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

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120 CHAPTER 6. HML WITH RECURSION ‘invariantly F ’) by means of the equation X max = F ∧ [Act]X , and that F possibly holds at some point (written Pos(F )) by Y min = F ∨ 〈Act〉Y . Intuitively, we use largest solutions for those properties that hold of a process unless it has a finite computation that disproves the property. For instance, process q does not have property Inv(〈a〉tt) because it can reach a state in which no alabelled transition is possible. Conversely, we use least solutions for those properties that hold of a process if it has a finite computation sequence which ‘witnesses’ the property. For instance, a process has property Pos(〈a〉tt) if it has a computation leading to a state that can perform an a-labelled transition. This computation is a witness for the fact that the process can perform an a-labelled transition at some point in its behaviour. We shall appeal to the intuition given above in the following section, where we present examples of recursively defined properties. Exercise 6.3 Give a formula, built using HML and the temporal operators Pos and/or Inv, that expresses a property satisfied by exactly one of the processes in Exercise 5.13. 6.1 Examples of recursive properties Adding recursive definitions to Hennessy-Milner logic gives us a very powerful language for specifying properties of processes. In particular this extension allows us to express different kinds of safety and liveness properties. Before developing the theory of HML with recursion, we give some more examples of its uses. Consider the formula Safe(F ) that is satisfied by a process p whenever it has a complete transition sequence p = p0 a1 a2 → p1 → p2 · · · , where each of the processes pi satisfies F . (A transition sequence is complete if it is infinite or its last state affords no transition.) This invariance of F under some computation can be expressed in the following way: X max = F ∧ ([Act]ff ∨ 〈Act〉X) .
6.1. EXAMPLES OF RECURSIVE PROPERTIES 121 It turns out to be the largest solution that is of interest here as we will argue for formally later. Intuitively, the recursively defined formula above states that a process p has a complete transition sequence all of whose states satisfy the formula F if, and only if, • p itself satisfies F , and • either p has no outgoing transition (in which case p will satisfy the formula [Act]ff) or p has a transition leading to a state that has a complete transition sequence all of whose states satisfy the formula F . A process p satisfies the property Even(F ), read ‘eventually F ’, if each of its complete transition sequences will contain at least one state that has the property F . This means that either p satisfies F , or p can perform some transition and every state that it can reach by performing a transition can itself eventually reach a state that has property F . This can be expressed by means of the following equation: Y min = F ∨ (〈Act〉tt ∧ [Act]Y ) . In this case we are interested in the least solution because Even(F ) should only be satisfied by those processes that are guaranteed to reach a state satisfying F in all of their computation paths. Note that the definitions of Safe(F ) and Even(F ), respectively Inv(F ) and Pos(F ), are mutually dual, i.e., they can be obtained from one another by replacing ∨ by ∧, [Act] by 〈Act〉 and min = by max = . One can show that ¬Inv(F ) ≡ Pos(¬F ) and ¬Safe(F ) ≡ Even(¬F ), where we write ¬ for logical negation. It is also possible to express that F should be satisfied in each transition sequence until G becomes true. There are two well known variants of this construction: • F U s G, the so-called strong until, that says that sooner or later p reaches a state where G is true and in all the states it traverses before this happens F must hold; • F U w G, the so-called weak until, that says that F must hold in all states p traverses until it gets into a state where G holds (but maybe this will never happen!). We express these operators as follows:
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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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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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