Lecture Notes in Computer Science 3472

124 Stefan D. Bruda
Processes may be nondeterministic, so there may be different runs of a given
test on a process, with different outcomes. In effect, the (overall) outcome of an
observer testing a process is a set, and therefore we are led to use powerdomains
of O. In fact, we have three possible powerdomains:
Pmay
def
=
{⊤} = {⊥, ⊤}
|
{⊥}
Pconv
def
=
{⊤}
|
{⊥, ⊤}
|
{⊥}
Pmust
def
=
{⊤}
|
{⊥} = {⊥, ⊤}
The names of the three powerdomains are not chosen haphazardly. By considering
Pmay as possible outcomes we identify processes that may pass a test in
order to be considered successful. Similarly, Pmust identifies tests that must be
successful, and by using Pconv we combine the may and must properties. The
partial order relations induced by the lattices Pmay, Pmust, andPconv shall be
denoted by ⊆may, ⊆must, and⊆conv, respectively.
We also need to introduce the notion of refusal. A process refuses an action
if the respective action is not applicable in the current state of the process, and
there is no internal transition to change the state (so that we are sure that the
action will not be applicable unless some other visible action is taken first).
Definition 5.3. Process p ∈ Q refuses action a ∈ Act, written p ref a, iffp ↓B,
p τ
−−→/ ,andp a
−−→/ . ⊓⊔
We thus described the notions of test and test outcomes. We also introduce
at this point a syntax for tests. In fact tests are as we mentioned just processes
that interact with the process under test, so we can represent tests in the same
way as we represent processes. Still, we find convenient to use a “denotational”
representation for tests since we shall refer quite often to such objects. We do
this by defining a set O of test expressions.
While we are at it, we also define the “semantics” of tests, i.e., the way tests
are allowed to interact with the processes being tested. Such a semantics for tests
is defined using a function obs : O×Q →P,whereP∈{Pmay, Pconv, Pmust}
such that obs(o, p) is the set of all the possible outcomes.
To concretize the concepts of syntax and semantics, we introduce now our first
testing scenario (i.e., set of test expressions and their semantics), of observable
testing equivalence 2 [Abr87]. This is a rather comprehensive testing model, which
we will mostly restrict in order to introduce other models—indeed, we shall
restrict this scenario in all but one of our subsequent presentations. A concrete
model for tests also allows us to introduce our first preorder.
For the remainder of this section, we fix a transition system (Q, Act∪{τ}, −→ ,
↑B) together with its derived transition system (Q, Act ∪{ε}, ⇒ , ↑).
2 Just testing equivalence originally [Abr87]; we introduce the new, awkward terminology
because the original name clashes with the names of preorders introduced
subsequently.