Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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98 CHAPTER 4. THEORY OF FIXED POINTS We calculate the sequence F i (Proc × Proc) for i ≥ 1 thus: F 1 (Proc × Proc) = {(Q1, Q4), (Q4, Q1), (Q2, Q3), (Q3, Q2)} ∪ I F 2 (Proc × Proc) = {(Q2, Q3), (Q3, Q2)} ∪ I and finally F 3 (Proc × Proc) = F 2 (Proc × Proc) . Therefore, the only distinct processes that are related by the largest strong bisimulation over this labelled transition system are Q2 and Q3, and indeed Q2 ∼ Q3. Exercise 4.11 Using the iterative algorithm described above, compute the largest strong bisimulation over the labelled transition system described by the following defining equations in CCS: P1 = a.P2 P2 = a.P1 P3 = a.P2 + a.P4 P4 = a.P3 + a.P5 P5 = 0 . You may find it useful to draw the labelled transition system associated with the above CCS definition first. Exercise 4.12 Use the iterative algorithm described above to compute the largest bisimulation over the labelled transition system in Example 3.7. Exercise 4.13 What is the worst case complexity of the algorithm outlined above when run on a labelled transition system consisting of n states and m transitions? Express your answer using O-notation, and compare it with the complexity of the algorithm due to Page and Tarjan mentioned in Section 3.6. Exercise 4.14 Let (Proc, Act, { α → | α ∈ Act}) be a labelled transition system. For each i ≥ 0, define the relation ∼i over Proc as follows: • s1 ∼0 s2 holds always; • s1 ∼i+1 s2 holds iff for each action α: - if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 ∼i s ′ 2 ; - if s2 α → s ′ 2 , then there is a transition s1 α → s ′ 1 such that s′ 1 ∼i s ′ 2 .
4.3. BISIMULATION AS A FIXED POINT 99 Prove that, for each i ≥ 0: 1. the relation ∼i is an equivalence relation, 2. ∼i+1 is included in ∼i, and 3. ∼i= F i (Proc × Proc). Exercise 4.15 1. Give a characterization for observational equivalence as a fixed point for a monotonic function similar to the one we presented above for strong bisimilarity. 2. Use your characterization to compute observational equivalence over the labelled transition system in Example 3.8. What is the worst case complexity of your algorithm?
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Reactive Systems: Modelling, Specif
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iv CONTENTS 5 Hennessy-Milner logic
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List of Figures 2.1 The interface f
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List of Tables 2.1 An alternative f
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xii PREFACE It has been very satisf
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xiv PREFACE a textbook, and offers
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xvi PREFACE studies in Edinburgh, a
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Part I A classic theory of reactive
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4 CHAPTER 1. INTRODUCTION After hav
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6 CHAPTER 1. INTRODUCTION • softw
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8 CHAPTER 1. INTRODUCTION prototype
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10 CHAPTER 2. THE LANGUAGE CCS ✬
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12 CHAPTER 2. THE LANGUAGE CCS Exer
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18 CHAPTER 2. THE LANGUAGE CCS CS d
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20 CHAPTER 2. THE LANGUAGE CCS Sinc
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22 CHAPTER 2. THE LANGUAGE CCS a p
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24 CHAPTER 2. THE LANGUAGE CCS •
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26 CHAPTER 2. THE LANGUAGE CCS Defi
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28 CHAPTER 2. THE LANGUAGE CCS we h
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30 CHAPTER 2. THE LANGUAGE CCS This
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32 CHAPTER 2. THE LANGUAGE CCS 2. D
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34 CHAPTER 2. THE LANGUAGE CCS To b
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Chapter 3 Behavioural equivalences
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3.1. CRITERIA FOR A GOOD BEHAVIOURA
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3.2. TRACE EQUIVALENCE: A FIRST ATT
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3.3. STRONG BISIMILARITY 43 trace e
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3.3. STRONG BISIMILARITY 45 Obvious
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Page 71 and 72: 3.3. STRONG BISIMILARITY 53 Conclud
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Page 75 and 76: 3.3. STRONG BISIMILARITY 57 Recall
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Page 79 and 80: 3.4. WEAK BISIMILARITY 61 Is there
Page 81 and 82: 3.4. WEAK BISIMILARITY 63 Start pu
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Page 87 and 88: 3.4. WEAK BISIMILARITY 69 Show that
Page 89 and 90: 3.4. WEAK BISIMILARITY 71 In light
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Page 125 and 126: 1. Decide whether the following sta
Page 127 and 128: 109 Note that logical negation is n
Page 129 and 130: 111 Another example of a process th
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Page 135 and 136: CHAPTER 6. HML WITH RECURSION 117 e
Page 137 and 138: CHAPTER 6. HML WITH RECURSION 119
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6.7. MIXING LARGEST AND LEAST FIXED
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6.8. FURTHER RESULTS ON MODEL CHECK
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Chapter 7 Modelling and analysis of
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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7.1. SPECIFYING MUTUAL EXCLUSION IN
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.3. TESTING MUTUAL EXCLUSION 169 i
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7.3. TESTING MUTUAL EXCLUSION 171 D
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7.3. TESTING MUTUAL EXCLUSION 173 1
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