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Contribution to the Verification of Timed Automata: Determinization, Quantitative Verification and Reachability in Networks of Automata Amélie Stainer

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Contribution to the Verification of Timed Automata: Determinization, Quantitative Verification and Reachability in Networks of Automata Amélie Stainer Contribution to the Verification of Timed Automata: Determinization, Quantitative Verification and Reachability in Networks of Automata Amélie Stainer To cite this version: Amélie Stainer. Contribution to the Verification of Timed Automata: Determinization, Quantita- tive Verification and Reachability in Networks of Automata. Computation and Language [cs.CL]. Université Rennes 1, 2013. English. tel-00926316 HAL Id: tel-00926316 https://tel.archives-ouvertes.fr/tel-00926316 Submitted on 16 Jan 2014 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. ANNÉE 2013 THÈSE / UNIVERSITÉ DE RENNES 1 sous le sceau de l’Université Européenne de Bretagne pour le grade de DOCTEUR DE L’UNIVERSITÉ DE RENNES 1 Mention : Informatique Ecole doctorale MATISSE présentée par Amélie Stainer Préparée à l’unité de recherche n°6074 IRISA Institut de recherche en informatique et systèmes aléatoires ISTIC Contribution à la Thèse soutenue à Rennes le 25 novembre 2013 vérification des devant le jury composé de : automates temporisés : Paul Gastin Professeur à l’ENS Cachan / rapporteur déterminisation, Joël Ouaknine Professeur à l’Université d’Oxford / rapporteur vérification quantitative Patricia Bouyer-Decître Directrice de recherche CNRS, LSV, ENS de Cachan / et accessibilité dans les examinatrice Didier Lime réseaux d’automates Maître de conférence à Centrale Nantes / examinateur Sophie Pinchinat Professeure à l’Université de Rennes 1 / examinatrice Thierry Jéron Directeur de recherche à INRIA Rennes-Bretagne Atlantique / directeur de thèse Nathalie Bertrand Chargée de recherche à INRIA Rennes-Bretagne Atlantique / co-encadrante de thèse 1 2 Contribution to the Verification of Timed Automata: Determinization, Quantitative Verification and Reachability in Networks of Automata Amélie Stainer University of Rennes 1 Supervised by: Nathalie Bertrand and Thierry Jéron INRIA Rennes - Bretagne atlantique Rennes - Septembre 2013 2 Contents I Contribution à la vérification des automates temporisés7 0.1 Introduction à la vérification des automates temporisés................9 0.2 Déterminisation des automates temporisés et application au test............ 12 0.3 Fréquences dans les automates temporisés....................... 13 0.4 Accessibilité dans les automates temporisés communicants.............. 15 0.5 Conclusion....................................... 16 II Introduction 17 1 Introduction 19 2 Technical preliminaries 27 III Determinization of Timed Automata 33 3 A Game Approach to Determinize Timed Automata 39 3.1 The game approach................................... 40 3.1.1 Game definition................................. 40 3.1.2 Example.................................... 43 3.1.3 Properties of the strategies........................... 45 3.2 Comparison both existing methods........................... 52 3.2.1 Comparison with [KT09]............................ 52 3.2.2 Comparison with [BBBB09].......................... 54 3.3 Extension to "-transitions and invariants........................ 57 3.3.1 "-transitions................................... 57 3.3.2 Invariants.................................... 59 3.3.3 Properties of the strategies in the extended game............... 61 3.3.4 Comparison with [KT09]............................ 64 3.4 Beyond over-approximation.............................. 65 3.4.1 Under-approximation.............................. 65 3.4.2 Combining over- and under-approximation.................. 65 3.5 Implementation of a prototype tool........................... 71 3.5.1 Using zones instead of regions......................... 71 3.5.2 Implementation of the prototype........................ 71 3.5.3 Execution of the program............................ 73 3 CONTENTS 4 Application of the Game Approach to Off-line Test Selection 77 4.1 A model of open timed automata with inputs / outputs................. 79 4.1.1 Timed automata with inputs/outputs...................... 79 4.1.2 The semantics of OTAIOs........................... 80 4.1.3 Properties and operations............................ 82 4.2 Conformance testing theory............................... 85 4.2.1 The tioco conformance theory......................... 85 4.2.2 Refinement preserving tioco .......................... 88 4.3 Off-line test case generation.............................. 90 4.3.1 Test purposes.................................. 