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Model Checking Stochastic Hybrid Systems

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Model Checking Stochastic Hybrid Systems Model Checking Stochastic Hybrid Systems Dissertation Zur Erlangung des Grades des Doktors der Ingenieurwissenschaften der Naturwissenschaftlich-Technischen Fakultäten der Universität des Saarlandes vorgelegt von Ernst Moritz Hahn Saarbrücken Oktober 2012 Tag des Kolloquiums: 21.12.2012 Dekan: Prof. Dr. Mark Groves Prüfungsausschuss: Vorsitzende: Prof. Dr. Verena Wolf Gutachter: Prof. Dr.-Ing. Holger Hermanns Prof. Dr. Martin Fränzle Prof. Dr. Marta Kwiatkowska Akademischer Mitarbeiter: Dr. Andrea Turrini Abstract The interplay of random phenomena with discrete-continuous dynamics deserves in- creased attention in many systems of growing importance. Their verification needs to consider both stochastic behaviour and hybrid dynamics. In the verification of classical hybrid systems, one is often interested in deciding whether unsafe system states can be reached. In the stochastic setting, we ask instead whether the probability of reaching particular states is bounded by a given threshold. In this thesis, we consider stochastic hybrid systems and develop a general abstraction framework for deciding such problems. This gives rise to the first mechanisable technique that can, in practice, formally verify safety properties of systems which feature all the relevant aspects of nondeterminism, general continuous-time dynamics, and probabilistic behaviour. Being based on tools for classical hybrid systems, future improvements in the effectiveness of such tools directly carry over to improvements in the effectiveness of our technique. We extend the method in several directions. Firstly, we discuss how we can handle continuous probability distributions. We then consider systems which we are in partial control of. Next, we consider systems in which probabilities are parametric, to analyse entire system families at once. Afterwards, we consider systems equipped with rewards, modelling costs or bonuses. Finally, we consider all orthogonal combinations of the extensions to the core model. Zusammenfassung In vielen Systemen wachsender Bedeutung tritt zufallsabhängiges Verhalten gleichzeitig mit diskret-kontinuierlicher Dynamik auf. Um solche Systeme zu verifizieren, müssen sowohl ihr stochastisches Verhalten als auch ihre hybride Dynamik betrachtet werden. In der Analyse klassischer hybrider Systeme ist eine wichtige Frage, ob unsichere Zustände erreicht werden können. Im stochastischen Fall fragen wir stattdessen nach garantierten Wahrscheinlichkeitsschranken. In dieser Arbeit betrachten wir stochastische hybride Sys- teme und entwickeln eine allgemeine Abstraktionsmethode um Probleme dieser Art zu entscheiden. Dies ermöglicht die erste automatische und praktisch anwendbare Methode, die Sicherheitseigenschaften von Systeme beweisen kann, in denen Nichtdeterminismus, komplexe Dynamik und probabilistisches Verhalten gleichzeitig auftreten. Da die Me- thode auf Analysetechniken für nichtstochastische hybride Systeme beruht, profitieren wir sofort von zukünftigen Verbesserungen dieser Verfahren. Wir erweitern diese Grundmethode in mehrere Richtungen: Zunächst ergänzen wir das Modell um kontinuierliche Wahrscheinlichkeitsverteilungen. Dann betrachten wir partiell kontrollierbare Systeme. Als nächstes untersuchen wir parametrische Systeme, um eine Klasse ähnlicher Modelle gleichzeitig behandeln. Anschließend betrachten wir Eigenschaften, die auf der Abwägung von Kosten und Nutzen beruhen. Schließlich zeigen wir, wie diese Erweiterungen orthogonal kombiniert werden können. Acknowledgements I would like to begin by thanking my advisor, Holger Hermanns. He shaped my un- derstanding of research and his stewardship has guided me through this dissertation. Furthermore, I would like to thank my coauthors and my colleagues at the Dependable Systems Group for providing a friendly and supportive environment. Especially, I would like to thank Lijun Zhang for his support, in particular at the initial phase of my thesis. I would like to thank Luis María Ferrer Fioriti for the time he devoted to reading the thesis and his valuable feedback. This thesis was supported by the Saarbrücken Graduate School of Computer Science, the Graduiertenkolleg “Leistungsgarantien für Rechnersysteme”, the exchange program DAAD-MINCyT: Programm zum Projektbezogenen Personenaustauschs PROALAR, the German Research Foundation (DFG) as part of the transregional research center SFB/TR 14 AVACS, the European Community’s Seventh Framework Programme under grant agreement no 214755 QUASIMODO, the European Union Seventh Framework Programme under grant agreement number 295261 as part of the MEALS project, and the NWO-DFG bilateral project ROCKS. Contents Preface 1 Abstract....................................... 