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Thesis and Analysis of Probabilistic Models

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Thesis and Analysis of Probabilistic Models Diagnosis, Synthesis and Analysis of Probabilistic Models Tingting Han Graduation committee: prof. dr. ir. L. van Wijngaarden University of Twente, The Netherlands (chairman) prof. dr. ir. J.-P. Katoen RWTH Aachen University / University of Twente, (promotor) Germany / The Netherlands prof. dr. P. R. D’Argenio Universidad Nacional de C´ordoba, Argentina prof. dr. ir. B. R. Haverkort University of Twente, The Netherlands prof. dr. S. Leue University of Konstanz, Germany prof. dr. J. C. van de Pol University of Twente, The Netherlands prof. dr. R. J. Wieringa University of Twente, The Netherlands prof. dr. F. W. Vaandrager Radboud University Nijmegen, The Netherlands IPA Dissertation Series 2009-21. CTIT Ph.D.-Thesis Series No. 09-149, ISSN 1381-3617. ISBN: 978-90-365-2858-0. The research reported in this dissertation has been carried out under the auspices of the Institute for Programming Research and Algorithmics (IPA) and within the context of the Center for Telematics and Information Technology (CTIT). The research funding was provided by the NWO Grant through the project: Verifying Quantitative Properties of Embedded Software (QUPES). Translation of the Dutch abstract: Viet Yen Nguyen. Translation of the German abstract: Martin R. Neuh¨außer. Typeset by LATEX. Cover design: Tingting Han. Picture from www.vladstudio.com. Publisher: W¨ohrmann Printing Service - www.wps.nl. Copyright c 2009 by Tingting Han, Aachen, Germany. DIAGNOSIS, SYNTHESIS AND ANALYSIS OF PROBABILISTIC MODELS PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus prof. dr. H. Brinksma, volgens besluit van het College voor Promoties, in het openbaar te verdedigen op vrijdag 25 september 2009 om 13:15 uur door Tingting Han geboren op 27 december 1980 te Hangzhou, Volksrepubliek China Dit proefschrift is goedgekeurd door de promotor, prof. dr. ir. Joost-Pieter Katoen. DIAGNOSIS, SYNTHESIS AND ANALYSIS OF PROBABILISTIC MODELS Der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westf¨alischen Technischen Hochschule Aachen vorgelegte Dissertation zur Erlangung des akademischen Grades einer Doktorin der Naturwissenschaften vorgelegt von Tingting Han, M. Eng aus Hangzhou, Volksrepublik China Berichter: Prof. Dr. Ir. Joost-Pieter Katoen Prof. Dr. Marta Z. Kwiatkowska Tag der m¨undlichen Pr¨ufung: 16. Oktober 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. Acknowledgements I still remember the summer back in 2005 when I took this PhD position. Four years were just like a snap of fingers, leaving me this booklet and many vivid pieces of snapshots deep and clear in my mind. I was a “risk” four years ago when Joost-Pieter Katoen, my promotor and supervi- sor, decided to offer me this job. After all, to Joost-Pieter at that time, I was nothing more than a two-page CV and a short international call. I do appreciate this trust, which in these years keeps urging me to make progress and “make profits”:). His ex- pertise, insights and far-reaching interests broadened my views and helped me find “shorter (if not the shortest:) paths” at each crossroad. I am grateful for his endur- ing guidance, his great support, understanding and flexibility. I am also thankful for his big effort and contribution going to China in 2008, visiting different universities and institutes, giving both tutorials and more advanced invited talks, attracting more Chinese students and researchers to the field of formal verification. Many results presented in this thesis are a product of joint work. Apart from Joost- Pieter, I am grateful to Berteun Damman, Alexandru Mereacre and Taolue Chen, who shared ideas, had (usually long and fruitful) discussions and transferred their expertise to me. I am also thankful to Christel Baier, David N. Jansen and Jeremy Sproston for their useful remark and insightful discussion on my papers. The peer review and exchange of ideas inside the group, usually by/with Henrik Bohnenkamp, Daniel Klink and Martin Neuh¨außer have provided me with their precious comments and enlightening thoughts. Besides, I would like to thank Prof. Marta Kwiatkowska for inviting me to visit her group in Birmingham. I also enjoyed the visit(s) from Husain Aljazzar, Miguel Andr´es, Lars Grunske, Ralf Wimmer and Lijun Zhang for the interesting talks and discussions. Besides, the regular VOSS and QUASIMODO meetings made me feel like being in a big and happy family. I would like also to thank my reading and defense committee in Twente: Prof. Pedro D’Argenio, Prof. Boudewijn Haverkort, Prof. Stefan Leue, Prof. Jaco van de Pol, Prof. Frits Vaandrager, Prof. Roel Wieringa and Prof. Leen van Wijngaarden as well as the examiners in Aachen: Prof. Gerhard Lakemeyer, Prof. Marta Kwiatkowska, Prof. Stefan Kowalewski and Prof. Wolfgang Thomas. Although I already knew in the beginning that I would move with Joost-Pieter from Twente to