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

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26 CHAPTER 2. THE LANGUAGE CCS Definition 2.3 The collection P of CCS expressions is given by the following grammar: P, Q ::= K | α.P | Pi | P | Q | P [f] | P \ L , where • K is a process name in K; • α is an action in Act; i∈I • I is a possibly infinite index set; • f : Act → Act is a relabelling function satisfying the following constraints: • L is a set of labels from L. f(τ) = τ and f(ā) = f(a) for each label a ; We write 0 for an empty sum of processes, i.e., 0 = i∈∅ Pi , and P1 + P2 for a sum of two processes, i.e., P1 + P2 = i∈{1,2} Pi . Moreover, we assume that the behaviour of each process constant K ∈ K is given by a defining equation K def = P , where P ∈ P. As it was already made clear by the previous informal discussion, the constant K may appear in P . We sometimes write [b1/a1, . . . , bn/an], where n ≥ 1, ai, bi ∈ A for each i ∈ {1, . . . , n} and the ai are distinct channel names, for the relabelling [f], where f is the relabelling function mapping each ai to bi, each ai to bi (i ∈ {1, . . . , n}) and acting like the identity function on all of the other actions. For each label a, we also often write \a in lieu of \{a}. To avoid the use of too many parentheses in writing CCS expressions, we use the convention that the operators have decreasing binding strength in the following
2.2. CCS, FORMALLY 27 order: restriction and relabelling (tightest binding), action prefixing, parallel composition and summation. For example, the expression a.0 | b.P \ L + c.0 stands for ((a.0) | (b.(P \ L))) + (c.0) . Exercise 2.6 Which of the following expressions are syntactically correct CCS expressions? Why? Assume that A, B are process constants and a, b are channel names. • a.b.A + B • (a.0 + a.A) \ {a, b} • (a.0 | a.A) \ {a, τ}, • a.B + [a/b] • τ.τ.B + 0 • (a.B + b.B)[a/b, b/a] • (a.B + τ.B)[a/τ, b/a] • (a.b.A + a.0) | B • (a.b.A + a.0).B • (a.b.A + a.0) + B • (0 | 0) + 0 Our readers can easily check that all of the processes presented in the previous section are indeed CCS expressions. Another example of a CCS expression is given by a counter, which is defined thus: Counter0 Countern def = up.Counter1 (2.5) def = up.Countern+1 + down.Countern−1 (n > 0) . (2.6) The behaviour of such a process is intuitively clear. For each non-negative integer n, the process Countern behaves like a counter whose value is n; the ‘up’ actions increase the value of the counter by one, and the ‘down’ actions decrease it by one. It would also be easy to construct the (infinite state) LTS for this process based on its syntactic description, and on the intuitive understanding of process behaviour
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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
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Page 63 and 64: 3.3. STRONG BISIMILARITY 45 Obvious
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Page 79 and 80: 3.4. WEAK BISIMILARITY 61 Is there
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Page 83 and 84: 3.4. WEAK BISIMILARITY 65 Our order
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CHAPTER 3.5. GAME CHARACTERIZATION
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CHAPTER 3.5. GAME CHARACTERIZATION
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3.6. FURTHER RESULTS ON EQUIVALENCE
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3.6. FURTHER RESULTS ON EQUIVALENCE
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86 CHAPTER 4. THEORY OF FIXED POINT
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Chapter 5 Hennessy-Milner logic In
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103 We are interested in using the
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105 of the one step transitions a
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1. Decide whether the following sta
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109 Note that logical negation is n
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111 Another example of a process th
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a s s1 b s2 b a a t b t1 b t
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Chapter 6 Hennessy-Milner logic wit
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CHAPTER 6. HML WITH RECURSION 117 e
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CHAPTER 6. HML WITH RECURSION 119
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6.1. EXAMPLES OF RECURSIVE PROPERTI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.3. LARGEST FIXED POINTS AND INVAR
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6.4. GAME CHARACTERIZATION FOR HML
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.6. CHARACTERISTIC PROPERTIES 139
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6.7. MIXING LARGEST AND LEAST FIXED
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6.8. FURTHER RESULTS ON MODEL CHECK
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Chapter 7 Modelling and analysis of
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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7.1. SPECIFYING MUTUAL EXCLUSION IN
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.3. TESTING MUTUAL EXCLUSION 169 i
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7.3. TESTING MUTUAL EXCLUSION 171 D
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Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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