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

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118 CHAPTER 6. HML WITH RECURSION implicitly defining the set of their solutions, and we are all familiar with equations that have no solutions at all. For instance, the equation x = x + 1 (6.2) has no solution over the set of natural numbers, and there is no X ⊆ N such that X = N \ X . (6.3) On the other hand, there are uncountably many X ⊆ N such that X = {2} ∪ X , (6.4) namely all of the sets of natural numbers that contain the number 2. There are also equations that have a finite number of solutions, but not a unique one. As an example, consider the equation X = {10} ∪ {n − 1 | n ∈ X, n = 0} . (6.5) The only finite set that is the solution for this equation is the set {0, 1, . . . , 10}, and the only infinite solution is N itself. Exercise 6.1 Check the claims that we have just made. Exercise 6.2 Reconsider equations (6.2)–(6.5). 1. Why doesn’t Tarski’s fixed point theorem apply to yield a solution to the first two of these equations? 2. Consider the structure introduced in the second bullet of Example 4.2 on page 88. For each d ∈ N ∪ {∞}, define ∞ + d = d + ∞ = ∞ . Does equation (6.2) have a solution in the resulting structure? How many solutions does that equation have? 3. Use Tarski’s fixed point theorem to find the largest and least solutions of (6.5). Since an equation like (6.1) is meant to describe a formula, it is therefore natural to ask ourselves the following questions.
CHAPTER 6. HML WITH RECURSION 119 • Does (6.1) have a solution? And what precisely do we mean by that? • If (6.1) has more than one solution, which one do we choose? • How can we compute whether a process satisfies the formula described by (6.1)? Precise answers to these questions will be given in the remainder of this chapter. However, to motivate our subsequent technical developments, it is appropriate here to discuss briefly the first two questions above. Recall that the meaning of a formula (with respect to a labelled transition system) is the set of processes that satisfy it. Therefore, it is natural to expect that a set S of processes that satisfy the formula described by equation (6.1) should be such that: S = 〈·a·〉Proc ∩ [·a·]S . It is clear that S = ∅ is a solution to the equation (as no process can satisfy both 〈a〉tt and [a]ff). However, the process p on Figure 6.1 can perform an a-transition invariantly and p ∈ ∅, so this cannot be the solution we are looking for. Actually it turns out that it is the largest solution we need here, namely S = {p}. The set S = ∅ is the least solution. In other cases it is the least solution we are interested in. For instance, we can express Pos([a]ff) by the following equation: Y ≡ [a]ff ∨ 〈a〉Y . Here the largest solution is Y = {p, q, r} but, as the process p on Figure 6.1 cannot terminate at all, this is clearly not the solution we are interested in. The least solution of the above equation over the labelled transition system on Figure 6.1 is Y = {q, r} and is exactly the set of processes in that labelled transition system that intuitively satisfy Pos([a]ff). When we write down a recursively defined property, we can indicate whether we desire the least or the largest solution by adding this information to the equality sign. For Inv(〈a〉tt) we want the largest solution, and in this case we write For Pos([a]ff) we will write X max = 〈a〉tt ∧ [a]X . Y min = [a]ff ∨ 〈a〉Y . More generally we can express that the formula F holds for each reachable state in a labelled transition system having set of actions Act (written Inv(F ), and read
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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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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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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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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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