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

More documents

Recommendations

Info

$Algorithmic melody composition based on fractal ... - cs.ioc.ee$

168 CHAPTER 7. MODELLING MUTUAL EXCLUSION ALGORITHMS Exercise 7.10 Assume that the CCS process Q weakly simulates P . Show that Q + R weakly simulates P and P + R, for each CCS process R. Exercise 7.11 1. Show that the processes α.P +α.Q and α.(P +Q) are weak trace equivalent for each action α, and terms P, Q. 2. Show that weak trace equivalence is preserved by all of the operators of CCS. 7.3 Testing mutual exclusion Another approach to establishing the correctness of Peterson’s algorithm is to use a notion of ‘testing’. Recall that what we mean by ensuring mutual exclusion is that at no point in the execution of process Peterson both processes will be in their critical section at the same time. Such a situation would arise if there is some execution of process Peterson in which two enter actions occur one after the other without any exit action in between them. For instance, process P1 might perform action enter1, and the next observable action might be enter2—causing both processes to be in their critical section at the same time. A way to check whether this undesirable situation can ever occur in the behaviour of process Peterson is to make it interact with a ‘monitor process’ that observes the behaviour of process Peterson, and reports an error if and when the undesirable situation arises. This is a conceptually simple, but very useful, technique that has arisen in various forms over and over again in the study of verification techniques for reactive systems, and probably finds its most theoretically satisfying embodiment in the classic automata-theoretic approach to verification—see, for instance, the references (Vardi, 1991; Vardi and Wolper, 1994). So, how can we construct a monitor process that reports a failure in ensuring mutual exclusion if any arises? Intuitively, such a process would observe the enter and exit actions performed by process Peterson. Whenever an enter action is observed, the monitor process reaches a state in which it is ready to report that something bad has happened if it observes that the other process can now enter its critical section as well. If our monitor process observes the relevant exit action as expected, it gladly returns to its initial state, ready to observe the next round of the execution of the algorithm. A CCS process term describing the above behaviour
7.3. TESTING MUTUAL EXCLUSION 169 is, for instance, MutexTest MutexTest1 MutexTest2 def = enter1.MutexTest1 + enter2.MutexTest2 def = exit1.MutexTest + enter2.bad.0 def = exit2.MutexTest + enter1.bad.0 , where we have assumed that our monitor process outputs on channel name bad, when it discovers that two enter actions have occurred without an intervening exit. In order to check whether process Peterson ensures mutual exclusion, it is now sufficient to let it interact with MutexTest, and ask whether the resulting system (Peterson | MutexTest) \ {enter1, enter2, exit1, exit2} can initially perform the action bad. Indeed, we have the following result. Proposition 7.2 Let P be a CCS process whose only visible actions are contained in the set L ′ = {enter1, enter2, exit1, exit2}. Then (P | MutexTest)\L ′ bad ⇒ iff either P σ ⇒ P ′ enter1 ⇒ P ′′ enter2 ⇒ P ′′′ σ or P ⇒ P ′ enter2 ⇒ P ′′ enter1 ⇒ P ′′′ , for some P ′ , P ′′ , P ′′′ and some sequence of actions σ in the regular language (enter1exit1+enter2exit2) ∗ . Proof: For the ‘if implication’, assume, without loss of generality, that P σ ⇒ P ′ enter1 ′′ enter2 ′′′ ⇒ P ⇒ P for some P ′ , P ′′ , P ′′′ and sequence of actions σ ∈ (enter1exit1 +enter2exit2) ∗ . We shall argue that (P | MutexTest) \ L ′ bad ⇒. To see this, note that, using induction on the length of the sequence σ, it is not hard to prove that (P | MutexTest) \ L ′ τ ⇒ (P ′ | MutexTest) \ L ′ Since P ′ enter1 ⇒ P ′′ enter2 ⇒ P ′′′ , we have that (P ′ | MutexTest) \ L ′ τ ⇒ (P ′′ | MutexTest1) \ L ′ τ ⇒ (P ′′′ | bad.0) \ L ′ bad → . Combining the above sequences of transitions, we may conclude that which was to be shown. (P | MutexTest) \ L ′ bad ⇒ , , .
Page 1:
Reactive Systems: Modelling, Specif
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 ✬
Page 30 and 31:
12 CHAPTER 2. THE LANGUAGE CCS Exer
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
18 CHAPTER 2. THE LANGUAGE CCS CS d
Page 38 and 39:
20 CHAPTER 2. THE LANGUAGE CCS Sinc
Page 40 and 41:
22 CHAPTER 2. THE LANGUAGE CCS a p
Page 42 and 43:
24 CHAPTER 2. THE LANGUAGE CCS •
Page 44 and 45:
26 CHAPTER 2. THE LANGUAGE CCS Defi
Page 46 and 47:
28 CHAPTER 2. THE LANGUAGE CCS we h
Page 48 and 49:
30 CHAPTER 2. THE LANGUAGE CCS This
Page 50 and 51:
32 CHAPTER 2. THE LANGUAGE CCS 2. D
Page 52 and 53:
34 CHAPTER 2. THE LANGUAGE CCS To b
Page 55 and 56:
Chapter 3 Behavioural equivalences
Page 57 and 58:
3.1. CRITERIA FOR A GOOD BEHAVIOURA
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
Page 89 and 90:
3.4. WEAK BISIMILARITY 71 In light
Page 91 and 92:
3.5. GAME CHARACTERIZATION OF BISIM
Page 93 and 94:
CHAPTER 3.5. GAME CHARACTERIZATION
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
3.6. FURTHER RESULTS ON EQUIVALENCE
Page 101:
3.6. FURTHER RESULTS ON EQUIVALENCE
Page 104 and 105:
86 CHAPTER 4. THEORY OF FIXED POINT
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 119 and 120:
Chapter 5 Hennessy-Milner logic In
Page 121 and 122:
103 We are interested in using the
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
Page 131 and 132:
a s s1 b s2 b a a t b t1 b t
Page 133 and 134:
Chapter 6 Hennessy-Milner logic wit
Page 135 and 136: CHAPTER 6. HML WITH RECURSION 117 e
Page 137 and 138: CHAPTER 6. HML WITH RECURSION 119
Page 139 and 140: 6.1. EXAMPLES OF RECURSIVE PROPERTI
Page 141 and 142: 6.2. SYNTAX AND SEMANTICS OF HML WI
Page 143 and 144: 6.2. SYNTAX AND SEMANTICS OF HML WI
Page 145 and 146: 6.3. LARGEST FIXED POINTS AND INVAR
Page 147 and 148: 6.4. GAME CHARACTERIZATION FOR HML
Page 153 and 154: 6.5. MUTUALLY RECURSIVE EQUATIONAL
Page 155 and 156: 6.5. MUTUALLY RECURSIVE EQUATIONAL
Page 157 and 158: 6.6. CHARACTERISTIC PROPERTIES 139
Page 167 and 168: 6.7. MIXING LARGEST AND LEAST FIXED
Page 173 and 174: 6.8. FURTHER RESULTS ON MODEL CHECK
Page 175 and 176: Chapter 7 Modelling and analysis of
Page 177 and 178: CHAPTER 7. MODELLING MUTUAL EXCLUSI
Page 179 and 180: CHAPTER 7. MODELLING MUTUAL EXCLUSI
Page 181 and 182: 7.1. SPECIFYING MUTUAL EXCLUSION IN
Page 183 and 184: 7.2. SPECIFYING MUTUAL EXCLUSION US
Page 185: 7.2. SPECIFYING MUTUAL EXCLUSION US
Page 189 and 190: 7.3. TESTING MUTUAL EXCLUSION 171 D
Page 191 and 192: 7.3. TESTING MUTUAL EXCLUSION 173 1
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?