Informal overview of discrete models in Software Deploy Data Matrix ECC200 in Software Informal overview of discrete models

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1.8 Informal overview of discrete models using barcode integrated for software control to generate, create data matrix barcode image in software applications. Developing with Visual Studio .NET the complexity of models Data Matrix barcode for None by means of three concepts: re nement, decomposition, and generic development.. 1.8.1 State and transitio ns Roughly speaking, a discrete model is made of a state represented by some constants and variables at a certain level of abstraction with regard to the real system under study.

Such variables are very much the same as those used in applied sciences (physics, biology, operational research) for studying natural systems. In such sciences, people also build models. It helps them to infer some laws about the real world by means of reasoning about these models.

Besides the state, the model also contains a number of transitions that can occur under certain circumstances. Such transitions are called here events. Each event is rst made of a guard , which is a predicate built on the state constants and variables.

It represents the necessary conditions for the event to occur. Each event is also made up of an action, which describes the way certain state variables are modi ed as a consequence of the event occurrence..

1.8.2 Operational interpr Data Matrix 2d barcode for None etation As can be seen, a discrete dynamical model thus indeed constitutes a kind of state transition machine.

We can give such a machine an extremely simple operational interpretation. Notice that such an interpretation should not be considered as providing any operational semantics in our models (this will be given later by means of a proof system), it is just given here to support their informal understanding. First of all, the execution of an event, which describes a certain observable transition of the state variables, is considered to take no time.

Moreover, no two events can occur simultaneously. The execution is then the following: When no event guards are true, then the model execution stops; it is said to have deadlocked . When some event guards are true, then one of the corresponding events necessarily occurs and the state is modi ed accordingly; subsequently, the guards are checked again, and so on.

This behavior clearly shows some possible non-determinism (called external nondeterminism) as several guards may be true simultaneously. We make no assumption concerning the speci c event which is indeed executed among those whose guards are true. When only one guard is true at all times, the model is said to be deterministic.

Notice that the fact that a model eventually nishes is not at all mandatory. As a matter of fact, most of the systems we study never deadlock; they run for ever..

Introduction 1.8.3 Formal reasoning Th Data Matrix ECC200 for None e very elementary transition machine we have described in the previous section, although primitive is nevertheless su ciently elaborate to allow us to undertake some interesting formal reasoning.

In the following, we envisage two kinds of discrete model properties. The rst kind of properties that we want to prove about our models, and hence ultimately about our real systems, are so-called invariant properties. An invariant is a condition on the state variables that must hold permanently.

In order to achieve this, it is just required to prove that, under the invariant in question and under the guard of each event, the invariant still holds after being modi ed according to the action associated with that event. We might also consider more complicated forms of reasoning, involving conditions which, in contrast to the invariants, do not hold permanently. The corresponding statements are called modalities.

In our approach, we only consider a very special form of modality, called reachability. What we would like to prove is that an event whose guard is not necessarily true now will nevertheless certainly occur within a certain nite time. 1.

8.4 Managing the complexity of closed models Note that the models we are going to construct will not just describe the control part of our intended system, they will also contain a certain representation of the environment within which the system we build is supposed to behave. In fact, we shall quite often essentially construct closed models, which are able to exhibit the actions and reactions taking place between a certain environment and a corresponding, possibly distributed, controller.

In doing so, we shall be able to insert the model of the controller within an abstraction of its environment, which is formalized as yet another model. The state of such a closed system thus contains physical variables, describing the environment state, as well as logical variables, describing the controller state. And, in the same way, the transitions will fall into two groups: those concerned with the environment and those concerned with the controller.

We shall also have to put into the model the way these two entities communicate. But, as we mentioned earlier, the number of transitions in the real systems under study is certainly enormous. And, needless to say, the number of variables describing the state of such systems is also extremely large.

How are we going to practically manage such complexity The answer to this question lies in three concepts: re nement (Section 1.8.5), decomposition (Section 1.

8.6), and generic instantiation (Section 1.8.

7). It is important to notice here that these concepts are linked together. As a matter of fact, we re ne a model to later decompose it, and, more importantly, we decompose it.

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