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Antichains: Alternative Algorithms for LTL Satisfiability and Model-Checking [chapter]

M. De Wulf, L. Doyen, N. Maquet, J. -F. Raskin
<span title="">2008</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper, we propose new efficient algorithms for LTL satisfiability and model-checking.  ...  Our algorithms do not construct nondeterministic automata from LTL formulas but work directly with alternating automata using efficient exploration techniques based on antichains.  ...  LTL model-checking Our algorithm for LTL model-checking is based on forward exploration and semisymbolic representations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-78800-3_6">doi:10.1007/978-3-540-78800-3_6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p63uoirflfandowrffppssmtka">fatcat:p63uoirflfandowrffppssmtka</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20080221161116/http://www.ulb.ac.be/di/ssd/madewulf/antichains_ltl.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/06/c4/06c46d9da9009cb13b65e63890f25b6763d87acc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-78800-3_6"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

An Antichain Algorithm for LTL Realizability [chapter]

Emmanuel Filiot, Naiyong Jin, Jean-François Raskin
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We have implemented this new algorithm and our first results are very encouraging. 1 Technically, we could have started with Player I, for modeling reason it is conservative to start with Player O.  ...  We show that such an observer can be made deterministic and that this deterministic observer has a nice structure which can be exploited by an incremental algorithm that manipulates antichains of game  ...  Another promising future work would be to combine our approach to the antichain algorithm for LTL satisfiability [20] , which does not construct an automaton explicitly.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02658-4_22">doi:10.1007/978-3-642-02658-4_22</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q4euvc43n5f2dizfsfnhnd6c2m">fatcat:q4euvc43n5f2dizfsfnhnd6c2m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110401155652/http://www.ulb.ac.be//di/ssd/cfv/TechReps/TechRep_CFV_2009_110.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/10/4a/104a12be99e06de19c29f67a5e1142ec406f98bc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02658-4_22"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Antichains and compositional algorithms for LTL synthesis

Emmanuel Filiot, Naiyong Jin, Jean-François Raskin
<span title="2011-03-22">2011</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fyhr5xuigbfmdabfoqp2f7lbum" style="color: black;">Formal methods in system design</a> </i> &nbsp;
In this paper, we present new monolithic and compositional algorithms to solve the LTL realizability problem.  ...  This partial order can be used to define an efficient antichain algorithm.  ...  First, given the LTL formula φ, we construct two UCWs: one that accepts all the models of φ, denoted by A φ , and one that accepts all the models of ¬φ, denoted by A ¬φ .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10703-011-0115-3">doi:10.1007/s10703-011-0115-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ul74nch74ne4pod6ifdpzofkem">fatcat:ul74nch74ne4pod6ifdpzofkem</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808183919/http://www.ulb.ac.be/di/ssd/filiot/ltlsynthesis-journal.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0b/cb/0bcb1d80b9ae5873781660d8995627010f2c0bf1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10703-011-0115-3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Antichains for the Automata-Based Approach to Model-Checking

Laurent Doyen, Jean-François Raskin, Orna Grumberg
<span title="2009-03-02">2009</span> <i title="Logical Methods in Computer Science e.V."> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c67srop7pbe3nciquoxzy3d6pm" style="color: black;">Logical Methods in Computer Science</a> </i> &nbsp;
We propose and evaluate antichain algorithms to solve the universality and language inclusion problems for nondeterministic Buechi automata, and the emptiness problem for alternating Buechi automata.  ...  We evaluate the performance of the algorithm to check the universality of Buechi automata using the random automaton model recently proposed by Tabakov and Vardi.  ...  We thank two anonymous reviewers for helpful comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2168/lmcs-5(1:5)2009">doi:10.2168/lmcs-5(1:5)2009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pcmazfmsqraczl4zlcohxrohwm">fatcat:pcmazfmsqraczl4zlcohxrohwm</a> </span>
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Simulations and Antichains for Efficient Handling of Finite Automata [article]

Lukáš Holík
<span title="2017-06-10">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We then show how the developed reduction and inclusion checking methods improve the method of abstract regular tree model checking, the method that was the original motivation for starting the work on  ...  We then generalise the so called antichain universality and language inclusion checking technique developed originally for finite word automata for tree automata.  ...  Simulations and Antichains in Abstract Regular Tree Model Checking.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1706.03208v1">arXiv:1706.03208v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qmzaknz5xnfy5lmjp426jjlf5a">fatcat:qmzaknz5xnfy5lmjp426jjlf5a</a> </span>
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Abstraction Refinement and Antichains for Trace Inclusion of Infinite State Systems [chapter]

