Found 3 Possible Name Variants:

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the approximation.
As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the "boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations.
We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees.

Kush Grover, Jan Křetínský, Tobias Meggendorfer, and Maximilian Weininger. Anytime Guarantees for Reachability in Uncountable Markov Decision Processes. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grover_et_al:LIPIcs.CONCUR.2022.11, author = {Grover, Kush and K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias and Weininger, Maximilian}, title = {{Anytime Guarantees for Reachability in Uncountable Markov Decision Processes}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.11}, URN = {urn:nbn:de:0030-drops-170743}, doi = {10.4230/LIPIcs.CONCUR.2022.11}, annote = {Keywords: Uncountable system, Markov decision process, discrete-time Markov control process, probabilistic verification, anytime guarantee} }

Document

**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Restarts are used in many computer systems to improve performance. Examples include reloading a webpage, reissuing a request, or restarting a randomized search. The design of restart strategies has been extensively studied by the performance evaluation community. In this paper, we address the problem of designing universal restart strategies, valid for arbitrary finite-state Markov chains, that enforce a given ω-regular property while not knowing the chain. A strategy enforces a property φ if, with probability 1, the number of restarts is finite, and the run of the Markov chain after the last restart satisfies φ. We design a simple "cautious" strategy that solves the problem, and a more sophisticated "bold" strategy with an almost optimal number of restarts.

Javier Esparza, Stefan Kiefer, Jan Křetínský, and Maximilian Weininger. Enforcing ω-Regular Properties in Markov Chains by Restarting. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2021.5, author = {Esparza, Javier and Kiefer, Stefan and K\v{r}et{\'\i}nsk\'{y}, Jan and Weininger, Maximilian}, title = {{Enforcing \omega-Regular Properties in Markov Chains by Restarting}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {5:1--5:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.5}, URN = {urn:nbn:de:0030-drops-143824}, doi = {10.4230/LIPIcs.CONCUR.2021.5}, annote = {Keywords: Markov chains, omega-regular properties, runtime enforcement} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

We introduce a framework for approximate analysis of Markov decision processes (MDP) with bounded-, unbounded-, and infinite-horizon properties. The main idea is to identify a "core" of an MDP, i.e., a subsystem where we provably remain with high probability, and to avoid computation on the less relevant rest of the state space. Although we identify the core using simulations and statistical techniques, it allows for rigorous error bounds in the analysis. Consequently, we obtain efficient analysis algorithms based on partial exploration for various settings, including the challenging case of strongly connected systems.

Jan Křetínský and Tobias Meggendorfer. Of Cores: A Partial-Exploration Framework for Markov Decision Processes. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2019.5, author = {K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias}, title = {{Of Cores: A Partial-Exploration Framework for Markov Decision Processes}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {5:1--5:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.5}, URN = {urn:nbn:de:0030-drops-109076}, doi = {10.4230/LIPIcs.CONCUR.2019.5}, annote = {Keywords: Markov Decision Processes, Reachability, Approximation} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We formalize the problem of maximizing the mean-payoff value with high probability while satisfying a parity objective in a Markov decision process (MDP) with unknown probabilistic transition function and unknown reward function. Assuming the support of the unknown transition function and a lower bound on the minimal transition probability are known in advance, we show that in MDPs consisting of a single end component, two combinations of guarantees on the parity and mean-payoff objectives can be achieved depending on how much memory one is willing to use. (i) For all epsilon and gamma we can construct an online-learning finite-memory strategy that almost-surely satisfies the parity objective and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. (ii) Alternatively, for all epsilon and gamma there exists an online-learning infinite-memory strategy that satisfies the parity objective surely and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. We extend the above results to MDPs consisting of more than one end component in a natural way. Finally, we show that the aforementioned guarantees are tight, i.e. there are MDPs for which stronger combinations of the guarantees cannot be ensured.

Jan Kretínský, Guillermo A. Pérez, and Jean-François Raskin. Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.8, author = {Kret{\'\i}nsk\'{y}, Jan and P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, title = {{Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.8}, URN = {urn:nbn:de:0030-drops-95468}, doi = {10.4230/LIPIcs.CONCUR.2018.8}, annote = {Keywords: Markov decision processes, Reinforcement learning, Beyond worst case} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We investigate the satisfiability and finite satisfiability problem for probabilistic computation-tree logic (PCTL) where operators are not restricted by any step bounds. We establish decidability for several fragments containing quantitative operators and pinpoint the difficulties arising in more complex fragments where the decidability remains open.

