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

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30 CHAPTER 2. THE LANGUAGE CCS This rule states that every transition of a term P determines a transition of the expression P \ L, provided that neither the action producing the transition nor its complement are in L. For example, as you can check, this side condition prevents us from proving the existence of the transition (coffee.CS) \ coffee coffee → CS \ coffee . Finally, note that, when considering the binary version of the summation operator, the family of rules SUMj reduces to the following two rules. SUM1 P1 α → P ′ 1 P1 + P2 α → P ′ 1 SUM2 P2 α → P ′ 2 P1 + P2 α → P ′ 2 To get a feeling for the power of recursive definitions of process behaviours, consider the process C defined thus: C def = up.(C | down.0) . (2.8) What are the transitions that this process affords? Using the rules for constants and action prefixing, you should have little trouble in arguing that the only initial transition for C is C up → C | down.0 . (2.9) What next? Observing that down.0 down → 0, using rule COM2 in Table 2.2 we can infer that C | down.0 down → C | 0 . Since it is reasonable to expect that the process C | 0 exhibits the same behaviour as C—and we shall see later on that this does hold true—, the above transition effectively brings our process back to its initial state, at least up to behavioural equivalence. However, this is not all, because, as we have already proven (2.9), using rule COM1 in Table 2.2 we have that the transition C | down.0 up → (C | down.0) | down.0 is also possible. You might find it instructive to continue building a little more of the transition graph for process C. As you may begin to notice, the LTS giving the operational semantics of the process expression C looks very similar to that for Counter0, as given in (2.5). Indeed, we shall prove later on that these two processes exhibit the same behaviour in a very strong sense. Exercise 2.7 Use the rules of the SOS semantics for CCS to derive the LTS for the process SmUni defined by equation 2.4 on page 13. (Use the definition of CS in Table 2.1.)
2.2. CCS, FORMALLY 31 Exercise 2.8 Assume that A def = b.a.B. By using the SOS rules for CCS prove the existence of the following transitions: • (A | b.0) \ {b} τ → (a.B | 0) \ {b}, • (A | b.a.B) + (b.A)[a/b] b → (A | a.B), and • (A | b.a.B) + (b.A)[a/b] a → A[a/b]. Exercise 2.9 Draw (part of) the transition graph for the process name A whose behaviour is given by the defining equation A def = (a.A) \ b . The resulting transition graph should have infinitely many states. Can you think of a CCS term that generates a finite labelled transition system that should intuitively have the same behaviour as A? Exercise 2.10 Draw (part of) the transition graph for the process name A whose behaviour is given by the defining equation A def = (a0.A)[f] where we assume that the set of channel names is {a0, a1, a2, . . .}, and f(ai) = ai+1 for each i. The resulting transition graph should (again!) have infinitely many states. Can you give an argument showing that there is no finite state labelled transition system that could intuitively have the same behaviour as A? Exercise 2.11 1. Draw the transition graph for the process name Mutex1 whose behaviour is given by the defining equation Mutex1 User def = (User | Sem) \ {p, v} def = ¯p.enter.exit.¯v.User Sem def = p.v.Sem .
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 ✬
Page 30 and 31: 12 CHAPTER 2. THE LANGUAGE CCS Exer
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 52 and 53: 34 CHAPTER 2. THE LANGUAGE CCS To b
Page 55 and 56: Chapter 3 Behavioural equivalences
Page 57 and 58: 3.1. CRITERIA FOR A GOOD BEHAVIOURA
Page 59 and 60: 3.2. TRACE EQUIVALENCE: A FIRST ATT
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
Page 73 and 74: 3.3. STRONG BISIMILARITY 55 3. the
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
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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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