90 4.3.2 Principle of test generation........................... 91 4.3.3 Test suite properties.............................. 97 4.4 Discussion and related work.............................. 100 IV Frequencies in Timed Automata 105 5 Preliminaries 113 5.1 Frequencies in timed automata............................. 114 5.2 Frequency-based semantics............................... 116 5.2.1 Frequencies and timed automata........................ 116 5.2.2 A brief comparison with the usual semantics................. 117 5.2.3 A particular case of double-priced timed automata.............. 118 5.3 The corner-point abstraction.............................. 119 5.3.1 Definition and examples............................ 119 5.3.2 Ratios in the corner-point abstraction..................... 122 5.3.3 Set of ratios in the corner-point abstraction.................. 123 5.4 Forgetfulness...................................... 126 5.4.1 Forgetfulness and aperiodicity......................... 126 5.4.2 Comparison with the forgetfulness of [BA11]................. 129 6 Frequencies in One-Clock Timed Automata 131 6.1 From A to Acp ..................................... 131 6.1.1 Contraction and dilatation........................... 132 6.1.2 Proposition 6.1 does not extend to timed automata with two clocks...... 136 6.2 From Acp to A ..................................... 137 6.2.1 Reward-diverging case............................. 137 6.2.2 Proposition 6.2 extends neither to timed automata with two clocks, nor to Zeno runs.................................... 138 6.2.3 Proposition 6.2 does not extend to reward-converging runs.......... 139 6.2.4 Reward-converging case............................ 139 6.3 Set of frequencies in A ................................. 140 6.3.1 Set of frequencies of non-Zeno runs in A ................... 140 6.3.2 Realizability of bounds by Zeno runs in A ................... 141 6.3.3 Proof of Theorem 6.1.............................. 144 4 CONTENTS 7 Frequencies in Forgetful Timed Automata 147 7.1 Frequencies in one-clock forgetful timed automata.................. 148 7.2 Extension to several clock forgetful timed automata.................. 151 7.2.1 Inclusion of the set of frequencies in the set of ratios............. 152 7.2.2 Techniques to compute the frequencies.................... 153 7.2.3 Inclusion of the set of ratios in the set of frequencies............. 155 7.2.4 Discussion about assumptions......................... 157 8 Emptiness and Universality Problems in Timed Automata with Frequency 159 8.1 Consequences of Chapters6 and7........................... 159 8.2 Lower bound for the universality problem....................... 160 8.3 Decidability of the universality problem for Zeno words with positive frequency in one-clock timed automata................................ 161 V Reachability of Communicating Timed Automata 165 9 Communicating Timed Processes: a Uniform Semantics 173 9.1 Definition of communicating timed processes..................... 173 9.2 Communicating timed or tick automata........................ 175 9.2.1 Communicating timed automata........................ 175 9.2.2 Communicating tick automata......................... 176 9.3 Discussion about the models.............................. 177 9.3.1 Modeling urgency with emptiness test..................... 177 9.3.2 On the power of time.............................. 178 9.3.3 Undecidability of multi-tick automata..................... 179 10 Reachability Problem in Communicating Tick Automata 181 10.1 Communicating counter automata........................... 181 10.2 From tick automata to counter automata......................... 182 10.3 From counter automata to tick automata........................ 185 10.4 Characterization of the decidable topologies...................... 187 11 Reachability Problem in Communicating Timed Automata 189 11.1 From continuous time to discrete time......................... 189 11.2 Correctness of the reduction.............................. 191 11.2.1 Proof of the correctness using a rescheduling lemma............. 191 11.2.2 Proof of the rescheduling lemma........................ 193 11.2.3 Consequences.................................. 196 11.3 Reciprocal reduction and its consequences....................... 196 11.4 Abstraction of communicating timed automata with emptiness tests is difficult.... 197 11.4.1 Our construction is not sound for emptiness test................ 197 11.4.2 Why soundness is hard to achieve....................... 198 VI Conclusion and Future Works 201 Bibliography 211 5 CONTENTS 6 Part I Contributions à la vérification des automates temporisés : déterminisation, vérification quantitative et accessibilité dans les réseaux d’automates
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