Jan Kretínský and Alexej Rotar. The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.32, author = {Kret{\'\i}nsk\'{y}, Jan and Rotar, Alexej}, title = {{The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.32}, URN = {urn:nbn:de:0030-drops-95708}, doi = {10.4230/LIPIcs.CONCUR.2018.32}, annote = {Keywords: temporal logic, probabilistic verification, probabilistic computation tree logic, satisfiability} }

Document

**Published in:** Dagstuhl Reports, Volume 8, Issue 3 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 18121 "Machine Learning and Model Checking Join Forces". This Dagstuhl Seminar brought together researchers working in the fields of machine learning and model checking. It helped to identify new research topics on the one hand and to help with current problems on the other hand.

Nils Jansen, Joost-Pieter Katoen, Pusmeet Kohli, and Jan Kretinsky. Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121). In Dagstuhl Reports, Volume 8, Issue 3, pp. 74-93, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{jansen_et_al:DagRep.8.3.74, author = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, title = {{Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121)}}, pages = {74--93}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {3}, editor = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.8.3.74}, URN = {urn:nbn:de:0030-drops-92988}, doi = {10.4230/DagRep.8.3.74}, annote = {Keywords: artificial intelligence, cyber-physical systems, formal methods, formal verification, logics, machine learning, model checking, quantitative verification, safety-critical systems} }

Document

**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by temporal logics or automata. We investigate which of the distances can be approximated by observing the systems, i.e. by black-box testing or simulation, and we provide both negative and positive results.

Przemyslaw Daca, Thomas A. Henzinger, Jan Kretinsky, and Tatjana Petrov. Linear Distances between Markov Chains. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{daca_et_al:LIPIcs.CONCUR.2016.20, author = {Daca, Przemyslaw and Henzinger, Thomas A. and Kretinsky, Jan and Petrov, Tatjana}, title = {{Linear Distances between Markov Chains}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.20}, URN = {urn:nbn:de:0030-drops-61829}, doi = {10.4230/LIPIcs.CONCUR.2016.20}, annote = {Keywords: probabilistic systems, verification, statistical model checking, temporal logic, behavioural equivalence} }

Document

**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

We consider weighted automata with both positive and negative integer weights on edges and study the problem of synchronization using adaptive strategies that may only observe whether the current weight-level is negative or nonnegative. We show that the synchronization problem is decidable in polynomial time for deterministic weighted automata.

Jan Kretinsky, Kim Guldstrand Larsen, Simon Laursen, and Jiri Srba. Polynomial Time Decidability of Weighted Synchronization under Partial Observability. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 142-154, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2015.142, author = {Kretinsky, Jan and Larsen, Kim Guldstrand and Laursen, Simon and Srba, Jiri}, title = {{Polynomial Time Decidability of Weighted Synchronization under Partial Observability}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {142--154}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.142}, URN = {urn:nbn:de:0030-drops-53927}, doi = {10.4230/LIPIcs.CONCUR.2015.142}, annote = {Keywords: weighted automata, partial observability, synchronization, complexity} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Interactive Markov chains (IMC) are compositional behavioral models extending both labeled transition systems and continuous-time Markov chains. IMC pair modeling convenience - owed to compositionality properties - with effective verification algorithms and tools - owed to Markov properties. Thus far however, IMC verification did not consider compositionality properties, but considered closed systems. This paper discusses the evaluation of IMC in an open and thus compositional interpretation. For this we embed the IMC into a game that is played with the environment. We devise algorithms that enable us to derive bounds on reachability probabilities that are assured to hold in any composition context.

