Document

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

The Transience objective is not to visit any state infinitely often. While this is not possible in any finite Markov Decision Process (MDP), it can be satisfied in countably infinite ones, e.g., if the transition graph is acyclic.
We prove the following fundamental properties of Transience in countably infinite MDPs.
1) There exist uniformly ε-optimal MD strategies (memoryless deterministic) for Transience, even in infinitely branching MDPs.
2) Optimal strategies for Transience need not exist, even if the MDP is finitely branching. However, if an optimal strategy exists then there is also an optimal MD strategy.
3) If an MDP is universally transient (i.e., almost surely transient under all strategies) then many other objectives have a lower strategy complexity than in general MDPs. E.g., ε-optimal strategies for Safety and co-Büchi and optimal strategies for {0,1,2}-Parity (where they exist) can be chosen MD, even if the MDP is infinitely branching.

Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Transience in Countable MDPs. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kiefer_et_al:LIPIcs.CONCUR.2021.11, author = {Kiefer, Stefan and Mayr, Richard and Shirmohammadi, Mahsa and Totzke, Patrick}, title = {{Transience in Countable MDPs}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {11:1--11:15}, 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.2021.11}, URN = {urn:nbn:de:0030-drops-143881}, doi = {10.4230/LIPIcs.CONCUR.2021.11}, annote = {Keywords: Markov decision processes, Parity, Transience} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study a class of reachability problems in weighted graphs with constraints on the accumulated weight of paths. The problems we study can equivalently be formulated in the model of vector addition systems with states (VASS). We consider a version of the vertex-to-vertex reachability problem in which the accumulated weight of a path is required always to be non-negative. This is equivalent to the so-called control-state reachability problem (also called the coverability problem) for 1-dimensional VASS. We show that this problem lies in NC: the class of problems solvable in polylogarithmic parallel time. In our main result we generalise the problem to allow disequality constraints on edges (i.e., we allow edges to be disabled if the accumulated weight is equal to a specific value). We show that in this case the vertex-to-vertex reachability problem is solvable in polynomial time even though a shortest path may have exponential length. In the language of VASS this means that control-state reachability is in polynomial time for 1-dimensional VASS with disequality tests.

Shaull Almagor, Nathann Cohen, Guillermo A. Pérez, Mahsa Shirmohammadi, and James Worrell. Coverability in 1-VASS with Disequality Tests. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{almagor_et_al:LIPIcs.CONCUR.2020.38, author = {Almagor, Shaull and Cohen, Nathann and P\'{e}rez, Guillermo A. and Shirmohammadi, Mahsa and Worrell, James}, title = {{Coverability in 1-VASS with Disequality Tests}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {38:1--38:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.38}, URN = {urn:nbn:de:0030-drops-128501}, doi = {10.4230/LIPIcs.CONCUR.2020.38}, annote = {Keywords: Reachability, Vector addition systems with states, Weighted graphs} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study countably infinite MDPs with parity objectives. Unlike in finite MDPs, optimal strategies need not exist, and may require infinite memory if they do. We provide a complete picture of the exact strategy complexity of ε-optimal strategies (and optimal strategies, where they exist) for all subclasses of parity objectives in the Mostowski hierarchy. Either MD-strategies, Markov strategies, or 1-bit Markov strategies are necessary and sufficient, depending on the number of colors, the branching degree of the MDP, and whether one considers ε-optimal or optimal strategies. In particular, 1-bit Markov strategies are necessary and sufficient for ε-optimal (resp. optimal) strategies for general parity objectives.

Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Strategy Complexity of Parity Objectives in Countable MDPs. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kiefer_et_al:LIPIcs.CONCUR.2020.39, author = {Kiefer, Stefan and Mayr, Richard and Shirmohammadi, Mahsa and Totzke, Patrick}, title = {{Strategy Complexity of Parity Objectives in Countable MDPs}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {39:1--39:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.39}, URN = {urn:nbn:de:0030-drops-128513}, doi = {10.4230/LIPIcs.CONCUR.2020.39}, annote = {Keywords: Markov decision processes, Parity objectives, Levy’s zero-one law} }

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Invited Talk

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Markov decision processes (MDPs) are a standard model for dynamic systems that exhibit both stochastic and nondeterministic behavior. For MDPs with finite state space it is known that for a wide range of objectives there exist optimal strategies that are memoryless and deterministic. In contrast, if the state space is infinite, optimal strategies may not exist, and optimal or ε-optimal strategies may require (possibly infinite) memory. In this paper we consider qualitative objectives: reachability, safety, (co-)Büchi, and other parity objectives. We aim at giving an introduction to a collection of techniques that allow for the construction of strategies with little or no memory in countably infinite MDPs.

Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, Patrick Totzke, and Dominik Wojtczak. How to Play in Infinite MDPs (Invited Talk). In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kiefer_et_al:LIPIcs.ICALP.2020.3, author = {Kiefer, Stefan and Mayr, Richard and Shirmohammadi, Mahsa and Totzke, Patrick and Wojtczak, Dominik}, title = {{How to Play in Infinite MDPs}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {3:1--3:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.3}, URN = {urn:nbn:de:0030-drops-124103}, doi = {10.4230/LIPIcs.ICALP.2020.3}, annote = {Keywords: Markov decision processes} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal n-step payoff by iterating n times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon n. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon n in binary and an MDP, computing an optimal policy is EXPTIME-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. To obtain this main result, we develop several stepping stones that yield results of an independent interest. For instance, we show that it is EXPTIME-complete to compute the n-fold iteration (with n in binary) of a function given by a straight-line program over the integers with max and + as operators. We also provide new complexity results for the bounded halting problem in linear-update counter machines.

Nikhil Balaji, Stefan Kiefer, Petr Novotný, Guillermo A. Pérez, and Mahsa Shirmohammadi. On the Complexity of Value Iteration (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 102:1-102:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{balaji_et_al:LIPIcs.ICALP.2019.102, author = {Balaji, Nikhil and Kiefer, Stefan and Novotn\'{y}, Petr and P\'{e}rez, Guillermo A. and Shirmohammadi, Mahsa}, title = {{On the Complexity of Value Iteration}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {102:1--102:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.102}, URN = {urn:nbn:de:0030-drops-106782}, doi = {10.4230/LIPIcs.ICALP.2019.102}, annote = {Keywords: Markov decision processes, Value iteration, Formal verification} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We study countably infinite Markov decision processes with Büchi objectives, which ask to visit a given subset F of states infinitely often. A question left open by T.P. Hill in 1979 [Theodore Preston Hill, 1979] is whether there always exist epsilon-optimal Markov strategies, i.e., strategies that base decisions only on the current state and the number of steps taken so far. We provide a negative answer to this question by constructing a non-trivial counterexample. On the other hand, we show that Markov strategies with only 1 bit of extra memory are sufficient.

Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Büchi Objectives in Countable MDPs (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 119:1-119:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kiefer_et_al:LIPIcs.ICALP.2019.119, author = {Kiefer, Stefan and Mayr, Richard and Shirmohammadi, Mahsa and Totzke, Patrick}, title = {{B\"{u}chi Objectives in Countable MDPs}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {119:1--119:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.119}, URN = {urn:nbn:de:0030-drops-106959}, doi = {10.4230/LIPIcs.ICALP.2019.119}, annote = {Keywords: Markov decision processes} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs (to be minimised) and rewards (to be maximised) along a computation. Each observer has a constant non-negative derivative which may depend on the location of the MPTA.
We study the Pareto Domination Problem, which asks whether it is possible to reach a target location via a run in which the accumulated costs and rewards Pareto dominate a given objective vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete. We also consider an epsilon-approximate Pareto Domination Problem that is decidable without restricting the number and types of observers.
We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10^{th} Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.

Martin Fränzle, Mahsa Shirmohammadi, Mani Swaminathan, and James Worrell. Costs and Rewards in Priced Timed Automata. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 125:1-125:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{franzle_et_al:LIPIcs.ICALP.2018.125, author = {Fr\"{a}nzle, Martin and Shirmohammadi, Mahsa and Swaminathan, Mani and Worrell, James}, title = {{Costs and Rewards in Priced Timed Automata}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {125:1--125:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.125}, URN = {urn:nbn:de:0030-drops-91297}, doi = {10.4230/LIPIcs.ICALP.2018.125}, annote = {Keywords: Priced Timed Automata, Pareto Domination, Diophantine Equations} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n*m matrix M into a product of a nonnegative n*d matrix W and a nonnegative d*m matrix H. Restricted NMF requires in addition that the column spaces of M and W coincide.
Finding the minimal inner dimension d is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question negatively, thus falsifying a positive answer claimed in 1974.
Furthermore, we investigate whether a rational matrix M always has a restricted NMF of minimal inner dimension whose factors W and H are also rational. We show that this holds for matrices M of rank at most 3 and we exhibit a rank-4 matrix for which W and H require irrational entries.

