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

We study graphs and two-player games in which rewards are assigned to states, and the goal of the players is to satisfy or dissatisfy certain property of the generated outcome, given as a mean payoff property. Since the notion of mean-payoff does not reflect possible fluctuations from the mean-payoff along a run, we propose definitions and algorithms for capturing the stability of the system, and give algorithms for deciding if a given mean payoff and stability objective can be ensured in the system.

Tomas Brazdil, Vojtech Forejt, Antonin Kucera, and Petr Novotny. Stability in Graphs and Games. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{brazdil_et_al:LIPIcs.CONCUR.2016.10, author = {Brazdil, Tomas and Forejt, Vojtech and Kucera, Antonin and Novotny, Petr}, title = {{Stability in Graphs and Games}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {10:1--10:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.10}, URN = {urn:nbn:de:0030-drops-61784}, doi = {10.4230/LIPIcs.CONCUR.2016.10}, annote = {Keywords: Games, Stability, Mean-Payoff, Window Objectives} }

Document

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

We study frequency linear-time temporal logic (fLTL) which extends the linear-time temporal logic (LTL) with a path operator G^p expressing that on a path, certain formula holds with at least a given frequency p, thus relaxing the semantics of the usual G operator of LTL. Such logic is particularly useful in probabilistic systems, where some undesirable events such as random failures may occur and are acceptable if they are rare enough. Frequency-related extensions of LTL have been previously studied by several authors, where mostly the logic is equipped with an extended "until" and "globally" operator, leading to undecidability of most interesting problems.
For the variant we study, we are able to establish fundamental decidability results. We show that for Markov chains, the problem of computing the probability with which a given fLTL formula holds has the same complexity as the analogous problem for LTL. We also show that for Markov decision processes the problem becomes more delicate, but when restricting the frequency bound p to be 1 and negations not to be outside any G^p operator, we can compute the maximum probability of satisfying the fLTL formula. This can be again performed with the same time complexity as for the ordinary LTL formulas.

Vojtech Forejt and Jan Krcal. On Frequency LTL in Probabilistic Systems. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 184-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{forejt_et_al:LIPIcs.CONCUR.2015.184, author = {Forejt, Vojtech and Krcal, Jan}, title = {{On Frequency LTL in Probabilistic Systems}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {184--197}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.184}, URN = {urn:nbn:de:0030-drops-53789}, doi = {10.4230/LIPIcs.CONCUR.2015.184}, annote = {Keywords: Markov chains, Markov decision processes, LTL, controller synthesis} }

Document

**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Solvency games, introduced by Berger et al., provide an abstract framework for modelling decisions of a risk-averse investor, whose goal is to avoid ever going broke. We study a new variant of this model, where, in addition to stochastic environment and fixed increments and decrements to the investor's wealth, we introduce
interest, which is earned or paid on the current level of savings or debt, respectively.
We study problems related to the minimum initial wealth sufficient to avoid bankruptcy (i.e. steady decrease of the wealth) with probability at least p. We present an exponential time algorithm which approximates this minimum initial wealth, and show that a polynomial time approximation is not possible unless P=NP.
For the qualitative case, i.e. p=1, we show that the problem whether a given number is larger than or equal to the minimum initial wealth belongs to NP \cap coNP, and show that a polynomial time algorithm would yield a polynomial time algorithm for mean-payoff games, existence of which is a longstanding open problem. We also identify some classes of solvency MDPs for which this problem is in P. In all above cases the algorithms also give corresponding bankruptcy avoiding strategies.

Tomás Brázdil, Taolue Chen, Vojtech Forejt, Petr Novotný, and Aistis Simaitis. Solvency Markov Decision Processes with Interest. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 487-499, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2013.487, author = {Br\'{a}zdil, Tom\'{a}s and Chen, Taolue and Forejt, Vojtech and Novotn\'{y}, Petr and Simaitis, Aistis}, title = {{Solvency Markov Decision Processes with Interest}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {487--499}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.487}, URN = {urn:nbn:de:0030-drops-43959}, doi = {10.4230/LIPIcs.FSTTCS.2013.487}, annote = {Keywords: Markov decision processes, algorithms, complexity, market models.} }

Document

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

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). Our first contribution is a general construction that reduces checking bisimilarity of probabilistic transition systems to checking bisimilarity of non-deterministic transition systems. This construction directly yields decidability of bisimilarity for pPDA, as well as an elementary upper bound for the bisimilarity problem on the subclass of probabilistic basic process algebras, i.e., single-state pPDA. We further show that, with careful analysis, the general reduction can be used to prove an EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown automata. Here we also provide a matching lower bound, establishing EXPTIME-completeness. Finally we prove that deciding bisimilarity of probabilistic one-counter automata, another subclass of pPDA, is PSPACE-complete. Here we use a more specialised argument to obtain optimal complexity bounds.

Vojtech Forejt, Petr Jancar, Stefan Kiefer, and James Worrell. Bisimilarity of Probabilistic Pushdown Automata. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 448-460, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{forejt_et_al:LIPIcs.FSTTCS.2012.448, author = {Forejt, Vojtech and Jancar, Petr and Kiefer, Stefan and Worrell, James}, title = {{Bisimilarity of Probabilistic Pushdown Automata}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {448--460}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.448}, URN = {urn:nbn:de:0030-drops-38800}, doi = {10.4230/LIPIcs.FSTTCS.2012.448}, annote = {Keywords: bisimilarity, probabilistic systems, pushdown automata} }

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.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} }