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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: The best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006).
In this paper, we consider problems that lie between the initialized and uninitialized variant. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighborhood is given as part of the input, then robust Skolem and robust positivity are Diophantine-hard, i.e., solving either would entail major breakthrough in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine-hardness result on a variant of the Skolem problem, to the best of our knowledge. On the other hand, if one asks whether such a neighborhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations.

S. Akshay, Hugo Bazille, Blaise Genest, and Mihir Vahanwala. On Robustness for the Skolem and Positivity Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 5:1-5:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akshay_et_al:LIPIcs.STACS.2022.5, author = {Akshay, S. and Bazille, Hugo and Genest, Blaise and Vahanwala, Mihir}, title = {{On Robustness for the Skolem and Positivity Problems}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.5}, URN = {urn:nbn:de:0030-drops-158156}, doi = {10.4230/LIPIcs.STACS.2022.5}, annote = {Keywords: Skolem problem, verification, dynamical systems, robustness} }

Document

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

This paper addresses reliability of timed systems in the setting of resilience, that considers the behaviors of a system when unspecified timing errors such as missed deadlines occur. Given a fault model that allows transitions to fire later than allowed by their guard, a system is universally resilient (or self-resilient) if after a fault, it always returns to a timed behavior of the non-faulty system. It is existentially resilient if after a fault, there exists a way to return to a timed behavior of the non-faulty system, that is, if there exists a controller which can guide the system back to a normal behavior. We show that universal resilience of timed automata is undecidable, while existential resilience is decidable, in EXPSPACE. To obtain better complexity bounds and decidability of universal resilience, we consider untimed resilience, as well as subclasses of timed automata.

S. Akshay, Blaise Genest, Loïc Hélouët, S. Krishna, and Sparsa Roychowdhury. Resilience of Timed Systems. 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. 33:1-33:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{akshay_et_al:LIPIcs.FSTTCS.2021.33, author = {Akshay, S. and Genest, Blaise and H\'{e}lou\"{e}t, Lo\"{i}c and Krishna, S. and Roychowdhury, Sparsa}, title = {{Resilience of Timed Systems}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {33:1--33:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.33}, URN = {urn:nbn:de:0030-drops-155442}, doi = {10.4230/LIPIcs.FSTTCS.2021.33}, annote = {Keywords: Timed automata, Fault tolerance, Integer-resets, Resilience} }

Document

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

In [Dana Angluin et al., 2006], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula φ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with 2^?(poly(|φ|)) states that computes φ. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree.
In this paper, we prove that every formula φ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with ?(poly(|φ|)) states. Our proof is based on several new constructions, which may be of independent interest. Given a formula φ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with ?(|φ|³) leaders) that computes φ; this completes the work initiated in [Michael Blondin et al., 2018], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes φ. Our last construction gets rid of this leader for small inputs.

Michael Blondin, Javier Esparza, Blaise Genest, Martin Helfrich, and Stefan Jaax. Succinct Population Protocols for Presburger Arithmetic. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 40:1-40:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blondin_et_al:LIPIcs.STACS.2020.40, author = {Blondin, Michael and Esparza, Javier and Genest, Blaise and Helfrich, Martin and Jaax, Stefan}, title = {{Succinct Population Protocols for Presburger Arithmetic}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.40}, URN = {urn:nbn:de:0030-drops-119018}, doi = {10.4230/LIPIcs.STACS.2020.40}, annote = {Keywords: Population protocols, Presburger arithmetic, state complexity} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

An important task in AI is one of classifying an observation as belonging to one class among several (e.g. image classification). We revisit this problem in a verification context: given k partially observable systems modeled as Hidden Markov Models (also called labeled Markov chains), and an execution of one of them, can we eventually classify which system performed this execution, just by looking at its observations? Interestingly, this problem generalizes several problems in verification and control, such as fault diagnosis and opacity. Also, classification has strong connections with different notions of distances between stochastic models.
In this paper, we study a general and practical notion of classifiers, namely limit-sure classifiers, which allow misclassification, i.e. errors in classification, as long as the probability of misclassification tends to 0 as the length of the observation grows. To study the complexity of several notions of classification, we develop techniques based on a simple but powerful notion of stationary distributions for HMMs. We prove that one cannot classify among HMMs iff there is a finite separating word from their stationary distributions. This provides a direct proof that classifiability can be checked in PTIME, as an alternative to existing proofs using separating events (i.e. sets of infinite separating words) for the total variation distance. Our approach also allows us to introduce and tackle new notions of classifiability which are applicable in a security context.

