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

More documents

Recommendations

Info

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

44 CHAPTER 3. BEHAVIOURAL EQUIVALENCES processes. Intuitively, a strong bisimulation is a kind of invariant relation between processes that is preserved by transitions in the sense of Definition 3.2. Before beginning to explore the properties of strong bisimilarity, let us remark one of its most appealing features, namely a proof technique that it supports to show that two processes are strongly bisimilar. Since two processes are strongly bisimilar if there is a strong bisimulation that relates them, to prove that they are related by ∼ it suffices only to exhibit a strong bisimulation that relates them. Example 3.1 Consider the labelled transition system where • Proc = {s, s1, s2, t, t1}, • Act = {a, b}, • a →= {(s, s1), (s, s2), (t, t1)}, and • b →= {(s1, s2), (s2, s2), (t1, t1)}. (Proc, Act, { α →| α ∈ Act}) , Here is a graphical representation of this labelled transition system. s a t a a b s1 s2 t1 We will show that s ∼ t. In order to do so, we have to define a strong bisimulation R such that (s, t) ∈ R. Let us define it as R = {(s, t), (s1, t1), (s2, t1)} . The binary relation R can be graphically depicted by dotted lines like in the following picture. s a t a a b s1 s2 t1 b b b b
3.3. STRONG BISIMILARITY 45 Obviously, (s, t) ∈ R. We have to show that R is a strong bisimulation, i.e., that it meets the requirements stated in Definition 3.2. To this end, for each pair of states from R, we have to investigate all the possible transitions from both states and see whether they can be matched by corresponding transitions from the other state. Note that a transition under some label can be matched only by a transition under the same label. We will now present a complete analysis of all of the steps needed to show that R is a strong bisimulation, even though they are very simple and tedious. • Let us consider first the pair (s, t): – transitions from s: ∗ s a → s1 can be matched by t a → t1 and (s1, t1) ∈ R, ∗ s a → s2 can be matched by t a → t1 and (s2, t1) ∈ R, and ∗ these are all the transitions from s; – transitions from t: ∗ t a → t1 can be matched, e.g, by s a → s2 and (s2, t1) ∈ R (another possibility would be to match it by s a → s1 but finding one matching transition is enough), and ∗ this is the only transition from t. • Next we consider the pair (s1, t1): – transitions from s1: ∗ s1 b → s2 can be matched by t1 b → t1 and (s2, t1) ∈ R, and ∗ this is the only transition from s1; – transitions from t1: ∗ t1 b → t1 can be matched by s1 b → s2 and (s2, t1) ∈ R, and ∗ this is the only transition from t1. • Finally we consider the pair (s2, t1): – transitions from s2: ∗ s2 b → s2 can be matched by t1 b → t1 and (s2, t1) ∈ R, and ∗ this is the only transition from s2; – transitions from t1: ∗ t1 b → t1 can be matched by s2 b → s2 and (s2, t1) ∈ R, and ∗ this is the only transition from t1.
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 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: 3.3. STRONG BISIMILARITY 43 trace e
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 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 112 and 113:
94 CHAPTER 4. THEORY OF FIXED POINT
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 149 and 150:
Page 151 and 152:
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 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
6.7. MIXING LARGEST AND LEAST FIXED
Page 169 and 170:
Page 171 and 172:
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 and 186:
7.2. SPECIFYING MUTUAL EXCLUSION US
Page 187 and 188:
7.3. TESTING MUTUAL EXCLUSION 169 i
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

Create successful ePaper yourself

Delete template?

Save as template?