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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We show improved fine-grained hardness of two key lattice problems in the 𝓁_p norm: Bounded Distance Decoding to within an α factor of the minimum distance (BDD_{p, α}) and the (decisional) γ-approximate Shortest Vector Problem (GapSVP_{p,γ}), assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH). Specifically, we show:
1) For all p ∈ [1, ∞), there is no 2^{o(n)}-time algorithm for BDD_{p, α} for any constant α > α_kn, where α_kn = 2^{-c_kn} < 0.98491 and c_kn is the 𝓁₂ kissing-number constant, unless non-uniform Gap-ETH is false.
2) For all p ∈ [1, ∞), there is no 2^{o(n)}-time algorithm for BDD_{p, α} for any constant α > α^‡_p, where α^‡_p is explicit and satisfies α^‡_p = 1 for 1 ≤ p ≤ 2, α^‡_p < 1 for all p > 2, and α^‡_p → 1/2 as p → ∞, unless randomized Gap-ETH is false.
3) For all p ∈ [1, ∞) ⧵ 2 ℤ and all C > 1, there is no 2^{n/C}-time algorithm for BDD_{p, α} for any constant α > α^†_{p, C}, where α^†_{p, C} is explicit and satisfies α^†_{p, C} → 1 as C → ∞ for any fixed p ∈ [1, ∞), unless non-uniform Gap-SETH is false.
4) For all p > p₀ ≈ 2.1397, p ∉ 2ℤ, and all C > C_p, there is no 2^{n/C}-time algorithm for GapSVP_{p, γ} for some constant γ > 1, where C_p > 1 is explicit and satisfies C_p → 1 as p → ∞, unless randomized Gap-SETH is false.
Our results for BDD_{p, α} improve and extend work by Aggarwal and Stephens-Davidowitz (STOC, 2018) and Bennett and Peikert (CCC, 2020). Specifically, the quantities α_kn and α^‡_p (respectively, α^†_{p,C}) significantly improve upon the corresponding quantity α_p^* (respectively, α_{p,C}^*) of Bennett and Peikert for small p (but arise from somewhat stronger assumptions). In particular, Item 1 improves the smallest value of α for which BDD_{p, α} is known to be exponentially hard in the Euclidean norm (p = 2) to an explicit constant α < 1 for the first time under a general-purpose complexity assumption. Items 1 and 3 crucially use the recent breakthrough result of Vlăduţ (Moscow Journal of Combinatorics and Number Theory, 2019), which showed an explicit exponential lower bound on the lattice kissing number. Finally, Item 4 answers a natural question left open by Aggarwal, Bennett, Golovnev, and Stephens-Davidowitz (SODA, 2021), which showed an analogous result for the Closest Vector Problem.

Huck Bennett, Chris Peikert, and Yi Tang. Improved Hardness of BDD and SVP Under Gap-(S)ETH. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 19:1-19:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bennett_et_al:LIPIcs.ITCS.2022.19, author = {Bennett, Huck and Peikert, Chris and Tang, Yi}, title = {{Improved Hardness of BDD and SVP Under Gap-(S)ETH}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {19:1--19:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.19}, URN = {urn:nbn:de:0030-drops-156151}, doi = {10.4230/LIPIcs.ITCS.2022.19}, annote = {Keywords: lattices, lattice-based cryptography, fine-grained complexity, Bounded Distance Decoding, Shortest Vector Problem} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We show that for any n-dimensional lattice ℒ ⊆ ℝⁿ, the torus ℝⁿ/ℒ can be embedded into Hilbert space with O(√{nlog n}) distortion. This improves the previously best known upper bound of O(n√{log n}) shown by Haviv and Regev (APPROX 2010, J. Topol. Anal. 2013) and approaches the lower bound of Ω(√n) due to Khot and Naor (FOCS 2005, Math. Ann. 2006).

Ishan Agarwal, Oded Regev, and Yi Tang. Nearly Optimal Embeddings of Flat Tori. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agarwal_et_al:LIPIcs.APPROX/RANDOM.2020.43, author = {Agarwal, Ishan and Regev, Oded and Tang, Yi}, title = {{Nearly Optimal Embeddings of Flat Tori}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {43:1--43:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.43}, URN = {urn:nbn:de:0030-drops-126464}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.43}, annote = {Keywords: Lattices, metric embeddings, flat torus} }

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**Published in:** LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

Nondeterministic good-for-MDPs (GFM) automata are for MDP model checking and reinforcement learning what good-for-games automata are for reactive synthesis: a more compact alternative to deterministic automata that displays nondeterminism, but only so much that it can be resolved locally, such that a syntactic product can be analysed. GFM has recently been introduced as a property for reinforcement learning, where the simpler Büchi acceptance conditions it allows to use is key. However, while there are classic and novel techniques to obtain automata that are GFM, there has not been a decision procedure for checking whether or not an automaton is GFM. We show that GFM-ness is decidable and provide an EXPTIME decision procedure as well as a PSPACE-hardness proof.