S. Akshay, Hugo Bazille, Eric Fabre, and Blaise Genest. Classification Among Hidden Markov Models. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 29:1-29:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akshay_et_al:LIPIcs.FSTTCS.2019.29, author = {Akshay, S. and Bazille, Hugo and Fabre, Eric and Genest, Blaise}, title = {{Classification Among Hidden Markov Models}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.29}, URN = {urn:nbn:de:0030-drops-115917}, doi = {10.4230/LIPIcs.FSTTCS.2019.29}, annote = {Keywords: verification: probabilistic systems, partially observable systems} }

Document

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

We introduce a new setting where a population of agents, each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We study a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player chooses actions, and the second player resolves non-determinism for each agent. The game with m agents is called the m-population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can playerone control the m-population game for all m in N whatever playertwo does?
In this paper, we prove that the population control problem is decidable, and it is a EXPTIME-complete problem. As far as we know, this is one of the first results on parameterized control. Our algorithm, not based on cut-off techniques, produces winning strategies which are symbolic, that i they do not need to count precisely how the population is spread between states. We also show that if the is no winning strategy, then there is a population size cutoff such that playerone wins the m-population game if and only if m< \cutoff. Surprisingly, \cutoff can be doubly exponential in the number of states of the NFA, with tight upper and lower bounds.

Nathalie Bertrand, Miheer Dewaskar, Blaise Genest, and Hugo Gimbert. Controlling a Population. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 12:1-12:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bertrand_et_al:LIPIcs.CONCUR.2017.12, author = {Bertrand, Nathalie and Dewaskar, Miheer and Genest, Blaise and Gimbert, Hugo}, title = {{Controlling a Population}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {12:1--12:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.12}, URN = {urn:nbn:de:0030-drops-78000}, doi = {10.4230/LIPIcs.CONCUR.2017.12}, annote = {Keywords: Model-checking, control, parametric systems} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

The quantitative verification of Probabilistic Automata (PA) is undecidable in general. Unary PA are a simpler model where the choice of action is fixed. Still, the quantitative verification problem is open and known to be as hard as Skolem's problem, a problem on linear recurrence sequences, whose decidability is open for at least 40 years. In this paper, we approach this problem by studying the languages generated by unary PAs (as defined below), whose regularity would entail the decidability of quantitative verification.
Given an initial distribution, we represent the trajectory of a unary PA over time as an infinite word over a finite alphabet, where the n-th letter represents a probability range after n steps. We extend this to a language of trajectories (a set of words), one trajectory for each initial distribution from a (possibly infinite) set. We show that if the eigenvalues of the transition matrix associated with the unary PA are all distinct positive real numbers, then the language is effectively regular. Further, we show that this result is at the boundary of regularity, as non-regular languages can be generated when the restrictions are even slightly relaxed. The regular representation of the language allows us to reason about more general properties, e.g., robustness of a regular property in a neighbourhood around a given distribution.

S. Akshay, Blaise Genest, Bruno Karelovic, and Nikhil Vyas. On Regularity of Unary Probabilistic Automata. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akshay_et_al:LIPIcs.STACS.2016.8, author = {Akshay, S. and Genest, Blaise and Karelovic, Bruno and Vyas, Nikhil}, title = {{On Regularity of Unary Probabilistic Automata}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.8}, URN = {urn:nbn:de:0030-drops-57093}, doi = {10.4230/LIPIcs.STACS.2016.8}, annote = {Keywords: Probabilistic automata, Symbolic dynamics, Markov chains, Skolem problem, Regularity} }

