2 Why We Need Model-Based Testing

242 Compositional Modeling
placeholders. We say that an action or a trace is concrete when it has no placeholders.
If a trace of a model program has placeholders, then any instance of it is a trace of
that model program as well, because a placeholder indicates that the corresponding
argument has no behavioral significance with respect to this model program.
For example,
ReqSetup(),ResSetup(_,_,STATUS("Completed")),ReqWork()
is a trace of the setup model program, whereas
ReqSetup(),ResSetup(),ReqWork()
is not, because the following instance of it is not a trace of the setup model program.
ReqSetup(),ResSetup(_,_,Status("Cancelled")),ReqWork()
Two traces match if they have a common instance. Notice that two traces that
have no placeholders match if and only if they are equal. For example, the trace
ReqSetup(1,2),ResSetup(1,_,Status("Completed")),ReqWork(_,3)
is an instance 2 of the trace
ReqSetup(1),ResSetup(_,_,Status("Completed")),ReqWork()
as well as the trace
ReqSetup(_,2),ResSetup(1),ReqWork(_,3)
Notice that none of the above traces are concrete.
14.3.1 Trace intersection
A desired property of M1 × M2 is that the set of all traces of M1 × M2 is the
intersection of the set of all traces of the loop extension of M1 for A2, say M ′ 1 , and
the set of traces of the loop extension of M2 for A1, say M ′ 2 . A reason why this
property is important is that it makes it possible to do compositional reasoning over
the traces in the following sense. If all traces of M ′ 1 satisfy a property p and all
traces of M ′ 2 satisfy a property q, then all traces of M1 × M2 satisfy both properties
p and q. A sufficient condition for the trace intersection property to be true is if M1
and M2 don’t share any state variables.
Let M1 be the setup model program in Figure 14.15 and let M2 be the cancellation
model program in Figure 14.7. First, let us see how the model program M1 × M2 is
constructed. The state variables of M1 × M2 is the union of the state variables of M1
and M2. Let us call the state variable of M1 setupMode that may take one of the three
2 It is in fact the most general common instance of both traces; i.e., any other instance of both
traces is also an instance of the most general common instance.
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