Sven Schewe, Qiyi Tang, and Tansholpan Zhanabekova. Deciding What Is Good-For-MDPs. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{schewe_et_al:LIPIcs.CONCUR.2023.35, author = {Schewe, Sven and Tang, Qiyi and Zhanabekova, Tansholpan}, title = {{Deciding What Is Good-For-MDPs}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-299-0}, ISSN = {1868-8969}, year = {2023}, volume = {279}, editor = {P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.35}, URN = {urn:nbn:de:0030-drops-190290}, doi = {10.4230/LIPIcs.CONCUR.2023.35}, annote = {Keywords: B\"{u}chi automata, Markov Decision Processes, Omega-regular objectives, Reinforcement learning} }

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

A labelled Markov decision process (MDP) is a labelled Markov chain with nondeterminism; i.e., together with a strategy a labelled MDP induces a labelled Markov chain. Motivated by applications to the verification of probabilistic noninterference in security, we study problems whether there exist strategies such that the labelled MDPs become bisimilarity equivalent/inequivalent. We show that the equivalence problem is decidable; in fact, it is EXPTIME-complete and becomes NP-complete if one of the MDPs is a Markov chain. Concerning the inequivalence problem, we show that (1) it is decidable in polynomial time; (2) if there are strategies for inequivalence then there are memoryless strategies for inequivalence; (3) such memoryless strategies can be computed in polynomial time.

Stefan Kiefer and Qiyi Tang. Strategies for MDP Bisimilarity Equivalence and Inequivalence. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kiefer_et_al:LIPIcs.CONCUR.2022.32, author = {Kiefer, Stefan and Tang, Qiyi}, title = {{Strategies for MDP Bisimilarity Equivalence and Inequivalence}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {32:1--32:22}, 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.2022.32}, URN = {urn:nbn:de:0030-drops-170955}, doi = {10.4230/LIPIcs.CONCUR.2022.32}, annote = {Keywords: Markov decision processes, Markov chains} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

We propose polynomial-time algorithms to minimise labelled Markov chains whose transition probabilities are not known exactly, have been perturbed, or can only be obtained by sampling. Our algorithms are based on a new notion of an approximate bisimulation quotient, obtained by lumping together states that are exactly bisimilar in a slightly perturbed system. We present experiments that show that our algorithms are able to recover the structure of the bisimulation quotient of the unperturbed system.

Stefan Kiefer and Qiyi Tang. Approximate Bisimulation Minimisation. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kiefer_et_al:LIPIcs.FSTTCS.2021.48, author = {Kiefer, Stefan and Tang, Qiyi}, title = {{Approximate Bisimulation Minimisation}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {48:1--48:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.48}, URN = {urn:nbn:de:0030-drops-155599}, doi = {10.4230/LIPIcs.FSTTCS.2021.48}, annote = {Keywords: Markov chains, Behavioural metrics, Bisimulation} }

Document

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

A labelled Markov decision process is a labelled Markov chain with nondeterminism, i.e., together with a strategy a labelled MDP induces a labelled Markov chain. The model is related to interval Markov chains. Motivated by applications of equivalence checking for the verification of anonymity, we study the algorithmic comparison of two labelled MDPs, in particular, whether there exist strategies such that the MDPs become equivalent/inequivalent, both in terms of trace equivalence and in terms of probabilistic bisimilarity. We provide the first polynomial-time algorithms for computing memoryless strategies to make the two labelled MDPs inequivalent if such strategies exist. We also study the computational complexity of qualitative problems about making the total variation distance and the probabilistic bisimilarity distance less than one or equal to one.

Stefan Kiefer and Qiyi Tang. Comparing Labelled Markov Decision Processes. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kiefer_et_al:LIPIcs.FSTTCS.2020.49, author = {Kiefer, Stefan and Tang, Qiyi}, title = {{Comparing Labelled Markov Decision Processes}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {49:1--49:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.49}, URN = {urn:nbn:de:0030-drops-132903}, doi = {10.4230/LIPIcs.FSTTCS.2020.49}, annote = {Keywords: Markov decision processes, Markov chains, Behavioural metrics} }

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

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch’s probabilistic bisimilarity for probabilistic automata. In this paper, we present a novel characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon’s simple policy iteration on these games. The correctness of Condon’s approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in UP cap coUP and PPAD. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. In the proofs of all the above-mentioned results, an alternative presentation of the Hausdorff distance due to Mémoli plays a central rôle.