Document

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

Zielonka's theorem, established 25 years ago, states that any regular
language closed under commutation is the language of an asynchronous
automaton (a tuple of automata, one per process, exchanging information when performing common actions). Since then, constructing asynchronous automata has been simplified and improved ([Cori/Métivier/Zielonka,1993],[Klarlund/Mukund/Sohoni,1994], [Diekert/Rozenberg,1995], [Genest/Muscholl,2006], [Genest/Gimbert/Muscholl/Walukiewicz,2010], [Baudru/Morin, 2006], [Baudru,2009], [Pighizzini,1993], [Stefanescu/Esparza/Muscholl,2003]).
We first survey these constructions and conclude that the synthesized
systems are not realistic in the following sense: existing constructions are either plagued by deadends, non deterministic guesses, or the acceptance condition or choice of actions are not distributed. We tackle this problem by giving (effectively testable) necessary and sufficient conditions which ensure that deadends can be avoided, acceptance condition and choices of action can be distributed, and determinism can be maintained. Finally, we implement our constructions, giving promising results when compared with the few other existing prototypes synthesizing asynchronous automata.

S. Akshay, Ionut Dinca, Blaise Genest, and Alin Stefanescu. Implementing Realistic Asynchronous Automata. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 213-224, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{akshay_et_al:LIPIcs.FSTTCS.2013.213, author = {Akshay, S. and Dinca, Ionut and Genest, Blaise and Stefanescu, Alin}, title = {{Implementing Realistic Asynchronous Automata}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {213--224}, 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.2013.213}, URN = {urn:nbn:de:0030-drops-43742}, doi = {10.4230/LIPIcs.FSTTCS.2013.213}, annote = {Keywords: Asynchronous automata, Zielonka construction, Implementability} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

For security and efficiency reasons, most systems do not give the
users a full access to their information. One key specification
formalism for these systems are the so called Partially Observable Markov Decision Processes (POMDP for short), which have been extensively studied in several research communities, among which AI and model-checking. In this paper we tackle the problem of the minimal information a user needs at runtime to achieve a simple goal, modeled as reaching an objective with probability one. More precisely, to achieve her goal, the user can at each step either choose to use the partial information, or pay a fixed cost and receive the full information. The natural question is then to minimize the cost the user needs to fulfill her objective. This optimization question gives rise to two different problems, whether we consider to minimize the worst case cost, or the average cost. On
the one hand, concerning the worst case cost, we show that efficient
techniques from the model checking community can be adapted to compute the optimal worst case cost and give optimal strategies for the users. On the other hand, we show that the optimal average price (a question typically considered in the AI community) cannot be computed in general, nor can it be approximated in polynomial time even up to a large approximation factor.

Nathalie Bertrand and Blaise Genest. Minimal Disclosure in Partially Observable Markov Decision Processes. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 411-422, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bertrand_et_al:LIPIcs.FSTTCS.2011.411, author = {Bertrand, Nathalie and Genest, Blaise}, title = {{Minimal Disclosure in Partially Observable Markov Decision Processes}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {411--422}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.411}, URN = {urn:nbn:de:0030-drops-33286}, doi = {10.4230/LIPIcs.FSTTCS.2011.411}, annote = {Keywords: Partially Observable Markov Decision Processes, Stochastic Games, Model-Checking, Worst-Case/Average-Case Analysis} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

This paper considers a tree-rewriting framework for modeling documents evolving through service calls. We focus on the automatic verification of properties of documents that may contain data from an infinite domain. We establish the boundaries of decidability: while verifying documents with recursive calls is undecidable, we obtain decidability as soon as either documents are in the positive-bounded fragment (while calls are unrestricted), or when there is a bound on the number of service calls (bounded model-checking of unrestricted documents). In the latter case, the complexity is NexpTime-complete. Our data tree-rewriting framework resembles Guarded Active XML, a platform handling XML repositories that evolve through web services. The model here captures the basic features of Guarded Active XML and extends it by node renaming and subtree deletion.

Blaise Genest, Anca Muscholl, and Zhilin Wu. Verifying Recursive Active Documents with Positive Data Tree Rewriting. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 469-480, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{genest_et_al:LIPIcs.FSTTCS.2010.469, author = {Genest, Blaise and Muscholl, Anca and Wu, Zhilin}, title = {{Verifying Recursive Active Documents with Positive Data Tree Rewriting}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {469--480}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.469}, URN = {urn:nbn:de:0030-drops-28877}, doi = {10.4230/LIPIcs.FSTTCS.2010.469}, annote = {Keywords: Active documents, Guarded Active XML, verification, data trees, tree rewriting, well-structured systems.} }

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