3 Zusammenfassung.................................. 4 Acknowledgements ................................. 5 1 Introduction 11 1.1 Contribution.................................. 12 1.2 LayoutandOriginoftheThesis . 14 2 Classical Hybrid Automata 17 2.1 SemanticalModels .............................. 17 2.1.1 Conventions.............................. 17 2.1.2 LabelledTransitionSystems . 18 2.2 HybridAutomata............................... 21 2.3 Abstractions.................................. 29 2.4 AlgorithmicConsideration . .. 32 2.5 CaseStudy .................................. 32 2.6 RelatedWork ................................. 33 2.7 Conclusion................................... 35 3 Probabilistic Hybrid Automata 37 3.1 StochasticModels............................... 38 3.1.1 StochasticRecapitulation . 38 3.1.2 ProbabilisticAutomata. 38 3.2 ProbabilisticHybridAutomata . .. 46 3.3 Abstractions.................................. 51 3.3.1 ComputingAbstractions . 56 3.4 AlgorithmicConsideration . .. 58 3.5 CaseStudies.................................. 62 3.5.1 Thermostat .............................. 63 3.5.2 BouncingBall............................. 63 3.5.3 WaterLevelControl . .. .. 65 3.5.4 AutonomousLawn-mower . 67 3.6 RelatedWork ................................. 68 3.7 Conclusion................................... 70 4 Continuous Distributions 71 4.1 SemanticModel................................ 71 4.1.1 StochasticRecapitulation . 71 7 Contents 4.1.2 Nondeterministic Labelled Markov Processes . ..... 73 4.2 StochasticHybridAutomata. 80 4.3 Abstractions.................................. 87 4.3.1 ComputingAbstractions . 92 4.4 CaseStudies.................................. 93 4.4.1 Thermostat .............................. 93 4.4.2 WaterLevelControl . .. .. 94 4.4.3 Moving-blockTrainControl . 96 4.5 RelatedWork ................................. 99 4.6 Conclusion................................... 100 5 Partial Control 103 5.1 Game Interpretation of Probabilistic Automata . ....... 103 5.2 Controlled Probabilistic Hybrid Automata . ...... 105 5.3 Abstractions.................................. 107 5.3.1 ComputingAbstractions . 115 5.4 AlgorithmicConsideration . 117 5.5 SynthesisingControllers . 117 5.6 CaseStudy .................................. 120 5.7 RelatedWork ................................. 121 5.8 Conclusion................................... 121 6 Parameters 123 6.1 Parametric Probabilistic Models . 123 6.2 ParametricHybridAutomata . 125 6.3 Abstractions.................................. 127 6.3.1 ComputingAbstractions . 128 6.4 AlgorithmicConsideration . 128 6.5 CaseStudies.................................. 136 6.5.1 Thermostat .............................. 136 6.5.2 WaterLevelControl . 136 6.6 RelatedWork ................................. 137 6.7 Conclusion................................... 138 7 Rewards 141 7.1 RewardsforProbabilisticAutomata. 141 7.1.1 StochasticRecap . .. .. 141 7.1.2 RewardStructuresandProperties . 142 7.2 Rewards for Probabilistic Hybrid Automata . ..... 151 7.3 Abstraction .................................. 156 7.3.1 ComputingAbstractions . 158 7.4 AlgorithmicConsideration . 159 7.5 CaseStudies.................................. 160 7.5.1 Thermostat .............................. 160 8 Contents 7.5.2 WaterLevelControl . 162 7.6 RelatedWork ................................. 163 7.7 Conclusion................................... 163 8 Orthogonal Combinations 165 8.1 Continuous Distributions and Partial Control . ....... 165 8.2 Continuous Distributions and Parametric Models . ....... 167 8.3 Continuous Distributions and Rewards . 168 8.4 PartialControlandParametricModels . 170 8.5 PartialControlandRewards. 171 8.6 ParametricModelsandRewards. 173 8.7 Conclusion................................... 175 9 Summary and Future Work 177 9.1 SummaryofContributions . 177 9.2 Conclusions .................................. 179 9.3 FutureWorks ................................. 179 Bibliography 181 9 1 Introduction Hybrid systems constitute a general and widely applicable class of dynamical systems with both discrete and continuous components. Classical hybrid system formalisms [Alu+95; Pre+98; Cla+03; RS07] capture many characteristics of real systems. However, in many modern application areas, also stochastic dynamics occurs. This is especially true for wireless sensing and control applications, where message loss probabilities and other random effects (node placement, node failure, battery drain) turn the overall control problem into a problem that can only be managed with a certain, hopefully sufficiently high, probability. Due to the influence of these random effects, the success (for example keeping the hybrid system in a safe region) can only be guaranteed with a probability less than one. Since these phenomena are important aspects when aiming at faithful models for
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