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

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86 CHAPTER 4. THEORY OF FIXED POINTS poset) is a pair (D, ⊑), where D is a set, and ⊑ is a binary relation over D (i.e., a subset of D × D) such that: • ⊑ is reflexive, i.e., d ⊑ d for all d ∈ D; • ⊑ is antisymmetric, i.e., d ⊑ e and e ⊑ d imply d = e for all d, e ∈ D; • ⊑ is transitive, i.e., d ⊑ e ⊑ d ′ implies d ⊑ d ′ for all d, d ′ , e ∈ D. We moreover say that (D, ⊑) is a totally ordered set if, for all d, e ∈ D, either d ⊑ e or e ⊑ d holds. Example 4.1 The following are examples of posets. • (N, ≤), where N denotes the set of natural numbers, and ≤ stands for the standard ordering over N. • (R, ≤), where R denotes the set of real numbers, and ≤ stands for the standard ordering over R. • (A ∗ , ≤), where A ∗ is the set of strings over alphabet A, and ≤ denotes the prefix ordering between strings, i.e., for all s, t ∈ A ∗ , s ≤ t iff there exists w ∈ A ∗ such that sw = t. (Check that this is indeed a poset!) • Let (A, ≤) be a finite totally ordered set. Then (A ∗ , ≺), the set of strings in A ∗ ordered lexicographically, is a poset. Recall that, for all s, t ∈ A ∗ , the relation s ≺ t holds with respect to the lexicographic order if one of the following conditions apply: 1. the length of s is smaller than that of t; 2. s and t have equal length, and either s = ε or there are strings u, v, z ∈ A ∗ and letters a, b ∈ A such that s = uav, t = ubz and a ≤ b. • Let (D, ⊑) be a poset and S be a set. Then the collection of functions from S to D is a poset when equipped with the ordering relation defined thus: f ⊑ g iff f(s) ⊑ g(s), for each s ∈ S . We encourage you to think of other examples of posets you are familiar with. Exercise 4.1 Convince yourselves that the structures mentioned in the above example are indeed posets. Which of the above posets is a totally ordered set?
4.1. POSETS AND COMPLETE LATTICES 87 As witnessed by the list of structures in Example 4.1 and by the many other examples that you have met in your discrete mathematics courses, posets are abundant in mathematics. Another example of a poset that will play an important role in the developments to follow is the structure (2 S , ⊆), where S is a set, 2 S stands for the set of all subsets of S, and ⊆ denotes set inclusion. For instance, the structure (2 Proc , ⊆) is a poset for each set of states Proc in a labelled transition system. Exercise 4.2 Is the poset (2 S , ⊆) totally ordered? Definition 4.2 [Least upper bounds and greatest lower bounds] Let (D, ⊑) be a poset, and take X ⊆ D. • We say that d ∈ D is an upper bound for X iff x ⊑ d for all x ∈ X. We say that d is the least upper bound (lub) of X, notation X, iff – d is an upper bound for X and, moreover, – d ⊑ d ′ for every d ′ ∈ D which is an upper bound for X. • We say that d ∈ D is a lower bound for X iff d ⊑ x for all x ∈ X. We say that d is the greatest lower bound (glb) of X, notation X, iff – d is a lower bound for X and, moreover, – d ′ ⊑ d for every d ′ ∈ D which is a lower bound for X. In the poset (N, ≤), all finite subsets of N have least upper bounds. Indeed, the least upper bound of such a set is its largest element. On the other hand, no infinite subset of N has an upper bound. All subsets of N have a least element, which is their greatest lower bound. In (2 S , ⊆), every subset X of 2 S has a lub and a glb given by X and X, respectively. For example, consider the poset (2 N , ⊆), consisting of the family of subsets of the set of natural numbers N ordered by inclusion. Take X to be the collection of finite sets of even numbers. Then X is the set of even numbers and X is the empty set. (Can you see why?) Exercise 4.3 (Strongly recommended) Let (D, ⊑) be a poset, and take X ⊆ D. Prove that the lub and the glb of X are unique, if they exist. Exercise 4.4 1. Prove that the lub and the glb of a subset X of 2 S are indeed X and X, respectively.
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Reactive Systems: Modelling, Specif
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iv CONTENTS 5 Hennessy-Milner logic
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List of Figures 2.1 The interface f
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List of Tables 2.1 An alternative f
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xii PREFACE It has been very satisf
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xiv PREFACE a textbook, and offers
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xvi PREFACE studies in Edinburgh, a
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Part I A classic theory of reactive
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4 CHAPTER 1. INTRODUCTION After hav
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10 CHAPTER 2. THE LANGUAGE CCS ✬
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Page 63 and 64: 3.3. STRONG BISIMILARITY 45 Obvious
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Page 67 and 68: 3.3. STRONG BISIMILARITY 49 Theorem
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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
Page 85 and 86: 3.4. WEAK BISIMILARITY 67 Send def
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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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7.3. 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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