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

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20 CHAPTER 2. THE LANGUAGE CCS Since τ actions are supposed to be unobservable, the following process seems to be an appropriate high level specification of the behaviour exhibited by process SmUni: Spec def = pub.Spec . Indeed, we expect that SmUni and Spec describe the same observable behaviour, albeit at different levels of abstraction. We shall see in the remainder of this book that one of the big questions in process theory is to come up with notions of ‘behavioural equivalence’ between processes that will allow us to establish formally that, for instance, SmUni and Spec do offer the same behaviour. But this is getting ahead of our story. 2.2 CCS, formally Having introduced CCS by example, we now proceed to present formal definitions for its syntax and semantics. 2.2.1 The model of labelled transition systems We have already indicated in our examples how the operational semantics for CCS can be given in terms of automata—which we have called labelled transition systems, as customary in concurrency theory. These we now proceed to define, for the sake of clarity. We first introduce the ingredients in the model of labelled transition systems informally, and then provide its formal definition. In the model of labelled transition systems, processes are represented by vertices of certain edge-labelled directed graphs (the labelled transition systems themselves) and a change of process state caused by performing an action is understood as moving along an edge, labelled by the action name, that goes out of that state. A labelled transition system consists therefore of a set of states (also referred to as processes or configurations), a set of labels (or actions), and a transition relation → describing changes in process states: if a process p can perform an action a and become a process p ′ , we write p a → p ′ . Sometimes a state is singled out as the start state in the labelled transition system under consideration. In that case, we say that the labelled transition system is rooted. Example 2.1 Let us start with a variation on the classic example of a tea/coffee vending machine. The very simplified behaviour of the process which determines the interaction of the machine with a customer can be described as follows. From the initial state—say, p—representing the situation ‘waiting for a request’, two possible actions are enabled. Either the tea button or the coffee button can be pressed
2.2. CCS, FORMALLY 21 (the corresponding action ‘tea’ or ‘coffee’ is executed) and the internal state of the machine changes accordingly to p1 or p2. Formally, this can be described by the transitions p tea → p1 and p coffee → p2 . The target state p1 records that the customer has requested tea, whereas p2 describes the situation in which coffee has been selected. Now the customer is asked to insert the corresponding amount of money, let us say one euro for a cup of tea and two euros for a cup of coffee. This is reflected by corresponding changes in the control state of the vending machine. These state changes can be modelled by the transitions p1 1C= → p3 and p2 2C= → p3 , whose target state p3 records that the machine has received payment for the chosen drink. Finally, the drink is collected and the machine returns to its initial state p, ready to accept the request of another customer. This corresponds to the transition p3 collect → p . It is often convenient and suggestive to use a graphical representation for labelled transition systems. The following picture represents the tea/coffee machine described above. p1 p tea coffee 1C= 2C= collect p3 Sometimes, when referring only to the process p, we do not have to give names to the other process states (in our example p1, p2 and p3) and it is sufficient to provide the following labelled transition system for the process p. p2
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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
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Page 44 and 45: 26 CHAPTER 2. THE LANGUAGE CCS Defi
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Page 55 and 56: Chapter 3 Behavioural equivalences
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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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3.4. 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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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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