Radu Iosif, Adam Rogalewicz, Tomáš Vojnar
<span title="">2016</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
A trace of a data automaton is an alternating sequence of alphabet symbols and values taken by the counters during an execution of the automaton.  ...  Since the problem is undecidable in general, we give a semi-algorithm based on abstraction refinement, which is proved to be sound and complete, but whose termination is not guaranteed.  ...  This work was supported by the Czech Science Foundation project 14-11384S, the BUT FIT project FIT-S-14-2486, and the French ANR-14-CE28-0018 project Vecolib.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-49674-9_5">doi:10.1007/978-3-662-49674-9_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tpyqicivjvcvtbmqvzbyucmbri">fatcat:tpyqicivjvcvtbmqvzbyucmbri</a> </span>
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Fixpoint Guided Abstraction Refinement for Alternating Automata [chapter]

Pierre Ganty, Nicolas Maquet, Jean-François Raskin
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper, we develop and evaluate two new algorithms for checking emptiness of alternating automata. Those algorithms build on previous works.  ...  First, they rely on antichains to efficiently manipulate the state-spaces underlying the analysis of alternating automata.  ...  Those techniques have also been applied with success to the satisfiability and model-checking of LTL specifications [8] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02979-0_19">doi:10.1007/978-3-642-02979-0_19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4y2ikuv6obekdii57z4qszoqxu">fatcat:4y2ikuv6obekdii57z4qszoqxu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110401163219/http://www.ulb.ac.be/di/ssd/cfv/TechReps/TechRep_CFV_2009_109.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/60/2b/602be681ce552718963d6093ca723fb9e61d4b65.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02979-0_19"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A Symbolic Decision Procedure for Symbolic Alternating Finite Automata [article]

Loris D'Antoni, Zachary Kincaid, Fang Wang
<span title="2016-10-06">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce Symbolic Alternating Finite Automata (s-AFA) as an expressive, succinct, and decidable model for describing sets of finite sequences over arbitrary alphabets.  ...  Due to this succinctness, emptiness and equivalence checking are PSpace-hard. We introduce an algorithm for checking the equivalence of two s-AFAs based on bisimulation up to congruence.  ...  Satisfiability checking We evaluate the performance of our algorithm for checking satisfiability of the S-AFAs corresponding to the given LTL-F formulas.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1610.01722v1">arXiv:1610.01722v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3b3czuwwyncihguk54r2f2zouy">fatcat:3b3czuwwyncihguk54r2f2zouy</a> </span>
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More Anti-chain Based Refinement Checking [chapter]

Ting Wang, Songzheng Song, Jun Sun, Yang Liu, Jin Song Dong, Xinyu Wang, Shanping Li
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this work, we study the problem of adopting anti-chain for stable failures refinement checking, failures-divergence refinement checking and probabilistic refine checking (i.e., a probabilistic implementation  ...  Though applying anti-chain for probabilistic refinement checking is more complicated, we manage to show improvements in some cases.  ...  Their fol-lowing works show that significant improvements can also be brought to the model checking problem of LTL by using anti-chain based algorithms [8, 17] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-34281-3_26">doi:10.1007/978-3-642-34281-3_26</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g34wz4xe4zag7cxmzdy36eiuoq">fatcat:g34wz4xe4zag7cxmzdy36eiuoq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170830044024/http://people.sutd.edu.sg/~sunjun/Publications/icfem12a.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e1/6a/e16a3985fa25bee20b27d6f50afa4483dc1b53b0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-34281-3_26"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Synthesising Succinct Strategies in Safety and Reachability Games [chapter]

Gilles Geeraerts, Joël Goossens, Amélie Stainer
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Our techniques adapt the antichain framework to the setting of games, and rely on the notion of turn-based alternating simulation, which is used to formalise natural relations that exist between the states  ...  In particular, our techniques apply to the realisability problem of LTL [8] , to the synthesis of real-time schedulers for multiprocessor platforms [4] , and to the determinisation of timed automata [3  ...  They have been applied to several problems, such as LTL model-checking [7] or multi-processor schedulability [9] with remarkable performance improvements of several orders of magnitude.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-11439-2_8">doi:10.1007/978-3-319-11439-2_8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/67wmthe57nambhgagoy6tnot5m">fatcat:67wmthe57nambhgagoy6tnot5m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200309071429/https://dipot.ulb.ac.be/dspace/bitstream/2013/160787/1/main.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b3/77/b3778720a3de1c38b88d20ea58b32eef261f3573.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-11439-2_8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Fixed point guided abstraction refinement for alternating automata

