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

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34 CHAPTER 2. THE LANGUAGE CCS To become familiar with these rules, you should apply them to the one-place buffer B, and derive its possible transitions. In what follows, we shall restrict ourselves to CCS expressions that have no free occurrences of value variables—that is, to CCS expressions in which each occurrence of a value variable, say y, is within the scope of an input prefix of the form a(y) or of a parameterized constant A(x1, . . . , xn) with y = xi for some 1 ≤ i ≤ n. For instance, the expression a(x). ¯ b(y + 1).0 is disallowed because the single occurrence of the value variable y is bound neither by an input prefixing nor by a parameterized constant. Since processes in value passing CCS may manipulate data, it is natural to add an if bexp then P else Q construct to the language, where bexp is a boolean expression. Assume, by way of example, that we wish to define a one-place buffer Pred that computes the predecessor function on the non-negative integers. This may be defined thus: Pred def = in(x).Pred(x) Pred(x) def = if x = 0 then out(0).Pred else out(x − 1).Pred . We expect Pred(0) to output the value 0 on channel ‘out’, and Pred(n + 1) to output n on the same channel for each non-negative integer n. The SOS rules for if bexp then P else Q will allow us to prove this formally. They are the expected ones, namely and P α → P ′ if bexp then P else Q α → P ′ Q α → Q ′ if bexp then P else Q α → Q ′ Exercise 2.12 Consider a one place buffer defined by Cell Cell(x) def = in(x).Cell(x) def = out(x).Cell . bexp is true bexp is false . Use the Cell to define a two-place bag and a two-place FIFO queue. (Recall that a bag, also known as multiset, is a set whose elements have multiplicity.) Give specifications of the expected behaviour of these processes, and use the operational rules given above to convince yourselves that your implementations are correct.
2.2. CCS, FORMALLY 35 Exercise 2.13 Consider the process B defined thus: C(x) B def = push(x).(C(x) ⌢ B) + empty.B def = push(y).(C(y) ⌢ C(x)) + pop(x).D D def = o(x).C(x) + ē.B , where the linking combinator P ⌢ Q is as follows: P ⌢ Q = (P[p ′ /p, e ′ /e, o ′ /o] | Q[p ′ /push, e ′ /empty, o ′ /pop]) \ {p ′ , o ′ , e ′ } . Draw an initial fragment of the transition graph for this process. What behaviour do you think B implements? Exercise 2.14 (For the theoretically minded) Prove that the operational semantics for value passing CCS we have given above is in complete agreement with the semantics for this language via translation into the pure calculus given by Milner in (Milner, 1989, Section 2.8).
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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
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Page 63 and 64: 3.3. STRONG BISIMILARITY 45 Obvious
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Page 75 and 76: 3.3. STRONG BISIMILARITY 57 Recall
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Page 83 and 84: 3.4. WEAK BISIMILARITY 65 Our order
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Page 87 and 88: 3.4. WEAK BISIMILARITY 69 Show that
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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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