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Qualitative Analysis of Synchronizing Probabilistic Systems Mahsa Shirmohammadi Qualitative analysis of synchronizing probabilistic systems Mahsa Shirmohammadi To cite this version: Mahsa Shirmohammadi. Qualitative analysis of synchronizing probabilistic systems. Numerical Anal- ysis [cs.NA]. École normale supérieure de Cachan - ENS Cachan; Université libre de Bruxelles (1970-..), 2014. English. NNT : 2014DENS0054. tel-01153942 HAL Id: tel-01153942 https://tel.archives-ouvertes.fr/tel-01153942 Submitted on 20 May 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ LIBRE DE BRUXELLES ÉCOLE NORMALE SUPÉRIEURE Service des méthodes formelles DE CACHAN et vérification Laboratoire spécification et vérification Qualitative Analysis of Probabilistic Synchronizing Systems Mahsa Shirmohammadi Dissertation submitted for the joint degree of Doctor of Philosophy in computer science from Université Libre de Bruxelles, Belgium and Ecole Normale Supérieure de Cachan, France Committee in charge: Laurent Doyen: Supervisor Thierry Massart: Supervisor Stefan Kiefer: Reviewer Jeremy Sproston: Reviewer Joost-Pieter Katoen: Examiner Jean François Raskin: Examiner Béatrice Bérard: Examiner Emmanuel Filiot: Examiner Defense date: 10th December 2014 To Maryam, my mother. Acknowledgment I gratefully acknowledge all who supported me to accomplish my wish. Being so lucky to have supports from several countries and in several languages; I try to mention all, however I am surely bound to forget a few. Laurent, I admire the way you turn science into art. Your seriousness, patience, insights, enthusiasm and not to forget your expectations motivated me to work hard, to be thirsty to learn how to carry out deep research, write clear proofs and give good talks. I am so grateful that you made me rewrite proofs several times, for some more than there are fingers in a hand, to reformulate and simplify my ideas and to turn a problem up and down, and look from one angle to another. Wishing that I hadfilled my backpack with more of your knowledge and advices, I am at the same time grateful for the freedom I had while writing my thesis; I found my academic personality those days, thanks Laurent. Thierry, we have heard “doubt your beliefs and believe in your doubts” more than once from Laurent; however the one who taught it to me is you. I will always remain amazed by your capability offinding mistakes in proofs and counter examples for conjectures. You taught me to question and challenge myself, to not be easily convinced, thanks for all of those, and much more for the social support you never forgot to offer me. Thanks for welcoming me atfirst in Brussels, for helping me integrate in my new life, for simply being ’there’ when the sensitivity of a girl overlapped with her professionalism. Thanks Thierry. I am so pleased that Stefan Kiefer, Jeremy Sproston, Jean François Raskin, Joost-Pieter Katoen, Béatrice Bérard and Emmanuel Filiot accepted to be the other jury members of my thesis. A further thank for all asked questions during my private defense, some of which spark new directions of research. I am grateful to the Belgian national fund for scientific research (F.N.R.S.) and Inter- national relation office of ENS de Cachan which provided me with two scholarships during my study: one allowed me to fully concentrate on my thesis, and the other one facilitated my dual life between France and Belgium over three years. I should not forget any member of LSV (ENS de Cachan) and the Verification group at ULB, with a special mention to Benedikt, Frédéric, Gilles, Dietmar, Luc, Cristina, Thida, Thomas and Julien who smiled at me and made refreshing small talks every time I met them during coffee breaks. I also appreciate all the helps I received from the secretaries of both labs: Pascaline, Maryka, Véronique and Virgine several times took care of administrative papers instead of the lost-in-French Mahsa. I wish to say thanks to Nicolas Markey (a big thanks), Stéphanie Delaune and Alain Finkel for constructive advice; to Kim Larsen and Line for welcoming me in Aalborg university while collaborating; to Alexander for teaching me how to teach; to Sylvain for saving my introduction; to Nathann for knocking the door of my office in a gloomy winter day; to all adorable girls who surrounded both me and my surviving feminine trait: Sinem jan janam, dear Aina, delicate Marie, small mommy Sandie, strong Raffaella, happiest ever Barbara, lively Aiswarya, organized Laure, fun Juliette and the other fun Juliette, kind Eli, i sportive Line, creative Carla, hard-working Atefeh, backgammon-champion Öykü, eastern- soul Xiajou, sweet Agathe, the only-musician-friend Nona and far-away Eleni (girls, I lived moments with you, your weddings, your 25th, 30th birthdays, your