Aachen, I had never anticipated what this “double identity” would mean to me. Actually I always feel proud when I can fill in two affiliations. I also feel lucky that I can get to know both top research groups. (Of course this acknowledgement will become twice as long as it would have been.:) i First, I really appreciate the liberty that Twente rendered me during my PhD study. The FMT group is so supportive, helpful and flexible that I do feel in debt to it. I would like to thank Ed Brinksma for introducing me to Joost-Pieter; and Jaco van de Pol for his understanding and support; also many thanks to Axel Belinfante, Rom Langerak, Arend Rensink, Theo Ruys, Mari¨elle Stoelinga, Mark Timmer, Michael Weber and the former members Hichem Boudali and Ivan Zapreev. Special thanks go to Joke Lammerink for everything she has done for me. Bedankt! On the other hand, I witnessed the growth and fast developments of the MOVES group, which offers me a relaxed and productive working atmosphere. As my roommate, Martin Neuh¨außer is always there and ready to help in every respect. I owe you a lot! I am also very lucky to have Haidi Yue, Viet Yen Nguyen, Alexandru Mereacre and Henrik Bohnenkamp around who make my laughters loud and lasting. Martin and Viet Yen are specially thanked for translating the abstract. Also many thanks to Erika Abr´aham,´ Xin Chen, Fabian Emmes, Carsten (Fuhs Kern Otto), Arnd Gehrmann, | | J¨urgen Giesl, Jonathan Heinen, Nils Jansen, Daniel Klink, Ulrich Loup, Thomas Noll, Elke Ohlenforst and Stefan Rieger and the former members Ren´eThiemann, Peter Schneider-Kamp and Stephan Swiderski. You made my time in the group more colorful! Dank sehr! My life in Aachen would have not been as pleasant if there were no cheerful com- pany and encouragements from friends in and outside Aachen. Just to name a few: Sebastian Bitzen, Andreas Brand, Yiwei Cao, Qingxia Gong, Olivier Guillard, Jianwei Han, Yingchun He, Xuan Li, Fang Liu, Leiqin Lu, Mikhail Pletyukhov, Kuangyu Shi, Leyi Shi, Hailin Wang, Wei Wang, Yan Wang, Yutian Wang, Shaochen Wei, Wei Wu, Ping Yu, Haidi Yue, Yuqi Zhang, Ziyun Zhang and many other friends. I am forever indebted to my family, especially my parents in China. Without their ÛÛåå unconditional love, encouragement and support, I won’t reach this far. a ! Also many thanks go to my little grandpa and my cousins Bin, Jian and Xiaoyun, who share experience, give advices and send me many pictures of my lovely little nephew ¤k|±³±ó[<ڶРand nieces. a ! Finally, to Taolue, well, due to the small space here, we can do this offline...:) ÜÜ Tingting Han (ô ) Amsterdam, May 2, 2009 ii Abstract This dissertation considers three important aspects of model checking Markov mod- els: diagnosis — generating counterexamples, synthesis — providing valid parameter values and analysis — verifying linear real-time properties. The three aspects are rela- tively independent while all contribute to developing new theory and algorithms in the research field of probabilistic model checking. We start by introducing a formal definition of counterexamples in the setting of probabilistic model checking. We transform the problem of finding informative coun- terexamples to shortest path problems. A framework is explored and provided for generating such counterexamples. We then investigate a more compact representation of counterexamples by regular expressions. Heuristic based algorithms are applied to obtain short regular expression counterexamples. In the end of this part, we extend the definition and counterexample generation algorithms to various combinations of probabilistic models and logics. We move on to the problem of synthesizing values for parametric continuous-time Markov chains (pCTMCs) wrt. time-bounded reachability specifications. The rates in the pCTMCs are expressed by polynomials over reals with parameters and the main question is to find all the parameter values (forming a synthesis region) with which the specification is satisfied. We first present a symbolic approach where the intersection points are computed by solving polynomial equations and then connected to approximate the synthesis region. An alternative non-symbolic approach based on interval arithmetic is investigated, where pCTMCs are instantiated. The error bound, time complexity as well as some experimental results have been provided, followed by a detailed comparison of the two approaches. In the last part, we focus on verifying CTMCs against linear real-time properties specified by deterministic timed automata (DTAs). The model checking problem aims at computing the probability of the set of paths in CTMC that can be accepted C by DTA , denoted PathsC( ). We consider DTAs with reachability (finite, DTA ) A A ♦ and Muller (infinite, DTAω) acceptance conditions, respectively. It is shown that PathsC( ) is measurable and computing its probability for DTA can be reduced to A ♦ computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of PDEs.
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