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

More documents

Recommendations

Info

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

162 CHAPTER 7. MODELLING MUTUAL EXCLUSION ALGORITHMS while true do begin ‘noncritical section’; bi := true; while k = j do begin while bj do skip; k:=i end; ‘critical section’; bi := false; end Figure 7.1: The pseudocode for Hyman’s algorithm How can we formalize this requirement? There are at least two options for doing so, depending on whether we wish to use HML with recursion or CCS processes/labelled transition systems as our specification formalism. In order to gain experience in the use of both approaches to specification and verification, in what follows we shall present specifications for mutual exclusion using HML with recursion and CCS. 7.1 Specifying mutual exclusion in HML Hennessy-Milner logic with recursion is an excellent formalism for specifying our informal correctness condition for Peterson’s algorithm. To see this, observe, first of all, that the aforementioned desideratum is really a safety property in that it intuitively states that a desirable state of affairs—namely that ‘it is not possible for both processes to be in their critical sections at the same time’—is maintained throughout the execution of the process Peterson. We already saw in Chapter 6 that safety properties can be specified in HML with recursion using formulae of the form Inv(F ), where F is the ‘desirable property’ that we wish to hold at all points in the execution of the process. Recall that Inv(F ) is nothing but a shorthand for the recursively defined formula Inv(F ) max = F ∧ [Act]Inv(F ) . So all that we are left to do in order to formalize our requirement for mutual exclusion is to give a formula F in HML describing the requirement that:
7.1. SPECIFYING MUTUAL EXCLUSION IN HML 163 It is not possible for both processes to be in their critical sections at the same time. In light of our CCS formalization of the processes P1 and P2, we know that process Pi (i ∈ {1, 2}) is in its critical section precisely when it can perform action exiti. So our formula F can be taken to be F def = [exit1]ff ∨ [exit2]ff . The formula Inv(F ) now states that it is invariantly the case that either P1 is not in the critical section or that P2 is not in the critical section, which is an equivalent formulation of our correctness criterion. Throughout this chapter, we are interpreting the modalities in HML over the transition system whose states are CCS processes, and whose transitions are weak transitions of the form α ⇒ for any action α including τ. So a formula like [exit1]ff is satisfied by all processes that do not afford an exit1 ⇒ -labelled transition—that is, by those processes that cannot perform action exit1 no matter how many internal steps they do before. Exercise 7.4 Consider the formula Inv(G), where G is ([enter1][enter2]ff) ∧ ([enter2][enter1]ff) . Would such a formula be a good specification for our correctness criterion? What if we took G to be the formula (〈enter1〉[enter2]ff) ∧ (〈enter2〉[enter1]ff) ? Argue for your answers! Now that we have a formal description of Peterson’s algorithm, and a specification of a correctness criterion for it, we could try to establish whether process Peterson satisfies the formula Inv(F ) or not. With some painstaking effort, this could be done manually either by showing that the set of states of the process Peterson is a post-fixed point of the set function associated with the mapping S ↦→ [F ] ∩ [·Act·]S , or by iteratively computing the largest fixed point of the above mapping. The good news, however, is that we do not need to do so! One of the benefits of having formal specifications of systems and of their correctness criteria is that, at least in
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: CHAPTER 7. MODELLING MUTUAL EXCLUSI
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?