Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We consider simple stochastic games G with energy-parity objectives, a combination of quantitative rewards with a qualitative parity condition. The Maximizer tries to avoid running out of energy while simultaneously satisfying a parity condition.
We present an algorithm to approximate the value of a given configuration in 2-NEXPTIME. Moreover, ε-optimal strategies for either player require at most O(2-EXP(|G|)⋅log(1/ε)) memory modes.

Mohan Dantam and Richard Mayr. Approximating the Value of Energy-Parity Objectives in Simple Stochastic Games. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dantam_et_al:LIPIcs.MFCS.2023.38, author = {Dantam, Mohan and Mayr, Richard}, title = {{Approximating the Value of Energy-Parity Objectives in Simple Stochastic Games}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.38}, URN = {urn:nbn:de:0030-drops-185724}, doi = {10.4230/LIPIcs.MFCS.2023.38}, annote = {Keywords: Energy-Parity Games, Simple Stochastic Games, Parity, Energy} }

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**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} }

Document

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

We study countably infinite Markov decision processes (MDPs) with real-valued transition rewards. Every infinite run induces the following sequences of payoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2. Total payoff (the sequence of the sums of all rewards so far), and 3. Mean payoff. For each payoff type, the objective is to maximize the probability that the liminf is non-negative. We establish the complete picture of the strategy complexity of these objectives, i.e., how much memory is necessary and sufficient for ε-optimal (resp. optimal) strategies. Some cases can be won with memoryless deterministic strategies, while others require a step counter, a reward counter, or both.

Richard Mayr and Eric Munday. Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mayr_et_al:LIPIcs.CONCUR.2021.12, author = {Mayr, Richard and Munday, Eric}, title = {{Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {12:1--12:15}, 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.2021.12}, URN = {urn:nbn:de:0030-drops-143893}, doi = {10.4230/LIPIcs.CONCUR.2021.12}, annote = {Keywords: Markov decision processes, Strategy complexity, Mean payoff} }

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)

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} }

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

A timed network consists of an arbitrary number of initially identical 1-clock timed automata, interacting via hand-shake communication. In this setting there is no unique central controller, since all automata are initially identical. We consider the universal safety problem for such controller-less timed networks, i.e., verifying that a bad event (enabling some given transition) is impossible regardless of the size of the network.
This universal safety problem is dual to the existential coverability problem for timed-arc Petri nets, i.e., does there exist a number m of tokens, such that starting with m tokens in a given place, and none in the other places, some given transition is eventually enabled.
We show that these problems are PSPACE-complete.

Parosh Aziz Abdulla, Mohamed Faouzi Atig, Radu Ciobanu, Richard Mayr, and Patrick Totzke. Universal Safety for Timed Petri Nets is PSPACE-complete. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abdulla_et_al:LIPIcs.CONCUR.2018.6, author = {Abdulla, Parosh Aziz and Atig, Mohamed Faouzi and Ciobanu, Radu and Mayr, Richard and Totzke, Patrick}, title = {{Universal Safety for Timed Petri Nets is PSPACE-complete}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {6:1--6:15}, 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.2018.6}, URN = {urn:nbn:de:0030-drops-95447}, doi = {10.4230/LIPIcs.CONCUR.2018.6}, annote = {Keywords: timed networks, safety checking, Petri nets, coverability} }

Document

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

Freeze LTL is a temporal logic with registers that is suitable for specifying properties of data words. In this paper we study the model checking problem for Freeze LTL on one-counter automata. This problem is known to be undecidable in full generality and PSPACE-complete for the special case of deterministic one-counter automata. Several years ago, Demri and Sangnier investigated the model checking problem for the flat fragment of Freeze LTL on several classes of counter automata and posed the decidability of model checking flat Freeze LTL on one-counter automata as an open problem. In this paper we resolve this problem positively, utilising a known reduction to a reachability problem on one-counter automata with parameterised equality and disequality tests. Our main technical contribution is to show decidability of the latter problem by translation to Presburger arithmetic.

Antonia Lechner, Richard Mayr, Joël Ouaknine, Amaury Pouly, and James Worrell. Model Checking Flat Freeze LTL on One-Counter Automata. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lechner_et_al:LIPIcs.CONCUR.2016.29, author = {Lechner, Antonia and Mayr, Richard and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Model Checking Flat Freeze LTL on One-Counter Automata}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {29:1--29:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.29}, URN = {urn:nbn:de:0030-drops-61841}, doi = {10.4230/LIPIcs.CONCUR.2016.29}, annote = {Keywords: one-counter automata, disequality tests, reachability, freeze LTL, Presburger arithmetic} }

Document

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

One-counter nets (OCN) are Petri nets with exactly one unbounded place.
They are equivalent to a subclass of one-counter automata with just a weak test for zero. Unlike many other semantic equivalences, strong and weak simulation preorder are decidable for OCN, but the computational complexity was an open problem. We show that both strong and weak simulation preorder on OCN are Pspace-complete.

Piotr Hofman, Slawomir Lasota, Richard Mayr, and Patrick Totzke. Simulation Over One-counter Nets is PSPACE-Complete. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 515-526, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{hofman_et_al:LIPIcs.FSTTCS.2013.515, author = {Hofman, Piotr and Lasota, Slawomir and Mayr, Richard and Totzke, Patrick}, title = {{Simulation Over One-counter Nets is PSPACE-Complete}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {515--526}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.515}, URN = {urn:nbn:de:0030-drops-43970}, doi = {10.4230/LIPIcs.FSTTCS.2013.515}, annote = {Keywords: Simulation preorder; one-counter nets; complexity} }

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