Tomas Brazdil, Holger Hermanns, Jan Krcal, Jan Kretinsky, and Vojtech Rehak. Verification of Open Interactive Markov Chains. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 474-485, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2012.474, author = {Brazdil, Tomas and Hermanns, Holger and Krcal, Jan and Kretinsky, Jan and Rehak, Vojtech}, title = {{Verification of Open Interactive Markov Chains}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {474--485}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.474}, URN = {urn:nbn:de:0030-drops-38826}, doi = {10.4230/LIPIcs.FSTTCS.2012.474}, annote = {Keywords: IMC, compositional verification, synthesis, time bounded reachability, discretization} }

Document

**Published in:** OASIcs, Volume 16, Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers (2011)

The formalism of modal transition systems (MTS) is a well established framework for systems specification as well as abstract interpretation. Nevertheless, due to incapability to capture some useful features, various extensions have been studied, such as e.g. mixed transition systems or disjunctive MTS. Thus a need to compare them has emerged. Therefore, we introduce transition system with obligations as a general model encompassing all the aforementioned models, and equip it with a process algebra description. Using these instruments, we then compare the previously studied subclasses and characterize their relationships.

Nikola Benes and Jan Kretinsky. Process Algebra for Modal Transition Systemses. In Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers. Open Access Series in Informatics (OASIcs), Volume 16, pp. 9-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benes_et_al:OASIcs.MEMICS.2010.9, author = {Benes, Nikola and Kretinsky, Jan}, title = {{Process Algebra for Modal Transition Systemses}}, booktitle = {Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers}, pages = {9--18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-22-4}, ISSN = {2190-6807}, year = {2011}, volume = {16}, editor = {Matyska, Ludek and Kozubek, Michal and Vojnar, Tomas and Zemcik, Pavel and Antos, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.MEMICS.2010.9}, URN = {urn:nbn:de:0030-drops-30701}, doi = {10.4230/OASIcs.MEMICS.2010.9}, annote = {Keywords: modal transition systems, process algebra, specification} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We study continuous-time stochastic games with time-bounded
reachability objectives. We show that each vertex in such a game
has a \emph{value} (i.e., an equilibrium probability), and we
classify the conditions under which optimal strategies exist.
Finally, we show how to compute optimal strategies in finite uniform
games, and how to compute $\varepsilon$-optimal strategies in
finitely-branching games with bounded rates (for finite games, we
provide detailed complexity estimations).

Tomas Brazdil, Vojtech Forejt, Jan Krcal, Jan Kretinsky, and Antonin Kucera. Continuous-Time Stochastic Games with Time-Bounded Reachability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 61-72, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2009.2307, author = {Brazdil, Tomas and Forejt, Vojtech and Krcal, Jan and Kretinsky, Jan and Kucera, Antonin}, title = {{Continuous-Time Stochastic Games with Time-Bounded Reachability}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {61--72}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2307}, URN = {urn:nbn:de:0030-drops-23077}, doi = {10.4230/LIPIcs.FSTTCS.2009.2307}, annote = {Keywords: Continuous time stochastic systems, time bounded reachability, stochastic games} }

Document

**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the approximation.
As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the "boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations.
We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees.

Kush Grover, Jan Křetínský, Tobias Meggendorfer, and Maximilian Weininger. Anytime Guarantees for Reachability in Uncountable Markov Decision Processes. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grover_et_al:LIPIcs.CONCUR.2022.11, author = {Grover, Kush and K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias and Weininger, Maximilian}, title = {{Anytime Guarantees for Reachability in Uncountable Markov Decision Processes}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.11}, URN = {urn:nbn:de:0030-drops-170743}, doi = {10.4230/LIPIcs.CONCUR.2022.11}, annote = {Keywords: Uncountable system, Markov decision process, discrete-time Markov control process, probabilistic verification, anytime guarantee} }

Document

**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Restarts are used in many computer systems to improve performance. Examples include reloading a webpage, reissuing a request, or restarting a randomized search. The design of restart strategies has been extensively studied by the performance evaluation community. In this paper, we address the problem of designing universal restart strategies, valid for arbitrary finite-state Markov chains, that enforce a given ω-regular property while not knowing the chain. A strategy enforces a property φ if, with probability 1, the number of restarts is finite, and the run of the Markov chain after the last restart satisfies φ. We design a simple "cautious" strategy that solves the problem, and a more sophisticated "bold" strategy with an almost optimal number of restarts.

