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

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104 CHAPTER 5. HENNESSY-MILNER LOGIC We write p |= F , read ‘p satisfies F ’, iff p ∈ [F ]. Two formulae are equivalent if, and only if, they are satisfied by the same processes in every transition system. Example 5.1 In order to understand the definition of the set operators 〈·a·〉, [·a·] introduced above, it is instructive to look at an example. Consider the following labelled transition system. Then s a t a a b s1 s2 t1 〈·a·〉{s1, t1} = {s, t} . This means that 〈·a·〉{s1, t1} is the collection of states from which it is possible to perform an a-labelled transition ending up in either s1 or t1. On the other hand, [·a·]{s1, t1} = {s1, s2, t, t1} . The idea here is that [·a·]{s1, t1} consists of the set of all processes that become either s1 or t1 no matter how they perform an a-labelled transition. Clearly, s does not have this property because it can perform the transition s a → s2, whereas t does because its only a-labelled transition ends up in t1. But why are s1, s2 and t1 in [·a·]{s1, t1}? To see this, look at the formal definition of the set [·a·]{s1, t1} = {p ∈ Proc | p a → p ′ implies p ′ ∈ {s1, t1}, for each p ′ } . Since s1, s2 and t1 do not afford a-labelled transitions, it is vacuously true that all of their a-labelled transitions end up in either s1 or t1! This is the reason why those states are in the set [·a·]{s1, t1}. We shall come back to this important point repeatedly in what follows. Exercise 5.1 Consider the labelled transition system in the example above. What are 〈·b·〉{s1, t1} and [·b·]{s1, t1}? Let us now re-examine the properties of our computer scientist that we mentioned earlier, and let us see whether we can express them using HML. First of all, note that, for the time being, we have defined the semantics of formulae in M in terms b b
105 of the one step transitions a →. This means, in particular, that we are not considering τ actions as unobservable. So, if we say that ‘a process P can do action a now’, then we really mean that the process can perform a transition of the form P a → Q for some Q. How can we express, for instance, that our computer scientist is willing to drink coffee now? Well, one way to say so using our logic is to say that the computer scientist has the possibility of doing a coffee-labelled transition. This suggests that we use a formula of the form 〈coffee〉F for some formula F that should be satisfied by the state reached by the computer scientist after having drunk her coffee. What should this F be? Since we are not requiring anything of the subsequent behaviour of the computer scientist, it makes sense to set F = tt. So, it looks as if we can express our natural language requirement in terms of the formula 〈coffee〉tt. In fact, since our property language has a formal semantics, we can actually prove that our proposed formula is satisfied exactly by all the processes that have an outgoing coffee-labelled transition. This can be done as follows: [〈coffee〉tt] = 〈·coffee·〉[tt] = 〈·coffee·〉Proc = {P | P coffee → P ′ for some P ′ ∈ Proc} = {P | P coffee → } . So the formula we came up with does in fact say what we wanted. Can we express using HML that the computer scientist cannot drink tea now? Consider the formula [tea]ff. Intuitively this formula says that all the states that a process can reach by doing a tea-labelled transition must satisfy the formula ff, i.e., false. Since no state has the property ‘false’, the only way that a process can satisfy the property [tea]ff is that it has no tea-labelled transition. To prove formally that our proposed formula is satisfied exactly by all the processes that have no outgoing tea-labelled transition, we proceed as follows: [[tea]ff ] = [·tea·][ff ] = [·tea·]∅ = {P | P tea → P ′ implies P ′ ∈ ∅, for each P ′ } = {P | P tea } . The last equality above follows from the fact that, for each process P , P tea iff (P tea → P ′ implies P ′ ∈ ∅, for each 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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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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Page 75 and 76: 3.3. STRONG BISIMILARITY 57 Recall
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Page 79 and 80: 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 104 and 105: 86 CHAPTER 4. THEORY OF FIXED POINT
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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
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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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