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

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150 CHAPTER 6. HML WITH RECURSION the system. So a state p in a labelled transition system has a livelock if it affords a computation of the form p = p0 τ → p1 τ → p2 τ → p3 τ → · · · for some sequence of states p1, p2, p3 . . .. In other words, a state p has a livelock now if it affords a τ-labelled transition leading to a state p1 which has a livelock now. This immediately suggests the following recursive specification of the property LivelockNow: LivelockNow = 〈τ〉LivelockNow . As usual, we are faced with a choice in selecting a suitable solution for the above equation. Since we are specifying a state of affairs that should hold forever, in this case we should select the largest solution to the equation above. It follows that our HML specification of the property ‘the state has a livelock’ is LivelockNow max = 〈τ〉LivelockNow . Exercise 6.15 What would be the least solution of the above equation? Exercise 6.16 (Mandatory) Consider the labelled transition system below. s a p τ τ τ q r Use the iterative algorithm for computing the set of states in that labelled transition system that satisfies the formula LivelockNow defined above. Exercise 6.17 This exercise is for those amongst you who feel they need more practice in computing fixed points using the iterative algorithm. Consider the labelled transition system below. s τ τ s1 a τ s2 s3 Use the iterative algorithm for computing the set of states in that labelled transition system that satisfies the formula LivelockNow defined above. τ
6.7. MIXING LARGEST AND LEAST FIXED POINTS 151 In light of the above discussion, a specification of the property mentioned at the beginning of this section using HML with recursive definitions can be given using the following system of equations: P os(LivelockNow) min = LivelockNow ∨ 〈Act〉P os(LivelockNow) LivelockNow max = 〈τ〉LivelockNow . This looks natural and innocuous. However, first appearances can be deceiving! Indeed, the equational systems we have considered so far have only allowed us to express formulae purely in terms of largest or least solutions to systems of recursion equations. (See Section 6.5.) For instance, in defining the characteristic formulae for bisimulation equivalence, we only used systems of equations in which the largest solution was sought for all of the equations in the system. Our next question is whether we can extend our framework in such a way that it can treat systems of equations with mixed solutions like the one describing the formula P os(LivelockNow) above. How can we, for instance, compute the set of processes in the labelled transition system s a p τ τ τ q r that satisfy the formula P os(LivelockNow)? In this case, the answer is not overly difficult. In fact, you might have already noted that we can compute the set of processes satisfying the formula P os(LivelockNow) once we have in our hands the collection of processes satisfying the formula LivelockNow. As you saw in Exercise 6.16, the only state in the above labelled transition system satisfying the formula LivelockNow is p. Therefore, we may obtain the collection of states satisfying the formula P os(LivelockNow) as the least solution of the set equation S = {p} ∪ 〈·Act·〉S , (6.16) where S ranges over subsets of {s, p, q, r}. We can calculate the least solution of this equation using the iterative methods we introduced in Section 6.2. Since we are looking for the least solution of the above equation, we begin by obtaining our first approximation S (1) to the solution by computing the value of the expression on the right-hand side of the equation when S = ∅, which is the least element in the complete lattice consisting of the subsets of {s, p, q, r} ordered by inclusion. We have that S (1) = {p} ∪ 〈·Act·〉∅ = {p} .
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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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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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3.4. WEAK BISIMILARITY 63 Start pu
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3.4. WEAK BISIMILARITY 65 Our order
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3.4. WEAK BISIMILARITY 67 Send def
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3.4. WEAK BISIMILARITY 69 Show that
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3.4. WEAK BISIMILARITY 71 In light
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3.5. GAME CHARACTERIZATION OF BISIM
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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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Page 119 and 120: Chapter 5 Hennessy-Milner logic In
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Page 123 and 124: 105 of the one step transitions a
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 157 and 158: 6.6. CHARACTERISTIC PROPERTIES 139
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Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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