Javier Esparza, Stefan Kiefer, Jan Křetínský, and Maximilian Weininger. Enforcing ω-Regular Properties in Markov Chains by Restarting. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2021.5, author = {Esparza, Javier and Kiefer, Stefan and K\v{r}et{\'\i}nsk\'{y}, Jan and Weininger, Maximilian}, title = {{Enforcing \omega-Regular Properties in Markov Chains by Restarting}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {5:1--5:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.5}, URN = {urn:nbn:de:0030-drops-143824}, doi = {10.4230/LIPIcs.CONCUR.2021.5}, annote = {Keywords: Markov chains, omega-regular properties, runtime enforcement} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

We introduce a framework for approximate analysis of Markov decision processes (MDP) with bounded-, unbounded-, and infinite-horizon properties. The main idea is to identify a "core" of an MDP, i.e., a subsystem where we provably remain with high probability, and to avoid computation on the less relevant rest of the state space. Although we identify the core using simulations and statistical techniques, it allows for rigorous error bounds in the analysis. Consequently, we obtain efficient analysis algorithms based on partial exploration for various settings, including the challenging case of strongly connected systems.

Jan Křetínský and Tobias Meggendorfer. Of Cores: A Partial-Exploration Framework for Markov Decision Processes. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2019.5, author = {K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias}, title = {{Of Cores: A Partial-Exploration Framework for Markov Decision Processes}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {5:1--5:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.5}, URN = {urn:nbn:de:0030-drops-109076}, doi = {10.4230/LIPIcs.CONCUR.2019.5}, annote = {Keywords: Markov Decision Processes, Reachability, Approximation} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We formalize the problem of maximizing the mean-payoff value with high probability while satisfying a parity objective in a Markov decision process (MDP) with unknown probabilistic transition function and unknown reward function. Assuming the support of the unknown transition function and a lower bound on the minimal transition probability are known in advance, we show that in MDPs consisting of a single end component, two combinations of guarantees on the parity and mean-payoff objectives can be achieved depending on how much memory one is willing to use. (i) For all epsilon and gamma we can construct an online-learning finite-memory strategy that almost-surely satisfies the parity objective and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. (ii) Alternatively, for all epsilon and gamma there exists an online-learning infinite-memory strategy that satisfies the parity objective surely and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. We extend the above results to MDPs consisting of more than one end component in a natural way. Finally, we show that the aforementioned guarantees are tight, i.e. there are MDPs for which stronger combinations of the guarantees cannot be ensured.

Jan Kretínský, Guillermo A. Pérez, and Jean-François Raskin. Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.8, author = {Kret{\'\i}nsk\'{y}, Jan and P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, title = {{Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.8}, URN = {urn:nbn:de:0030-drops-95468}, doi = {10.4230/LIPIcs.CONCUR.2018.8}, annote = {Keywords: Markov decision processes, Reinforcement learning, Beyond worst case} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We investigate the satisfiability and finite satisfiability problem for probabilistic computation-tree logic (PCTL) where operators are not restricted by any step bounds. We establish decidability for several fragments containing quantitative operators and pinpoint the difficulties arising in more complex fragments where the decidability remains open.

Jan Kretínský and Alexej Rotar. The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.32, author = {Kret{\'\i}nsk\'{y}, Jan and Rotar, Alexej}, title = {{The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.32}, URN = {urn:nbn:de:0030-drops-95708}, doi = {10.4230/LIPIcs.CONCUR.2018.32}, annote = {Keywords: temporal logic, probabilistic verification, probabilistic computation tree logic, satisfiability} }

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**Published in:** Dagstuhl Reports, Volume 8, Issue 3 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 18121 "Machine Learning and Model Checking Join Forces". This Dagstuhl Seminar brought together researchers working in the fields of machine learning and model checking. It helped to identify new research topics on the one hand and to help with current problems on the other hand.

Nils Jansen, Joost-Pieter Katoen, Pusmeet Kohli, and Jan Kretinsky. Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121). In Dagstuhl Reports, Volume 8, Issue 3, pp. 74-93, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{jansen_et_al:DagRep.8.3.74, author = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, title = {{Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121)}}, pages = {74--93}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {3}, editor = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.8.3.74}, URN = {urn:nbn:de:0030-drops-92988}, doi = {10.4230/DagRep.8.3.74}, annote = {Keywords: artificial intelligence, cyber-physical systems, formal methods, formal verification, logics, machine learning, model checking, quantitative verification, safety-critical systems} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by temporal logics or automata. We investigate which of the distances can be approximated by observing the systems, i.e. by black-box testing or simulation, and we provide both negative and positive results.

