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

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40 CHAPTER 3. BEHAVIOURAL EQUIVALENCES The lack of consensus on what constitutes an appropriate notion of observable behaviour for reactive systems has led to a large number of proposals for behavioural equivalences for concurrent processes. (See the study (Glabbeek, 2001), where van Glabbeek presents the linear time-branching time spectrum—a lattice of known behavioural equivalences and preorders over labelled transition systems, ordered by inclusion.) In our search for a reasonable notion of behavioural relation to support implementation verification, we shall limit ourselves to presenting a tiny sample of these. So let’s begin our search! 3.2 Trace equivalence: a first attempt Labelled transition systems (LTSs) (Keller, 1976) are a fundamental model of concurrent computation, which is widely used in light of its flexibility and applicability. In particular, they are the prime model underlying Plotkin’s Structural Operational Semantics (Plotkin, 2004b) and, following Milner’s pioneering work on CCS (Milner, 1989), are by now the standard semantic model for various process description languages. As we have already seen, LTSs model processes by explicitly describing their states and their transitions from state to state, together with the actions that produced them. Since this view of process behaviours is very detailed, several notions of behavioural equivalence and preorder have been proposed for LTSs. The aim of such behavioural semantics is to identify those (states of) LTSs that afford the same ‘observations’, in some appropriate technical sense. Now, LTSs are essentially (possibly infinite state) automata, and the classic theory of automata suggests a ready made notion of equivalence for them, and thus for the CCS processes that denote them. Let us say that a trace of a process P is a sequence α1 · · · αk ∈ Act ∗ (k ≥ 0) such that there exists a sequence of transitions P = P0 α1 α2 αk → P1 → P2 · · · Pk−1 → Pk , for some P1, . . . , Pk. We write Traces(P ) for the collection of all traces of P . Since Traces(P ) describes all the possible finite sequences of interactions that we may have with process P , it is reasonable to require that our notion of behavioural equivalence only relates processes that afford the same traces, or else we should have a very good reason for telling them apart—namely a sequence of actions that can be performed with one, but not with the other. This means that, for all processes P and Q, we require that if P and Q are behaviourally equivalent, then Traces(P ) = Traces(Q) . (3.1)
3.2. TRACE EQUIVALENCE: A FIRST ATTEMPT 41 Taking the point of view of standard automata theory, and abstracting from the notion of ‘accept state’ that is missing altogether in our treatment, an automaton may be completely identified by its set of traces, and thus two processes are equivalent if, and only if, they afford the same traces. This point of view is totally justified and natural if we view our LTSs as nondeterministic devices that may generate or accept sequences of actions. However, is it still a reasonable one if we view our automata as reactive machines that interact with their environment? To answer this questions, consider the coffee and tea machine CTM defined in equation 2.2 on page 11, and compare it with the following one: CTM ′ def = coin.coffee.CTM ′ + coin.tea.CTM ′ . (3.2) You should be able to convince yourselves that CTM and CTM ′ afford the same traces. (Do so!) However, if you were a user of the coffee and tea machine who wants coffee and hates tea, which machine would you like to interact with? We certainly would prefer to interact with CTM as that machine will give us coffee after receiving a coin, whereas CTM ′ may refuse to deliver coffee after having accepted our coin! This informal discussion may be directly formalized within CCS by assuming that the behaviour of the coffee starved user is described by the process Consider now the terms and CA def = coin.coffee.CA . (CA | CTM) \ {coin, coffee, tea} (CA | CTM ′ ) \ {coin, coffee, tea} that we obtain by forcing interaction between the coffee addict CA and the two vending machines. Using the SOS rules for CCS, you should convince yourselves that the former term can only perform an infinite computation consisting of τlabelled transitions, whereas the second term can deadlock thus: (CA | CTM ′ ) \ {coin, coffee, tea} τ → (coffee.CA | tea.CTM ′ ) \ {coin, coffee, tea} . Note that the target term of this transition captures precisely the deadlock situation that we intuitively expected to have, namely that the user only wants coffee, but the machine is only willing to deliver tea. So trace equivalent terms may exhibit
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Reactive Systems: Modelling, Specif
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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
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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 63 and 64: 3.3. STRONG BISIMILARITY 45 Obvious
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90 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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