As said above, we rst require that M is included in L and that M is not empty: M L M = in Software Integrated datamatrix 2d barcode in Software As said above, we rst require that M is included in L and that M is not empty: M L M =

As said above, we rst require that M is included in L and that M is not empty: M L M = generate, create datamatrix 2d barcode none with software projects Microsoft Office Development. Microsoft Office 2000/2003/2007/2010 We have now to consider the Software ECC200 space on which the potential re nement states are moving: this is the union of M with the image of the initial set M under the transitive closure of the relation ae, namely M cl(ae)[M ]. Clearly, this set contains exactly the elements of all traces of the re ned model. Since we want that a re ned trace is also an abstract one, we thus have the following additional property: (M cl(ae)[M ]) re ae Finally, we do not want a re ned trace to be unable to be extended if the corresponding abstract one is able to be extended.

This requires that the domain of the abstract transition relation ae intersected with the set M cl(ae)[M ] is included in the domain of the re ned transition relation re also intersected with the same set M cl(ae)[M ]; formally: (M cl(ae)[M ]) dom(ae) (M cl(ae)[M ]) dom(re) The last two conditions are not so easy to deal with because of the presence of the set M cl(ae)[M ]. As a consequence, we forget about this set and get the following. Mathematical models for proof obligations slightly stronger (but far s Software Data Matrix ECC200 impler) conditions: M L M = re ae dom(ae) dom(re) Going back to our examples, we can now clearly prove that the second one is a re nement of the rst one: L = {0 0} ae = { (0 0) (1 0), (0 1) (1 1), (1 0) (0 0), (1 1) (0 1), (1 0) (1 1), (0 1) (0 0) } M = {0 0} re = { (0 0) (1 0), (1 1) (0 1), (1 0) (1 1), (0 1) (0 0) } (I). In conclusion, we have seen that traces allowed us to informally reason about a re nement by making explicit what can be observed in a re nement and in an abstraction. But, on the other hand, we end up with conditions (I), which do not depend on the traces, but rather on the initial sets and on the transition relations of both the abstraction and the re nement..

14.4.5 Considering the indiv Software Data Matrix idual events In the previous section, we considered the abstract and concrete relation ae and re obtained after taking the union of the various relations ae1 , .

. . , aen making the abstract events and re1 , .

. . , ren making the corresponding concrete events; formally: ae = ae1 aen re = re1 ren .

. The above condition re ae is made stronger by imposing that the containment is constrained at the ner level of each individual events, namely: re1 ae1 . . .

ren aen .. 14.4 Presentation of simple re nement by means of traces Such conditions clearly impl y re ae. We could have imposed some similar stronger conditions dealing with the domains as well, namely: dom(ae1 ) dom(re1 ) . .

. dom(aen ) dom(ren ). But this happens to be sometimes too strong.

Re nement conditions (I) can be rewritten as follows:. M L M = re1 ae1 (II) .. ECC200 for None .

ren aen dom(ae) dom(re). In doing so, we impose (for the moment) that to each concrete event formalized by the relation rei there corresponds an abstract event formalized by the relation aei . Notice that this constraint will be made more liberal in Section 14.6, where we shall study three possible extensions: (1) the splitting of an abstract event into several concrete ones, (2) the merging of several abstract events into a single concrete one, and (3) the introduction in a re nement of new events, which have no counterparts in the abstraction.

. 14.4.6 External and internal datamatrix 2d barcode for None variables In the previous section, we de ned a re nement by considering what we can observe from our discrete transition systems.

But, what we can observe is just a convention we give ourselves. In fact, the state can be more complicated than what we can observe of it. In the action/reaction examples, as they are developed in 3, we have a more complicated state containing, besides variables a and r, two additional variables ca and cr recording the number of times the action and the reaction are on (1) respectively.

Such variables ca and cr are said to be internal variables, whereas.
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