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

More documents

Recommendations

Info

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

152 CHAPTER 6. HML WITH RECURSION Intuitively, we have so far discovered the (obvious!) fact that p has a possibility of reaching a state where a livelock may arise because p has a livelock now. Our second approximation S (2) is obtained by computing the set obtained by evaluating the expression on the right-hand side of equation (6.16) when S = S (1) = {p}. The result is S (2) = {p} ∪ 〈·Act·〉{p} = {s, p} . Intuitively, we have now discovered the new fact that s has a possibility of reaching a state where a livelock may arise because s has a transition leading to p, which, as we found out in the previous approximation, has itself a possibility of reaching a livelock. You should now be able to convince yourselves that the set {s, p} is indeed a fixed point of equation (6.16)—that is, that {s, p} = {p} ∪ 〈·Act·〉{s, p} . It follows that {s, p} is the least solution of equation (6.16), and that the states s and p are the only ones in our example labelled transition system that satisfy the formula P os(LivelockNow). This makes perfect sense intuitively because s and p are the only states in that labelled transition system that afford a sequence of transitions leading to a state from which an infinite computation consisting of τlabelled transitions is possible. (In case of p, this sequence is empty since p can embark in a τ-loop immediately.) Note that we could find the set of states satisfying P os(LivelockNow) by first computing [LivelockNow], and then using this set to compute [P os(LivelockNow)] , because the specification of the formula LivelockNow was independent of that P os(LivelockNow). In general, we can apply this strategy when the collection of equations can be partitioned into a sequence of ‘blocks’ such that • the equations in the same block are all either largest fixed point equations or least fixed equations, and • equations in each block only use variables defined in that block or in preceding ones. The following definition formalizes this class of systems of equations. Definition 6.2 A n-nested mutually recursive equational system E is an n-tuple 〈 (D1, X1, m1), (D2, X2, m2), . . . , (Dn, Xn, mn) 〉, where the Xis are pairwise disjoint, finite sets of variables, and, for each 1 ≤ i ≤ n,
6.7. MIXING LARGEST AND LEAST FIXED POINTS 153 • Di is a declaration mapping the variables in the set Xi to formulae in HML with recursion that may use variables in the set 1≤j≤i Xj, • mi = max or mi = min, and • mi = mi+1. We refer to (Di, Xi, mi) as the ith block of E and say that it is a maximal block if mi = max and a minimal block otherwise. Observe that our earlier specification of the formula P os(LivelockNow) is given in terms of a 2-nested mutually recursive equational system. In fact, take X1 = {LivelockNow} and X2 = {P os(LivelockNow)}. You can now easily check that the constraints in the above definition are met. On the other hand, the mixed equational system X max = 〈a〉Y Y min = 〈b〉X does not meet these requirements because the variables X and Y are both defined in mutually recursive fashion, and their definitions refer to different types of fixed points. If we allow fixed points to be mixed completely freely we obtain the modal µ-calculus (Kozen, 1983), which was mentioned in Section 6.1. In this book we shall however not allow a full freedom in mixing fixed points in declarations but restrict ourselves to systems of equations satisfying the constraints in Definition 6.2. Note that employing the approach described above using our running example in this section, such systems of equations have a unique solution, obtained by solving the first block and then proceeding with the others using the solutions already obtained for the preceding blocks. Finally if F is a Hennessy-Milner formula defined over a set of variables Y = {Y1, . . . , Yk} that are declared by an n-nested mutually recursive equational system E, then [F ] is well-defined and can be expressed by [F ] = OF ([Y1 ], . . . , [Yk ]) , (6.17) where [Y1 ], . . . , [Yk ] are the sets of states satisfying the recursively defined formulae associated with the variables Y1, . . . , Yk. Exercise 6.18 Consider the labelled transition system in Exercise 6.17. Use equation (6.17) to compute the set of states satisfying the formula F = 〈Act〉P os(LivelockNow) .
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 169: 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?