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

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112 CHAPTER 5. HENNESSY-MILNER LOGIC Assume, towards a contradiction, that there is no S ′ such that S a → S ′ and S ′ satisfies the same properties as R ′ . Since S is image finite, the set of processes S can reach by performing an a-labelled transition is finite, say {S1, . . . , Sn} with n ≥ 0. By our assumption, none of the processes in the above set satisfies the same formulae as R ′ . So, for each i ∈ {1, . . . , n}, there is a formula Fi such that R ′ |= Fi and Si |= Fi . (Why? Could it not be that R ′ |= Fi and Si |= Fi, for some i ∈ {1, . . . , n}?) We are now in a position to construct a formula that is satisfied by R, but not by S—contradicting our assumption that R and S satisfy the same formulae. In fact, the formula 〈a〉(F1 ∧ F2 ∧ · · · ∧ Fn) is satisfied by R, but not by S. The easy verification is left to the reader. The proof of the theorem is now complete. ✷ Exercise 5.9 (Mandatory) Fill in the details that we have omitted in the above proof. What is the formula that we have constructed to distinguish R and S in the proof of the implication from right to left if n = 0? Remark 5.1 In fact, the implication from left to right in the above theorem holds for arbitrary processes, not just image finite ones. The above theorem has many applications in the theory of processes, and in verification technology. For example, a consequence of its statement is that if two image finite processes are not strongly bisimilar, then there is a formula in M that tells us one reason why they are not. Moreover, as the proof of the above theorem suggests, we can always construct this distinguishing formula. Note, moreover, that the above characterization theorem for strong bisimilarity is very general. For instance, in light of your answer to Exercise 3.30, it also applies to observational equivalence, provided that we interpret HML over the labelled transition system whose set of actions consists of all of the observable actions and of the label τ, and whose transitions are precisely the ‘weak transitions’ whose labels are either observable actions or τ. Exercise 5.10 Consider the following labelled transition system.
a s s1 b s2 b a a t b t1 b t2 a a v a b v1 v2 b b v3 Argue that s ∼ t, s ∼ v and t ∼ v. Next, find a distinguishing formula of Hennessy-Milner logic for the pairs • s and t, • s and v, and • t and v. b 113 Verify your claims in the Edinburgh Concurrency Workbench (use the strongeq and checkprop commands) and check whether you found the shortest distinguishing formula (use the dfstrong command). Exercise 5.11 For each of the following CCS expressions decide whether they are strongly bisimilar and, if they are not, find a distinguishing formula in Hennessy- Milner logic: • b.a.0 + b.0 and b.(a.0 + b.0), • a.(b.c.0 + b.d.0) and a.b.c.0 + a.b.d.0, • a.0 | b.0 and a.b.0 + b.a.0, and • (a.0 | b.0) + c.a.0 and a.0 | (b.0 + c.0). Verify your claims in the Edinburgh Concurrency Workbench (use the strongeq and checkprop commands) and check whether you found the shortest distinguishing formula (use the dfstrong command). Exercise 5.12 (For the theoretically minded) Let (Proc, Act, { a →| a ∈ Act}) be image finite. Show that ∼= ∼i , i≥0
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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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3.3. STRONG BISIMILARITY 47 So, by
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3.3. STRONG BISIMILARITY 49 Theorem
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3.3. STRONG BISIMILARITY 51 The imp
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3.3. STRONG BISIMILARITY 53 Conclud
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3.3. STRONG BISIMILARITY 55 3. the
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3.3. STRONG BISIMILARITY 57 Recall
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3.3. STRONG BISIMILARITY 59 B 2 0
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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 104 and 105: 86 CHAPTER 4. THEORY OF FIXED POINT
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Page 127 and 128: 109 Note that logical negation is n
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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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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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Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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