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

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114 CHAPTER 5. HENNESSY-MILNER LOGIC where ∼i (i ≥ 0) is the sequence of equivalence relations defined in Exercise 4.14. Exercise 5.13 (For the theoretically minded) Consider the process A ω given by: ω def A = a.A ω . Show that the processes A
Chapter 6 Hennessy-Milner logic with recursive definitions An HML formula can only describe a finite part of the overall behaviour of a process. In fact, as each modal operator allows us to explore the effect of taking one step in the behaviour of a process, using a single HML formula we can only describe properties of a fixed finite fragment of the computations of a process. As those of you who solved Exercise 5.13 already discovered, how much of the behaviour of a process we can explore using a single formula is entirely determined by its so-called modal depth—i.e., by the maximum nesting of modal operators in it. For example, the formula ([a]〈a〉ff) ∨ 〈b〉tt has modal depth 2, and checking whether a process satisfies it or not involves only an analysis of its sequences of transitions whose length is at most 2. (We will return to this issue in Section 6.6, where a formal definition of the modal depth of a formula will be given.) However, we often wish to describe properties that describe states of affairs that may or must occur in arbitrarily long computations of a process. If we want to express properties like, for example, that a process is always able to perform a given action, we have to extend the logic. As the following example indicates, one way of doing so is to allow for infinite conjunctions and disjunctions in our property language. Example 6.1 Consider the processes p and q in Figure 6.1. It is not hard to come up with an HML formula that p satisfies and q does not. In fact, after performing an a-action, p will always be able to perform another one, whereas q may fail to do so. This can be captured formally in HML as follows: p |= [a]〈a〉tt but q |= [a]〈a〉tt . 115
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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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3.4. WEAK BISIMILARITY 61 Is there
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Page 83 and 84: 3.4. WEAK BISIMILARITY 65 Our order
Page 85 and 86: 3.4. WEAK BISIMILARITY 67 Send def
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 93 and 94: CHAPTER 3.5. GAME CHARACTERIZATION
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Page 104 and 105: 86 CHAPTER 4. THEORY OF FIXED POINT
Page 119 and 120: Chapter 5 Hennessy-Milner logic In
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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
Page 139 and 140: 6.1. EXAMPLES OF RECURSIVE PROPERTI
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Page 145 and 146: 6.3. LARGEST FIXED POINTS AND INVAR
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Page 157 and 158: 6.6. CHARACTERISTIC PROPERTIES 139
Page 167 and 168: 6.7. MIXING LARGEST AND LEAST FIXED
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Page 175 and 176: Chapter 7 Modelling and analysis of
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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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