Przemyslaw Daca, Thomas A. Henzinger, Jan Kretinsky, and Tatjana Petrov. Linear Distances between Markov Chains. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{daca_et_al:LIPIcs.CONCUR.2016.20, author = {Daca, Przemyslaw and Henzinger, Thomas A. and Kretinsky, Jan and Petrov, Tatjana}, title = {{Linear Distances between Markov Chains}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.20}, URN = {urn:nbn:de:0030-drops-61829}, doi = {10.4230/LIPIcs.CONCUR.2016.20}, annote = {Keywords: probabilistic systems, verification, statistical model checking, temporal logic, behavioural equivalence} }

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**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

We consider weighted automata with both positive and negative integer weights on edges and study the problem of synchronization using adaptive strategies that may only observe whether the current weight-level is negative or nonnegative. We show that the synchronization problem is decidable in polynomial time for deterministic weighted automata.

Jan Kretinsky, Kim Guldstrand Larsen, Simon Laursen, and Jiri Srba. Polynomial Time Decidability of Weighted Synchronization under Partial Observability. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 142-154, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2015.142, author = {Kretinsky, Jan and Larsen, Kim Guldstrand and Laursen, Simon and Srba, Jiri}, title = {{Polynomial Time Decidability of Weighted Synchronization under Partial Observability}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {142--154}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.142}, URN = {urn:nbn:de:0030-drops-53927}, doi = {10.4230/LIPIcs.CONCUR.2015.142}, annote = {Keywords: weighted automata, partial observability, synchronization, complexity} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Interactive Markov chains (IMC) are compositional behavioral models extending both labeled transition systems and continuous-time Markov chains. IMC pair modeling convenience - owed to compositionality properties - with effective verification algorithms and tools - owed to Markov properties. Thus far however, IMC verification did not consider compositionality properties, but considered closed systems. This paper discusses the evaluation of IMC in an open and thus compositional interpretation. For this we embed the IMC into a game that is played with the environment. We devise algorithms that enable us to derive bounds on reachability probabilities that are assured to hold in any composition context.

Tomas Brazdil, Holger Hermanns, Jan Krcal, Jan Kretinsky, and Vojtech Rehak. Verification of Open Interactive Markov Chains. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 474-485, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2012.474, author = {Brazdil, Tomas and Hermanns, Holger and Krcal, Jan and Kretinsky, Jan and Rehak, Vojtech}, title = {{Verification of Open Interactive Markov Chains}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {474--485}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.474}, URN = {urn:nbn:de:0030-drops-38826}, doi = {10.4230/LIPIcs.FSTTCS.2012.474}, annote = {Keywords: IMC, compositional verification, synthesis, time bounded reachability, discretization} }

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**Published in:** OASIcs, Volume 16, Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers (2011)

The formalism of modal transition systems (MTS) is a well established framework for systems specification as well as abstract interpretation. Nevertheless, due to incapability to capture some useful features, various extensions have been studied, such as e.g. mixed transition systems or disjunctive MTS. Thus a need to compare them has emerged. Therefore, we introduce transition system with obligations as a general model encompassing all the aforementioned models, and equip it with a process algebra description. Using these instruments, we then compare the previously studied subclasses and characterize their relationships.

Nikola Benes and Jan Kretinsky. Process Algebra for Modal Transition Systemses. In Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers. Open Access Series in Informatics (OASIcs), Volume 16, pp. 9-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benes_et_al:OASIcs.MEMICS.2010.9, author = {Benes, Nikola and Kretinsky, Jan}, title = {{Process Algebra for Modal Transition Systemses}}, booktitle = {Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers}, pages = {9--18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-22-4}, ISSN = {2190-6807}, year = {2011}, volume = {16}, editor = {Matyska, Ludek and Kozubek, Michal and Vojnar, Tomas and Zemcik, Pavel and Antos, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.MEMICS.2010.9}, URN = {urn:nbn:de:0030-drops-30701}, doi = {10.4230/OASIcs.MEMICS.2010.9}, annote = {Keywords: modal transition systems, process algebra, specification} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We study continuous-time stochastic games with time-bounded
reachability objectives. We show that each vertex in such a game
has a \emph{value} (i.e., an equilibrium probability), and we
classify the conditions under which optimal strategies exist.
Finally, we show how to compute optimal strategies in finite uniform
games, and how to compute $\varepsilon$-optimal strategies in
finitely-branching games with bounded rates (for finite games, we
provide detailed complexity estimations).

