Lecture Notes in Computer Science 3472

5 Preorder Relations 143
for free. Indeed, the tests in the sets O1 and O2 described in the previous sections
are sequential, in the sense that unions always occur between a test whose result
that is immediately available (Succ or Fail) and some other, possibly longer
test. In testing preorders as well as in failure trace preorder we need to copy the
process while it runs; indeed, we may need to combine the outcomes of two (or
more) different runs of the process, which means that we need to run two copies
of the process to obtain these outcomes independently from each other. Because
of the sequential tests used by refusal preorder copying is no longer necessary
(but it becomes necessary once more in failure trace preorder). This being said,
the definition of the must operator from refusal preorder implies that processes
need to be copied anyway (since we have to apply many tests on them), so the
failure trace testing scenario is not that bad after all.
5.8 Fair Testing
Recall the processes depicted in Figure 5.6 on page 133 and our back and forth
argument that they should be considered equivalent (or not). When we considered
them under the testing preorder, s and t were not equivalent, whereas they
are so under the other preorders. Testing preorder, with its habit that the presence
of divergence may ruin a test, will differentiate between these two processes
as opposed to all the other preorders we have seen so far. As we mentioned,
whether such a behavior is a good or bad thing depends on one’s opinion about
divergences.
For those who prefer to ignore divergences as long as there is a hope that
the process will continue with its visible operation, i.e., for those who prefer to
consider the processes shown in Figure 5.6 equivalent, fair testing is available
[BRV95].
We have the same testing scenario for fair testing as the one used in Section
5.5, except that the operator must is enhanced, such that it chooses a visible
action whenever such an action is available. With the same set O of observers as
the one used to define the testing preorder, the new operator fmust is defined
as follows:
p fmust o iff for any σ ∈ Act ∗ and o ′ ∈Owith o = σo ′ , it holds that:
obs(o ′ , p ′ ) = obs(o, p) forsomep ′ ∈ Q, p σ ⇒ p ′ , implies that there
exists a ∈ Act ∪{Succ} such that o ′ = ao ′′ , o ′′ ∈O∪{ε}.
The preorder ⊑fmust, as well as the equivalence �fmust induced by the operator
fmust are defined in the usual manner.
The operator fmust is the “fair” variant of the operator must of testing
preorder lineage. It ignores the divergences as long as there is a visible action (a
in the above definition) accessible to the observer. The following characterization
of ⊑fmust in terms of other preorders is easily obtained from the results presented
elsewhere [BRV95]: