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

More documents

Recommendations

Info

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

124 CHAPTER 6. HML WITH RECURSION The second equation states the, equally obvious, fact that every state satisfies tt irrespective of the set of states that are assumed to satisfy X. The last equation instead says that to calculate the set of states satisfying the formula [a]F under the assumption that the states in S satisfy X, it is sufficient to 1. compute the set of states satisfying the formula F under the assumption that the states in S satisfy X, and then 2. find the collection of states that end up in that set no matter how they perform an a-labelled transition. Exercise 6.4 Given the transition graph from Example 6.2, use the above definition to calculate O [b]ff∧[a]X({p2}). One can show that for every formula F , the function OF is monotonic (see Definition 4.4) over the complete lattice (2 Proc , ⊆). In other words, for all subsets S1, S2 of Proc, if S1 ⊆ S2 then OF (S1) ⊆ OF (S2). Exercise 6.5 Show that OF is monotonic for all F . Consider what will happen if we introduce negation into our logic. Hint: Use structural induction on F . As mentioned before, the idea underlying the definition of the function OF is that if [X ] ⊆ Proc gives the set of processes that satisfy X, then OF ([X ]) will be the set of processes that satisfy F . What is this set [X ] then? Syntactically we shall assume that [X ] is implicitly given by a recursive equation for X of the form X min = FX or X max = FX . As shown in the previous section, such an equation can be interpreted as the set equation [X ] = OFX ([X ]) . (6.6) As OFX is a monotonic function over a complete lattice we know that (6.6) has solutions, i.e., that OFX has fixed points. In particular Tarski’s fixed point theorem (see Theorem 4.1) gives us that there is a unique largest fixed point, denoted by FIX OFX , and also a unique least one, denoted by fix OFX , given respectively by FIX OFX = {S ⊆ Proc | S ⊆ OFX (S)} and fix OFX = {S ⊆ Proc | OFX (S) ⊆ S} . A set S with the property that S ⊆ OFX (S) is called a post-fixed point for OFX . Correspondingly S is a pre-fixed point for OFX if OFX (S) ⊆ S.
6.2. SYNTAX AND SEMANTICS OF HML WITH RECURSION 125 In what follows, for a function f : 2 Proc −→ 2 Proc we define f 0 = id (the identity function on 2 Proc ), and f m+1 = f ◦ f m . When Proc is finite we have the following characterization of the largest and least fixed points. Theorem 6.1 If Proc is finite then FIX OFX = (OFX )M (Proc) for some M and fix OFX = (OFX )m (∅) for some m. Proof: This follows directly from the fixed point theorem for finite complete lattices. See Theorem 4.2 for the details. ✷ The above theorem gives us an algorithm for computing the least and largest set of processes solving an equation of the form (6.6). Consider, by way of example, the formula X max = FX , where FX = 〈b〉tt ∧ [b]X. The set of processes in the labelled transition system s a t a a b s1 s2 t1 that satisfy this property is the largest solution to the equation [X ] = (〈·b·〉{s, s1, s2, t, t1}) ∩ [·b·][X ] . This solution is nothing but the largest fixed point of the set function defined by the right-hand side of the above equation—that is, the function mapping each set of states S to the set OFX (S) = (〈·b·〉{s, s1, s2, t, t1}) ∩ [·b·]S . Since we are looking for the largest fixed point of this function, we begin the iterative algorithm by taking S = {s, s1, s2, t, t1}, the set of all states in our labelled b b
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 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 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: 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 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?