Tomas Brazdil, Vojtech Forejt, Jan Krcal, Jan Kretinsky, and Antonin Kucera. Continuous-Time Stochastic Games with Time-Bounded Reachability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 61-72, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2009.2307, author = {Brazdil, Tomas and Forejt, Vojtech and Krcal, Jan and Kretinsky, Jan and Kucera, Antonin}, title = {{Continuous-Time Stochastic Games with Time-Bounded Reachability}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {61--72}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2307}, URN = {urn:nbn:de:0030-drops-23077}, doi = {10.4230/LIPIcs.FSTTCS.2009.2307}, annote = {Keywords: Continuous time stochastic systems, time bounded reachability, stochastic games} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the approximation.
As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the "boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations.
We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees.

Kush Grover, Jan Křetínský, Tobias Meggendorfer, and Maximilian Weininger. Anytime Guarantees for Reachability in Uncountable Markov Decision Processes. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grover_et_al:LIPIcs.CONCUR.2022.11, author = {Grover, Kush and K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias and Weininger, Maximilian}, title = {{Anytime Guarantees for Reachability in Uncountable Markov Decision Processes}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.11}, URN = {urn:nbn:de:0030-drops-170743}, doi = {10.4230/LIPIcs.CONCUR.2022.11}, annote = {Keywords: Uncountable system, Markov decision process, discrete-time Markov control process, probabilistic verification, anytime guarantee} }

Document

**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Restarts are used in many computer systems to improve performance. Examples include reloading a webpage, reissuing a request, or restarting a randomized search. The design of restart strategies has been extensively studied by the performance evaluation community. In this paper, we address the problem of designing universal restart strategies, valid for arbitrary finite-state Markov chains, that enforce a given ω-regular property while not knowing the chain. A strategy enforces a property φ if, with probability 1, the number of restarts is finite, and the run of the Markov chain after the last restart satisfies φ. We design a simple "cautious" strategy that solves the problem, and a more sophisticated "bold" strategy with an almost optimal number of restarts.

Javier Esparza, Stefan Kiefer, Jan Křetínský, and Maximilian Weininger. Enforcing ω-Regular Properties in Markov Chains by Restarting. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2021.5, author = {Esparza, Javier and Kiefer, Stefan and K\v{r}et{\'\i}nsk\'{y}, Jan and Weininger, Maximilian}, title = {{Enforcing \omega-Regular Properties in Markov Chains by Restarting}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {5:1--5:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.5}, URN = {urn:nbn:de:0030-drops-143824}, doi = {10.4230/LIPIcs.CONCUR.2021.5}, annote = {Keywords: Markov chains, omega-regular properties, runtime enforcement} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

We introduce a framework for approximate analysis of Markov decision processes (MDP) with bounded-, unbounded-, and infinite-horizon properties. The main idea is to identify a "core" of an MDP, i.e., a subsystem where we provably remain with high probability, and to avoid computation on the less relevant rest of the state space. Although we identify the core using simulations and statistical techniques, it allows for rigorous error bounds in the analysis. Consequently, we obtain efficient analysis algorithms based on partial exploration for various settings, including the challenging case of strongly connected systems.

