Lecture Notes in Computer Science 3472

s
a b
t
5 Preorder Relations 133
y
a b
Fig. 5.6. Processes equivalent under observation preorder (s �B t; s �R t; s ��must t;
s �fmust t).
the state, so no matter how many times we go through it we end where we left
from. Still, the τ-loop is not without significance in practice since such a loop
may produce divergence (if the process keeps iterating through it). However, it
can also be argued that the τ-loopisexecutedanarbitrarybutfinitenumberof
times and so the process executes b eventually (under some notion of fairness).
We shall actually argue back and forth about these two processes as we go along
with the description of other preorder relations, so you do not have to make up
your mind just yet.
5.5 Testing Preorders
Testing preorders [dNH84] are coarser than observation preorder. Essentially,
testing preorders differentiate between processes based on differences in deadlock
behavior. We may differentiate by the ability to respond positively to a test, or
the ability to respond negatively to a test, or both. In practical cases this is often
sufficient.
Recall the concept of outcome of a test presented in Section 5.2.2. For a test
o and a process p the result of applying o to p is the set of runs Runs(o, p)
with outcomes from the set {⊥, ⊤}. Also recall the lattices Pmay, Pmust, and
Pconv over the powerset of {⊥, ⊤}, together with the corresponding partial order
relations.
We then have the following testing scenario for testing preorders: We run
a test in parallel to the process being tested, such that they perform the same
actions. If the test reaches a success state, then the test succeeds; if on the other
hand the process reaches a deadlock state (i.e., a state with no way out), or if
the process diverges before the test has reached a success state, the test fails.
Sometimes we are interested in running the same test repeatedly and collect all
of the possible outcomes; we need this when we want to make sure that a test
succeeds no matter what.
Formally, we change in what follows (simplify in fact) the semantics of Expression
(5.3) from Definition 5.4 on page 125 to
obs(ao, p) = � {obs(o, p ′ ) | p a ⇒ p ′ }∪{⊥|p ↑} ∪ {⊥ | p �⇒} (5.11)
Then we look at two alternative ways to restrict the set of tests O:
(1) Let Omay be defined only by expressions of form (5.1), (5.3), and (5.5). We
do not need any test that signifies failure; instead, failure under test happens
τ