pregnancy worried days, your graduations, your love or hate days, simply normal days, thank you, thank you for all); to Gabriel and Tristan to be myfirst and so good friends in Europe; to Guillermo, Romain and the other Romain for being such unique charming office-mates, and for enjoying summer BBQs where I have to mention Ocan and Mathieu, the delicious- sauce maker; to Chris, Hernan, Vince and César for never rejecting to try what I cook and even complimenting it, and more than that for the friendship and respect between us; to Ali and Hani my best ever friends that keep our relation fresh and deep from such a long distance; to my soul-mates Mina, Mona and Somayeh for not letting me disconnect from the other world of mine by long and late phone conversations; to Mani for lifting spirit words; and to dear Hossein for encouraging me to pursue my post-graduate study and be more independent and more honest. Vincent and Christoph, I am indebted to you guys. Ask me why, and I need to think minutes before taking out one of the hundreds cards from the deck of friendly supports, scientific advices, stressful critical times, relaxed laughs and smiles, sunny and rainy but all memorable days. Vince, you have been so generous to me, so more than what others can guess, so perfectly careful with the fragile glass of my soul, thanks. Chris, you make me look into myself and hunt clues of peace and braveness whose existences were lost over years, thanks. Special thanks to Vince for going through all my thesis, and to Chris for commenting my introduction and my post-doctorate application to Oxford. Thanks Vince, thanks Chris. Madaram Maryam, my mother Maryam, one tip of afinger from your kindness and devotion was sufficient to make me who I am, though you lived your life in each of ours, your children. My eyes have always felt shy to look up at the height of the shadow of your support and to say thanks; I can only write it down as down as my short mount of achievements. Thanks Maryam, Madaram. Mahsa so thankful, after herfirst Thanksgiving dinner at Tony’s place Paris, FMSH library December 2014. ii Abstract Markov decision processes (MDPs) arefinite-state probabilistic systems with both strategic and random choices, hence well-established to model the interactions between a controller and its randomly responding environment. An MDP can be mathematically 1 viewed as a1 2 -player stochastic game played in rounds when the controller chooses an ac- tion, and the environment chooses a successor according to afixed probability distribution. There are two incomparable views on the behavior of an MDP, when the strategic choices arefixed. In the traditional view, an MDP is a generator of sequence of states, called the state-outcome; the winning condition of the player is thus expressed as a set of desired sequences of states that are visited during the game, e.g. Borel condition such as reachability. The computational complexity of related decision problems and memory requirement of winning strategies for the state-outcome conditions are well-studied. Recently, MDPs have been viewed as generators of sequences of probability distributions over states, called the distribution-outcome. We introduce synchronizing conditions defined on distribution-outcomes, which intuitively requires that the probability mass accumulates in some (group of) state(s), possibly in limit. A probability distribution isp-synchronizing if the probability mass is at leastp in some state, and a sequence of probability distributions is always, eventually, weakly, or stronglyp-synchronizing if respectively all, some, infinitely many, or all butfinitely many distributions in the sequence arep-synchronizing. For each synchronizing mode, an MDP can be(i) sure winning if there is a strategy that produces a1-synchronizing sequence;(ii) almost-sure winning if there is a strategy that produces a sequence that is, for allϵ>0, a (1-ϵ)-synchronizing sequence;(iii) limit-sure winning if for allϵ>0, there is a strategy that produces a (1-ϵ)-synchronizing sequence. We consider the problem of deciding whether an MDP is winning, for each synchronizing and winning mode: we establish matching upper and lower complexity bounds of the problems, as well as the memory requirement for optimal winning strategies. As a further contribution, we study synchronization in probabilistic automata (PAs), that are kind of MDPs where controllers are restricted to use only word-strategies; i.e. no ability to observe the history of the system execution, but the number of choices made so far. The synchronizing languages of a PA is then the set of all synchronizing word- strategies: we establish the computational complexity of the emptiness problems for all synchronizing languages in all winning modes.
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