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

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4 CHAPTER 1. INTRODUCTION After having worked through the material in this book, you will be able to describe non-trivial reactive systems and their specifications using the aforementioned models, and verify the correctness of a model of a system with respect to given specifications either manually or by using automatic verification tools like the Edinburgh Concurrency Workbench (Cleaveland et al., 1993) and the model checker for real-time systems UPPAAL (Behrmann et al., 2004). Our, somewhat ambitious, aim is therefore to present a model of reactive systems that supports their design, specification and verification. Moreover, since many real-life systems are hard to analyze manually, we should like to have computer support for our verification tasks. This means that all the models and languages that we shall use in this book need to have a formal syntax and semantics. (The syntax of a language consists of the rules governing the formation of statements, whereas its semantics assigns meaning to each of the syntactically correct statements in the language.) These requirements of formality are not only necessary in order to be able to build computer tools for the analysis of systems’ descriptions, but are also fundamental in agreeing upon what the terms in our models are actually intended to describe in the first place. Moreover, as Donald Knuth once wrote: A person does not really understand something until after teaching it to a computer, i.e. expressing it as an algorithm.. . . An attempt to formalize things as algorithms leads to a much deeper understanding than if we simply try to comprehend things in the traditional way. The pay-off of using formal models with an explicit formal semantics to describe our systems will therefore be the possibility of devising algorithms for the animation, simulation and verification of system models. These would be impossible to obtain if our models were specified only in an informal notation. Now that we know what to expect from this book, it is time to get to work. We shall begin our journey through the beautiful land of Concurrency Theory by introducing a prototype description language for reactive systems and its semantics. However, before setting off on such an enterprise, we should describe in more detail what we actually mean with the term ‘reactive system’. 1.1 What are reactive systems? The ‘standard’ view of computing systems is that, at a high level of abstraction, these may be considered as black boxes that take inputs and provide appropriate outputs. This view agrees with the description of algorithmic problems. An algorithmic problem is specified by a collection of legal inputs, and, for each legal
1.1. WHAT ARE REACTIVE SYSTEMS? 5 input, its expected output. In an imperative setting, an abstract view of a computing system may therefore be given by describing how it transforms an initial state— that is, a function from variables to their values—to a final state. This function will, in general, be partial—that is, it may be undefined for some initial states—to capture that the behaviour of a computing system may be non-terminating for some input states. For example, the effect of the program S = z ← x; x ← y; y ← z is described by the function [S ] from states to states defined thus: [S ] = λs. s[x ↦→ s(y), y ↦→ s(x), z ↦→ s(x)] , where the state s[x ↦→ s(y), y ↦→ s(x), z ↦→ s(x)] is the one in which the value of variable x is the value of y in state s and that of variables y and z is the value of x in state s. The values of all of the other variables are those they had in state s. This state transformation is a way of formally describing that the intended effect of S is essentially to swap the values of the variables x and y. On the other hand, the effect of the program U = while true do skip , where we use skip to stand for a ‘no operation’, is described by the partial function from states to states given by [U ] = λs. undefined , that is the always undefined function. This captures the fact that the computation of U never produces a result (final state) irrespective of the initial state. In this view of computing systems, non-termination is a highly undesirable phenomenon. An algorithm that fails to terminate on some inputs is not one the users of a computing system would expect to have to use. A moment of reflection, however, should make us realize that we already use many computing systems whose behaviour cannot be readily described as a function from inputs to outputs— not least because, at some level of abstraction, these systems are inherently meant to be non-terminating. Examples of such computing systems are • operating systems, • communication protocols, • control programs, and
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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 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 ✬
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Page 38 and 39: 20 CHAPTER 2. THE LANGUAGE CCS Sinc
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Page 44 and 45: 26 CHAPTER 2. THE LANGUAGE CCS Defi
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Page 55 and 56: Chapter 3 Behavioural equivalences
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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
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3.3. STRONG BISIMILARITY 55 3. the
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3.3. STRONG BISIMILARITY 57 Recall
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3.3. STRONG BISIMILARITY 59 B 2 0
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3.4. WEAK BISIMILARITY 61 Is there
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3.4. WEAK BISIMILARITY 63 Start pu
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3.4. WEAK BISIMILARITY 65 Our order
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3.4. WEAK BISIMILARITY 67 Send def
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3.4. WEAK BISIMILARITY 69 Show that
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3.4. WEAK BISIMILARITY 71 In light
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3.5. GAME CHARACTERIZATION OF BISIM
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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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7.3. TESTING MUTUAL EXCLUSION 173 1
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