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

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6 CHAPTER 1. INTRODUCTION • software running in embedded system devices like mobile telephones. At a high level of abstraction, the behaviour of a control program can be seen to be governed by the following pseudo-code algorithm skeleton. loop read the sensors’ values at regular intervals depending on the sensors’ values trigger the relevant actuators forever The aforementioned examples, and many others, are examples of computing systems that interact with their environment by exchanging information with it. Like the neurons in a human brain, these systems react to stimuli from their computing environment (in the example control program above these are variations in the values of the sensors) by possibly changing their state or mode of computation, and in turn influence their environment by sending back some signals to it, or initiating some operations whose effect it is to affect the computing environment (this is the role played by the actuators in the example control program). David Harel and Amir Pnueli coined the term reactive system in (Harel and Pnueli, 1985) to describe a system that, like the aforementioned ones, computes by reacting to stimuli from its environment. As the above examples and discussion indicate, reactive systems are inherently parallel systems, and a key role in their behaviour is played by communication and interaction with their computing environment. A ‘standard’ computing system can also be viewed as a reactive system in which interaction with the environment only takes place at the beginning of the computation (when inputs are fed to the computing device) and at the end (when the output is received). On the other hand, all the example systems given before maintain a continuous interaction with their environment, and we may think of both the computing system and its environment as parallel processes that communicate one with the other. In addition, as again nicely exemplified by the skeleton of a control program given above, non-termination is a desirable feature of some reactive systems. In contrast to the setting of ‘standard’ computing systems, we certainly do not expect the operating systems running on our computers or the control program monitoring a nuclear reactor to terminate! Now that we have an idea of what reactive systems are, and of the key aspects of their behaviour, we can begin to consider what an appropriate abstract model for this class of systems should offer. In particular, such a model should allow us to describe the behaviour of collections of (possibly non-terminating) parallel processes that may compute independently and interact with one another. It should provide us with facilities for the description of well-known phenomena that appear in the presence of concurrency and are familiar to us from the world of operating
1.2. PROCESS ALGEBRAS 7 systems and parallel computation in general (e.g., deadlock, livelock, starvation and so on). Finally, in order to abstract from implementation dependent issues having to do with, e.g., scheduling policies, the chosen model should permit a clean description of non-determinism—a most useful modelling tool in computer science. Our aim in the remainder of this book will be to present a general purpose theory that can be used to describe, and reason about, any collection of interacting processes. The approach we shall present will make use of a collection of models and formal techniques that is often referred to as Process Theory. The key ingredients in this approach are • (Process) Algebra, • Automata/labelled transition systems, • Structural Operational Semantics, and • Logic. These ingredients give the foundations for the development of (semi-)automatic verification tools for reactive systems that support various formal methods for validation and verification that can be applied to the analysis of highly non-trivial computing systems. The development of these tools requires in turn advances in algorithmics, and via the increasing complexity of the analyzed designs feeds back to the theory development phase by suggesting the invention of new languages and models for the description of reactive systems. Unlike in the setting of sequential programs, where we often kid ourselves into believing that the development of correct programs can be done without any recourse to ‘formalism’, it is a well-recognized fact of life that the behaviour of even very short parallel programs may be very hard to analyze and understand. Indeed, analyzing these programs requires a careful consideration of issues related to the interactions amongst their components, and even imagining all of these is often a mind-boggling task. As a result, the techniques and tools that we shall present in this book are becoming widely accepted in the academic and industrial communities that develop reactive systems. 1.2 Process algebras The first ingredient in the approach to the theory of reactive systems presented in this book is a prototypical example of a process algebra. Process algebras are
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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
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Page 19: Part I A classic theory of reactive
Page 22 and 23: 4 CHAPTER 1. INTRODUCTION After hav
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Page 55 and 56: Chapter 3 Behavioural equivalences
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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 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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