Giorgio Bacci, Giovanni Bacci, Kim G. Larsen, Radu Mardare, Qiyi Tang, and Franck van Breugel. Computing Probabilistic Bisimilarity Distances for Probabilistic Automata. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bacci_et_al:LIPIcs.CONCUR.2019.9, author = {Bacci, Giorgio and Bacci, Giovanni and Larsen, Kim G. and Mardare, Radu and Tang, Qiyi and van Breugel, Franck}, title = {{Computing Probabilistic Bisimilarity Distances for Probabilistic Automata}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {9:1--9:17}, 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.2019.9}, URN = {urn:nbn:de:0030-drops-109119}, doi = {10.4230/LIPIcs.CONCUR.2019.9}, annote = {Keywords: Probabilistic automata, Behavioural metrics, Simple stochastic games, Simple policy iteration algorithm} }

Document

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

Probabilistic bisimilarity, due to Segala and Lynch, is an equivalence relation that captures which states of a probabilistic automaton behave exactly the same. Deng, Chothia, Palamidessi and Pang proposed a robust quantitative generalization of probabilistic bisimilarity. Their probabilistic bisimilarity distances of states of a probabilistic automaton capture the similarity of their behaviour. The smaller the distance, the more alike the states behave. In particular, states are probabilistic bisimilar if and only if their distance is zero.
Although the complexity of computing probabilistic bisimilarity distances for probabilistic automata has already been studied and shown to be in NP cap coNP and PPAD, we are not aware of any practical algorithm to compute those distances. In this paper we provide several key results towards algorithms to compute probabilistic bisimilarity distances for probabilistic automata. In particular, we present a polynomial time algorithm that decides distance one. Furthermore, we give an alternative characterization of the probabilistic bisimilarity distances as a basis for a policy iteration algorithm.

Qiyi Tang and Franck van Breugel. Deciding Probabilistic Bisimilarity Distance One for Probabilistic Automata. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{tang_et_al:LIPIcs.CONCUR.2018.9, author = {Tang, Qiyi and van Breugel, Franck}, title = {{Deciding Probabilistic Bisimilarity Distance One for Probabilistic Automata}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {9:1--9:17}, 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.2018.9}, URN = {urn:nbn:de:0030-drops-95472}, doi = {10.4230/LIPIcs.CONCUR.2018.9}, annote = {Keywords: probabilistic automaton, probabilistic bisimilarity, distance} }

Document

**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

In the late nineties, Desharnais, Gupta, Jagadeesan and Panangaden presented probabilistic bisimilarity distances on the states of a labelled Markov chain. This provided a quantitative generalisation of probabilistic bisimilarity introduced by Larsen and Skou a decade earlier. In the last decade, several algorithms to approximate and compute these probabilistic bisimilarity distances have been put forward. In this paper, we correct, improve and generalise some of these algorithms. Furthermore, we compare their performance experimentally.

Qiyi Tang and Franck van Breugel. Algorithms to Compute Probabilistic Bisimilarity Distances for Labelled Markov Chains. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{tang_et_al:LIPIcs.CONCUR.2017.27, author = {Tang, Qiyi and van Breugel, Franck}, title = {{Algorithms to Compute Probabilistic Bisimilarity Distances for Labelled Markov Chains}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.27}, URN = {urn:nbn:de:0030-drops-77983}, doi = {10.4230/LIPIcs.CONCUR.2017.27}, annote = {Keywords: labelled Markov chain, probabilistic bisimilarity, pseudometric, policy iteration} }

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

A transformation mapping a labelled Markov chain to a simple stochastic game is presented. In the resulting simple stochastic game, each vertex corresponds to a pair of states of the labelled Markov chain. The value of a vertex of the simple stochastic game is shown to be equal to the probabilistic bisimilarity distance, a notion due to Desharnais, Gupta, Jagadeesan and Panangaden, of the corresponding pair of states of the labelled Markov chain. Bacci, Bacci, Larsen and Mardare introduced an algorithm to compute the probabilistic bisimilarity distances for a labelled Markov chain. A modification of a basic version of their algorithm for a labelled Markov chain is shown to be the policy iteration algorithm applied to the corresponding simple stochastic game. Furthermore, it is shown that this algorithm takes exponential time in the worst case.

Qiyi Tang and Franck van Breugel. Computing Probabilistic Bisimilarity Distances via Policy Iteration. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{tang_et_al:LIPIcs.CONCUR.2016.22, author = {Tang, Qiyi and van Breugel, Franck}, title = {{Computing Probabilistic Bisimilarity Distances via Policy Iteration}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {22:1--22:15}, 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.2016.22}, URN = {urn:nbn:de:0030-drops-61837}, doi = {10.4230/LIPIcs.CONCUR.2016.22}, annote = {Keywords: labelled Markov chain, simple stochastic game, probabilistic bisimilarity, pseudometric, value function, policy iteration} }

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