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

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28 CHAPTER 2. THE LANGUAGE CCS we have so far developed. However, intuition alone can lead us to wrong conclusions, and most importantly cannot be fed to a computer! To capture formally our understanding of the semantics of the language CCS, we therefore introduce the collection of SOS rules in Table 2.2. These rules are used to generate an LTS whose states are CCS expressions. In that LTS, a transition P α → Q holds for CCS expressions P, Q and action α if, and only if, it can be proven using the rules in Table 2.2. A rule like α.P α → P is an axiom, as it has no premises—that is, it has no transition above the solid line. This means that proving that a process of the form α.P affords the transition α.P α → P (the conclusion of the rule) can be done without establishing any further sub-goal. Therefore each process of the form α.P affords the transition α.P α → P . As an example, we have that the following transition pub.CS1 pub → CS1 is provable using the above rule for action prefixing. On the other hand, a rule like P α → P ′ K α → P ′ def K = P (2.7) has a non-empty set of premises. This rule says that to establish that constant K affords the transition mentioned in the conclusion of the rule, we have to prove first that the body of the defining equation for K, namely the process P , affords the transition P α → P ′ . Using this rule, pattern matching and transition (2.7), we can prove the transition CS pub → CS1 , which we had informally derived before for the version of process CS given in Table 2.1 on page 18. The aforementioned rule for constants has a side condition, namely K def = P , that describes a constraint that must be met in order for the rule to be applicable. In that specific example, the side condition states intuitively that the rule may be used to derive an initial transition for constant K if ‘K is declared to have body P ’. Another example of a rule with a side condition is that for restriction. P α → P ′ P \ L α → P ′ \ L α, ¯α ∈ L
2.2. CCS, FORMALLY 29 SUMj RES ACT α.P α → P Pj α → P ′ j i∈I Pi α → P ′ j COM1 COM2 P α → P ′ P | Q α → P ′ | Q Q α → Q ′ P | Q α → P | Q ′ where j ∈ I COM3 P a → P ′ Q ā → Q ′ P | Q τ → P ′ | Q ′ P α → P ′ P \ L α → P ′ \ L REL P α → P ′ P [f] f(α) → P ′ [f] where α, ¯α ∈ L CON P α → P ′ K α def where K = P → P ′ Table 2.2: SOS rules for CCS (α ∈ Act, a ∈ L)
Page 1: Reactive Systems: Modelling, Specif
Page 4 and 5: iv CONTENTS 5 Hennessy-Milner logic
Page 7 and 8: List of Figures 2.1 The interface f
Page 9: List of Tables 2.1 An alternative f
Page 12 and 13: xii PREFACE It has been very satisf
Page 14 and 15: xiv PREFACE a textbook, and offers
Page 16 and 17: xvi PREFACE studies in Edinburgh, a
Page 19: Part I A classic theory of reactive
Page 22 and 23: 4 CHAPTER 1. INTRODUCTION After hav
Page 24 and 25: 6 CHAPTER 1. INTRODUCTION • softw
Page 26 and 27: 8 CHAPTER 1. INTRODUCTION prototype
Page 28 and 29: 10 CHAPTER 2. THE LANGUAGE CCS ✬
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Page 36 and 37: 18 CHAPTER 2. THE LANGUAGE CCS CS d
Page 38 and 39: 20 CHAPTER 2. THE LANGUAGE CCS Sinc
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Page 55 and 56: Chapter 3 Behavioural equivalences
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Page 61 and 62: 3.3. STRONG BISIMILARITY 43 trace e
Page 63 and 64: 3.3. STRONG BISIMILARITY 45 Obvious
Page 65 and 66: 3.3. STRONG BISIMILARITY 47 So, by
Page 67 and 68: 3.3. STRONG BISIMILARITY 49 Theorem
Page 69 and 70: 3.3. STRONG BISIMILARITY 51 The imp
Page 71 and 72: 3.3. STRONG BISIMILARITY 53 Conclud
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Page 75 and 76: 3.3. STRONG BISIMILARITY 57 Recall
Page 77 and 78: 3.3. STRONG BISIMILARITY 59 B 2 0
Page 79 and 80: 3.4. WEAK BISIMILARITY 61 Is there
Page 81 and 82: 3.4. WEAK BISIMILARITY 63 Start pu
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
Page 91 and 92: 3.5. GAME CHARACTERIZATION OF BISIM
Page 93 and 94: CHAPTER 3.5. GAME CHARACTERIZATION
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CHAPTER 3.5. GAME CHARACTERIZATION
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3.6. FURTHER RESULTS ON EQUIVALENCE
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3.6. FURTHER RESULTS ON EQUIVALENCE
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86 CHAPTER 4. THEORY OF FIXED POINT
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Chapter 5 Hennessy-Milner logic In
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103 We are interested in using the
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105 of the one step transitions a
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1. Decide whether the following sta
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109 Note that logical negation is n
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111 Another example of a process th
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a s s1 b s2 b a a t b t1 b t
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Chapter 6 Hennessy-Milner logic wit
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CHAPTER 6. HML WITH RECURSION 117 e
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CHAPTER 6. HML WITH RECURSION 119
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6.1. EXAMPLES OF RECURSIVE PROPERTI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.3. LARGEST FIXED POINTS AND INVAR
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6.4. GAME CHARACTERIZATION FOR HML
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.6. CHARACTERISTIC PROPERTIES 139
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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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Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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