Dmitry Chistikov, Stefan Kiefer, Ines Marusic, Mahsa Shirmohammadi, and James Worrell. On Restricted Nonnegative Matrix Factorization. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 103:1-103:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chistikov_et_al:LIPIcs.ICALP.2016.103, author = {Chistikov, Dmitry and Kiefer, Stefan and Marusic, Ines and Shirmohammadi, Mahsa and Worrell, James}, title = {{On Restricted Nonnegative Matrix Factorization}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {103:1--103:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.103}, URN = {urn:nbn:de:0030-drops-62389}, doi = {10.4230/LIPIcs.ICALP.2016.103}, annote = {Keywords: nonnegative matrix factorization, nonnegative rank, probabilistic automata, labelled Markov chains, minimization} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Register automata (RAs) are finite automata extended with a finite set of registers to store and compare data. We study the concept of synchronizing data words in RAs: Does there exist a data word that sends all states of the RA to a single state?
For deterministic RAs with k registers (k-DRAs), we prove that inputting data words with 2k+1 distinct data, from the infinite data domain, is sufficient to synchronize. We show that the synchronizing problem for DRAs is in general PSPACE-complete, and is NLOGSPACE-complete for 1-DRAs. For nondeterministic RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of RA) might be necessary to synchronize. The synchronizing problem for NRAs is in general undecidable, however, we establish Ackermann-completeness of the problem for 1-NRAs. Our most substantial achievement is proving NEXPTIME-completeness of the length-bounded synchronizing problem in NRAs (length encoded in binary). A variant of this last construction allows to prove that the bounded universality problem in NRAs is co-NEXPTIME-complete.

Parvaneh Babari, Karin Quaas, and Mahsa Shirmohammadi. Synchronizing Data Words for Register Automata. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{babari_et_al:LIPIcs.MFCS.2016.15, author = {Babari, Parvaneh and Quaas, Karin and Shirmohammadi, Mahsa}, title = {{Synchronizing Data Words for Register Automata}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.15}, URN = {urn:nbn:de:0030-drops-64996}, doi = {10.4230/LIPIcs.MFCS.2016.15}, annote = {Keywords: data words, register automata, synchronizing problem, Ackermann-completeness, bounded universality, regular-like expressions with squaring} }

Document

**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

The problem of synchronizing automata is concerned with the existence of a word that sends all states of the automaton to one and the same state. This problem has classically been studied for complete deterministic finite automata, with the existence problem being NLOGSPACE-complete.
In this paper we consider synchronizing-word problems for weighted and timed automata. We consider the synchronization problem in several variants and combinations of these, including deterministic and non-deterministic timed and weighted automata, synchronization to unique location with possibly different clock valuations or accumulated weights, as well as synchronization with a safety condition forbidding the automaton to visit states outside a safety-set during synchronization (e.g. energy constraints). For deterministic weighted automata, the synchronization problem is proven PSPACE-complete under energy constraints, and in 3-EXPSPACE under general safety constraints. For timed automata the synchronization problems are shown to be PSPACE-complete in the deterministic case, and undecidable in the non-deterministic case.

Laurent Doyen, Line Juhl, Kim G. Larsen, Nicolas Markey, and Mahsa Shirmohammadi. Synchronizing Words for Weighted and Timed Automata. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 121-132, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{doyen_et_al:LIPIcs.FSTTCS.2014.121, author = {Doyen, Laurent and Juhl, Line and Larsen, Kim G. and Markey, Nicolas and Shirmohammadi, Mahsa}, title = {{Synchronizing Words for Weighted and Timed Automata}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {121--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.121}, URN = {urn:nbn:de:0030-drops-48370}, doi = {10.4230/LIPIcs.FSTTCS.2014.121}, annote = {Keywords: Synchronizing words, weighted automata, timed automata} }

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