from each position, one of the two players has a positional winning strategy.  ...  Determinacy of Muller Games. Theorem 5.12 classifies the Muller conditions that imply positional determinacy on all game graphs.  ... 
doi:10.2168/lmcs-2(4:6)2006 fatcat:yajsqi53hfdjddobhfo4wp4swy

On Parity Game Preorders and the Logic of Matching Plays [chapter]

M. W. Gazda, T. A. C. Willemse
2016 Lecture Notes in Computer Science  
Parity games: machinery for deciding (bi)simulations and model checking • modal µ-calculus: K |= νX .µY.  ...  Parity games: machinery for deciding (bi)simulations and model checking • modal µ-calculus: K |= νX .µY.  ...  Theorem (Positional determinacy) / Department of Mathematics and Computer Science Games ! Theorem (Positional determinacy) Every vertex is won by either ◊ or .  ... 
doi:10.1007/978-3-662-49192-8_23 fatcat:xozjfhfbzvf4rdu4jeg2akjm4i

An Optimal Value Iteration Algorithm for Parity Games [article]

Nathanaël Fijalkow
2018 arXiv   pre-print
This suggests that the succinct progress measure algorithm of Jurdzi\'nski and Lazi\'c is in this framework optimal, and that the polynomial time algorithm for parity games is hiding someplace else.  ...  The quest for a polynomial time algorithm for solving parity games gained momentum in 2017 when two different quasipolynomial time algorithms were constructed.  ...  Acknowledgments The notion of universal trees was hinted at me by Marcin Jurdziński and Ranko Lazić. They largely contributed to the making of this paper, and I thank them for their support.  ... 
arXiv:1801.09618v1 fatcat:sknboutetjakvaskzucyxsh64q

Facets of Synthesis: Revisiting Church's Problem [chapter]

Wolfgang Thomas
2009 Lecture Notes in Computer Science  
We recall the fundamental questions raised more than 50 years ago in "Church's Synthesis Problem" that led to the foundation of the algorithmic theory of infinite games.  ...  In this essay we discuss the origin, central results, and some perspectives of algorithmic synthesis of nonterminating reactive programs.  ...  The main problem of this approach is to show complementation for parity tree automata. For this the determinacy of parity games (in the version over infinite game graphs) can be used.  ... 
doi:10.1007/978-3-642-00596-1_1 fatcat:uhny7wacofafzl3s4fcd6sc37e

Fixpoints, games and the difference hierarchy

Julian C. Bradfield
2003 RAIRO - Theoretical Informatics and Applications  
Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference  ...  This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory.  ...  This appears to be inevitable: the proof of determinacy for Σ 0 3 proceeds via games in which the positions are themselves games.  ... 
doi:10.1051/ita:2003011 fatcat:4hwj5rljynabbm7vc7af2y4kim

Page 10269 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
The paper presents a new, elementary proof of the positional de- terminacy of parity games, and further adapts this proof to a more general class of mean payoff games.  ...  proof of the positional determinacy of mean payoff games, due to Ehrenfeucht and Mycielski, was rather involved and used mutual induction to show the result both for infinite games (mentioned above) and  ... 

What are Strategies in Delay Games? Borel Determinacy for Games with Lookahead [article]

Felix Klein, Martin Zimmermann
2015 arXiv   pre-print
First, we prove determinacy of such games with respect to a fixed evolution of the lookahead. However, strategies in such games may depend on information about the evolution.  ...  We investigate determinacy of delay games with Borel winning conditions, infinite-duration two-player games in which one player may delay her moves to obtain a lookahead on her opponent's moves.  ...  The work presented here was initiated by a discussion with Dietmar Berwanger at the Dagstuhl Seminar "Non-Zero-Sum-Games and Control" in 2015.  ... 
arXiv:1504.02627v1 fatcat:4beux5qhbzb5fhe4zsvzfwkmpa

From winning strategy to Nash equilibrium [article]

Stéphane Le Roux
2014 arXiv   pre-print
As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of Muller games.  ...  These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins.  ...  Acknowledgement I thank Achim Blumensath and Michael Ummels for discussions on parity games and the like, Alexander Kreuzer for explanations on Ramsey's theorem, an anonymous referee in particular for  ... 
arXiv:1203.1866v4 fatcat:mxpzvfv3dvgpbar34ergnrn46m

Cost-Parity and Cost-Streett Games

Nathanael Fijalkow, Martin Zimmermann, Marc Herbstritt
2012 Foundations of Software Technology and Theoretical Computer Science  
For cost-parity games we show that the first player has positional winning strategies and that determining the winner lies in NP ∩ coNP.  ...  This unifies the complexity results for the classical and finitary variants of these games. Both types of cost-games can be solved by solving linearly many instances of their classical variants.  ...  Finally, our results on half-positional determinacy for cost-parity games and finite-state determinacy for cost-Streett games shows that these two conditions can be expressed in parametric linear temporal  ... 
doi:10.4230/lipics.fsttcs.2012.124 dblp:conf/fsttcs/FijalkowZ12 fatcat:4nnmkdmxhzdo7cpwfsralntzey

Topological extension of parity automata

Michał Skrzypczak
2013 Information and Computation  
The paper presents a concept of a coloring -an extension of deterministic parity automata.  ...  A coloring K is a function A * → N satisfying Every coloring defines a subset of A ω by the standard parity condition We show that sets defined by colorings are exactly all ∆ 0 3 sets in the standard product  ...  This paper analyses parity games with infinitely many priorities. The question of positional determinacy of such games is investigated.  ... 
doi:10.1016/j.ic.2013.06.004 fatcat:nhbyf2w2mnbh7bp7wvqov7g5te
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