Jan Křetínský and Tobias Meggendorfer. Of Cores: A Partial-Exploration Framework for Markov Decision Processes. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2019.5, author = {K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias}, title = {{Of Cores: A Partial-Exploration Framework for Markov Decision Processes}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {5:1--5:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.5}, URN = {urn:nbn:de:0030-drops-109076}, doi = {10.4230/LIPIcs.CONCUR.2019.5}, annote = {Keywords: Markov Decision Processes, Reachability, Approximation} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We formalize the problem of maximizing the mean-payoff value with high probability while satisfying a parity objective in a Markov decision process (MDP) with unknown probabilistic transition function and unknown reward function. Assuming the support of the unknown transition function and a lower bound on the minimal transition probability are known in advance, we show that in MDPs consisting of a single end component, two combinations of guarantees on the parity and mean-payoff objectives can be achieved depending on how much memory one is willing to use. (i) For all epsilon and gamma we can construct an online-learning finite-memory strategy that almost-surely satisfies the parity objective and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. (ii) Alternatively, for all epsilon and gamma there exists an online-learning infinite-memory strategy that satisfies the parity objective surely and which achieves an epsilon-optimal mean payoff with probability at least 1 - gamma. We extend the above results to MDPs consisting of more than one end component in a natural way. Finally, we show that the aforementioned guarantees are tight, i.e. there are MDPs for which stronger combinations of the guarantees cannot be ensured.

Jan Kretínský, Guillermo A. Pérez, and Jean-François Raskin. Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.8, author = {Kret{\'\i}nsk\'{y}, Jan and P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, title = {{Learning-Based Mean-Payoff Optimization in an Unknown MDP under Omega-Regular Constraints}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.8}, URN = {urn:nbn:de:0030-drops-95468}, doi = {10.4230/LIPIcs.CONCUR.2018.8}, annote = {Keywords: Markov decision processes, Reinforcement learning, Beyond worst case} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We investigate the satisfiability and finite satisfiability problem for probabilistic computation-tree logic (PCTL) where operators are not restricted by any step bounds. We establish decidability for several fragments containing quantitative operators and pinpoint the difficulties arising in more complex fragments where the decidability remains open.

Jan Kretínský and Alexej Rotar. The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2018.32, author = {Kret{\'\i}nsk\'{y}, Jan and Rotar, Alexej}, title = {{The Satisfiability Problem for Unbounded Fragments of Probabilistic CTL}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.32}, URN = {urn:nbn:de:0030-drops-95708}, doi = {10.4230/LIPIcs.CONCUR.2018.32}, annote = {Keywords: temporal logic, probabilistic verification, probabilistic computation tree logic, satisfiability} }

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**Published in:** Dagstuhl Reports, Volume 8, Issue 3 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 18121 "Machine Learning and Model Checking Join Forces". This Dagstuhl Seminar brought together researchers working in the fields of machine learning and model checking. It helped to identify new research topics on the one hand and to help with current problems on the other hand.

Nils Jansen, Joost-Pieter Katoen, Pusmeet Kohli, and Jan Kretinsky. Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121). In Dagstuhl Reports, Volume 8, Issue 3, pp. 74-93, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{jansen_et_al:DagRep.8.3.74, author = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, title = {{Machine Learning and Model Checking Join Forces (Dagstuhl Seminar 18121)}}, pages = {74--93}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {3}, editor = {Jansen, Nils and Katoen, Joost-Pieter and Kohli, Pusmeet and Kretinsky, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.8.3.74}, URN = {urn:nbn:de:0030-drops-92988}, doi = {10.4230/DagRep.8.3.74}, annote = {Keywords: artificial intelligence, cyber-physical systems, formal methods, formal verification, logics, machine learning, model checking, quantitative verification, safety-critical systems} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by temporal logics or automata. We investigate which of the distances can be approximated by observing the systems, i.e. by black-box testing or simulation, and we provide both negative and positive results.

Przemyslaw Daca, Thomas A. Henzinger, Jan Kretinsky, and Tatjana Petrov. Linear Distances between Markov Chains. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{daca_et_al:LIPIcs.CONCUR.2016.20, author = {Daca, Przemyslaw and Henzinger, Thomas A. and Kretinsky, Jan and Petrov, Tatjana}, title = {{Linear Distances between Markov Chains}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.20}, URN = {urn:nbn:de:0030-drops-61829}, doi = {10.4230/LIPIcs.CONCUR.2016.20}, annote = {Keywords: probabilistic systems, verification, statistical model checking, temporal logic, behavioural equivalence} }

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**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

We consider weighted automata with both positive and negative integer weights on edges and study the problem of synchronization using adaptive strategies that may only observe whether the current weight-level is negative or nonnegative. We show that the synchronization problem is decidable in polynomial time for deterministic weighted automata.