Pierre Ganty, Nicolas Maquet, Jean-François Raskin
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
interpretation Antichains a b s t r a c t In this paper, we develop and evaluate two new algorithms for checking emptiness of alternating automata. These algorithms build on previous works.  ...  Those abstract domains and their refinement operators can be used both in forward and backward algorithms for checking emptiness of alternating automata.  ...  Acknowledgements The authors would like to thank Gilles Geeraerts for some fruitful discussions on the abstraction scheme. The second author is supported by an FNRS-FRIA grant.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2010.05.037">doi:10.1016/j.tcs.2010.05.037</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ojvndzw3zbavholhq7zfryptsi">fatcat:ojvndzw3zbavholhq7zfryptsi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171004003056/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/d56/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDMwNDM5NzUxMDAwMzI5NA%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3f/15/3f15ab3f5f304b06a49ed0cee80ebc91242f2115.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2010.05.037"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Symblicit algorithms for optimal strategy synthesis in monotonic Markov decision processes

Aaron Bohy, Véronique Bruyère, Jean-François Raskin
<span title="2014-07-18">2014</span> <i title="Open Publishing Association"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qzezkwolyvf33pglm2jldjklca" style="color: black;">Electronic Proceedings in Theoretical Computer Science</a> </i> &nbsp;
The aim of this paper is to show that the new data structure of pseudo-antichains (an extension of antichains) provides another interesting alternative, especially for the class of monotonic MDPs.  ...  For two practical applications coming from automated planning and LTL synthesis, we report promising experimental results w.r.t. both the run time and the memory consumption.  ...  We would like to thank Mickael Randour for his fruitful discussions, Marta Kwiatkowska, David Parker and Christian Von Essen for their help regarding the tool PRISM, and Holger Hermanns and Ernst Moritz  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4204/eptcs.157.8">doi:10.4204/eptcs.157.8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6g2xu5foiffefj6kw2mlaazgze">fatcat:6g2xu5foiffefj6kw2mlaazgze</a> </span>
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Synthesis from LTL Specifications with Mean-Payoff Objectives [article]

Aaron Bohy, Véronique Bruyère, Emmanuel Filiot, Jean-François Raskin
<span title="2013-01-09">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The synthesis problem then amounts to automatically construct (if possible) a reactive system whose executions all satisfy a given LTL formula and have mean-payoff values greater than or equal to some  ...  The latter problem is called LTLMP synthesis and the LTLMP realizability problem asks to check whether such a system exists.  ...  One of their extensions is similar to LTL MP . However the authors of [5] only study the complexity of model-checking problems whereas we consider realizability and synthesis problems.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1210.3539v3">arXiv:1210.3539v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/llonmlxghbgy3c5bvzavlciili">fatcat:llonmlxghbgy3c5bvzavlciili</a> </span>
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Evaluating LTL Satisfiability Solvers [chapter]

Viktor Schuppan, Luthfi Darmawan
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Related Work Rozier and Vardi compare several explicit state and symbolic BDD-based model checkers for LTL satisfiability checking [60] .  ...  We consider a wide range of solvers implementing three major classes of algorithms: reduction to model checking, tableau-based approaches, and temporal resolution.  ...  ALASKA performs model checking and satisfiability checking of LTL via symbolic computation of fixed points using antichains [1, 69] .  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808190720/http://www.schuppan.de/viktor/VSchuppanLDarmawan-ATVA-2011-preprint.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/02/20/02204eb8d844216800e7e9f7aed50233f373fb4f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-24372-1_28"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Synthesising Succinct Strategies in Safety Games [article]

Gilles Geeraerts, Joël Goossens, Amélie Stainer
<span title="2014-05-06">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
platforms, and also for the determinisation of timed automata.  ...  Finite turn-based safety games have been used for very different problems such as the synthesis of linear temporal logic (LTL), the synthesis of schedulers for computer systems running on multiprocessor  ...  It is easy to check that if α and β are maximal and minimal antichains of S respectively, then ↓ (α) =↓ (S ) and ↑ (β) =↑ (S ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1404.6228v2">arXiv:1404.6228v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/44cehvziura2vj2radvavfzbxi">fatcat:44cehvziura2vj2radvavfzbxi</a> </span>
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