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defined as follows: Tangling occurs when, in a mapping between source and target, a
target element is related to multiple source elements. In other words, an element t є
Target is tangled if and only if card(f -1 (t))> 1, where f -1 is the inverse application. In
Fig. 1 target element t3 is tangled with respect to the source elements s1 and s3.
There is a specific combination of scattering and tangling which we call
crosscutting, defined as follows: Crosscutting occurs when, in a mapping between
source and target, a source element is scattered over target elements and where in at
least one of these target elements, some other source element is tangled. In other
words, crosscutting can be defined as follows. For element s1, s2 є Source / (s1 ≠ s2),
s1 crosscuts s2 if and only if card ( f(s1) ) > 1 and t є f(s1) : card (f -1 (t)) > 1 and s2 є
f -1 (t). We do not require that the second source element is scattered. In that sense, our
definition is not symmetric, in contrast to the definition in [23] (see Sect. 7). In
Fig. 1 , source element s1 is crosscutting source element s3 with respect to the given
mapping between source and target but not the opposite. Following on with this
example, and according to our definition, this means that we should redesign s1 but
not s3 in order to remove crosscutting (i.e., through the use of aspect-oriented
techniques). On the other hand, assuming crosscutting as a symmetric property
implies that redesign of either s1 or s3 is feasible.
2.3 Case Analysis of Crosscutting
In the previous section we defined scattering, tangling and crosscutting for a mapping
between source and target. Now, we discuss a case analysis of possible combinations.
Assuming that the properties tangling, scattering, and crosscutting may be true or
false, there are eight combinations (see Table 2 ). Each case addresses a certain
mapping from source to target. However, crosscutting requires tangling and
scattering, which eliminates 3 of these combinations (Cases 6, 7 and 8: not feasible).
Table 2. Feasibility of combinations of tangling, scattering and crosscutting
tangling
scattering
crosscutting
feasibility
Case 1
no
no
no
feasible
Case 2
yes
no
no
feasible
Case 3
no
yes
no
feasible
Case 4
yes
yes
no
feasible
Case 5
yes
yes
yes
feasible
Case 6
no
no
yes
not feasible
Case 7
no
yes
yes
not feasible
Case 8
yes
no
yes
not feasible
There are five feasible cases listed in Table 4. In Case 4, we have scattering and
tangling in which no common elements are involved. With our definition of crosscutting
we clearly separate the cases with just tangling, just scattering and on the other hand
crosscutting. Our proposition is that tangling and scattering are necessary but not
sufficient conditions for crosscutting. An example of this situation is explained in one of
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