Jan Kretinsky, Kim Guldstrand Larsen, Simon Laursen, and Jiri Srba. Polynomial Time Decidability of Weighted Synchronization under Partial Observability. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 142-154, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kretinsky_et_al:LIPIcs.CONCUR.2015.142, author = {Kretinsky, Jan and Larsen, Kim Guldstrand and Laursen, Simon and Srba, Jiri}, title = {{Polynomial Time Decidability of Weighted Synchronization under Partial Observability}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {142--154}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.142}, URN = {urn:nbn:de:0030-drops-53927}, doi = {10.4230/LIPIcs.CONCUR.2015.142}, annote = {Keywords: weighted automata, partial observability, synchronization, complexity} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Interactive Markov chains (IMC) are compositional behavioral models extending both labeled transition systems and continuous-time Markov chains. IMC pair modeling convenience - owed to compositionality properties - with effective verification algorithms and tools - owed to Markov properties. Thus far however, IMC verification did not consider compositionality properties, but considered closed systems. This paper discusses the evaluation of IMC in an open and thus compositional interpretation. For this we embed the IMC into a game that is played with the environment. We devise algorithms that enable us to derive bounds on reachability probabilities that are assured to hold in any composition context.

Tomas Brazdil, Holger Hermanns, Jan Krcal, Jan Kretinsky, and Vojtech Rehak. Verification of Open Interactive Markov Chains. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 474-485, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2012.474, author = {Brazdil, Tomas and Hermanns, Holger and Krcal, Jan and Kretinsky, Jan and Rehak, Vojtech}, title = {{Verification of Open Interactive Markov Chains}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {474--485}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.474}, URN = {urn:nbn:de:0030-drops-38826}, doi = {10.4230/LIPIcs.FSTTCS.2012.474}, annote = {Keywords: IMC, compositional verification, synthesis, time bounded reachability, discretization} }

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**Published in:** OASIcs, Volume 16, Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers (2011)

The formalism of modal transition systems (MTS) is a well established framework for systems specification as well as abstract interpretation. Nevertheless, due to incapability to capture some useful features, various extensions have been studied, such as e.g. mixed transition systems or disjunctive MTS. Thus a need to compare them has emerged. Therefore, we introduce transition system with obligations as a general model encompassing all the aforementioned models, and equip it with a process algebra description. Using these instruments, we then compare the previously studied subclasses and characterize their relationships.

Nikola Benes and Jan Kretinsky. Process Algebra for Modal Transition Systemses. In Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers. Open Access Series in Informatics (OASIcs), Volume 16, pp. 9-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benes_et_al:OASIcs.MEMICS.2010.9, author = {Benes, Nikola and Kretinsky, Jan}, title = {{Process Algebra for Modal Transition Systemses}}, booktitle = {Sixth Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'10) -- Selected Papers}, pages = {9--18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-22-4}, ISSN = {2190-6807}, year = {2011}, volume = {16}, editor = {Matyska, Ludek and Kozubek, Michal and Vojnar, Tomas and Zemcik, Pavel and Antos, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.MEMICS.2010.9}, URN = {urn:nbn:de:0030-drops-30701}, doi = {10.4230/OASIcs.MEMICS.2010.9}, annote = {Keywords: modal transition systems, process algebra, specification} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We study continuous-time stochastic games with time-bounded
reachability objectives. We show that each vertex in such a game
has a \emph{value} (i.e., an equilibrium probability), and we
classify the conditions under which optimal strategies exist.
Finally, we show how to compute optimal strategies in finite uniform
games, and how to compute $\varepsilon$-optimal strategies in
finitely-branching games with bounded rates (for finite games, we
provide detailed complexity estimations).

Tomas Brazdil, Vojtech Forejt, Jan Krcal, Jan Kretinsky, and Antonin Kucera. Continuous-Time Stochastic Games with Time-Bounded Reachability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 61-72, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2009.2307, author = {Brazdil, Tomas and Forejt, Vojtech and Krcal, Jan and Kretinsky, Jan and Kucera, Antonin}, title = {{Continuous-Time Stochastic Games with Time-Bounded Reachability}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {61--72}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2307}, URN = {urn:nbn:de:0030-drops-23077}, doi = {10.4230/LIPIcs.FSTTCS.2009.2307}, annote = {Keywords: Continuous time stochastic systems, time bounded reachability, stochastic games} }

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