Lecture Notes in Computer Science 3472

5 Preorder Relations 127
Definition 5.6. The observable testing preorder is a relation ⊑⊆ Q × Q,
where p ⊑ q iff obs(o, p) ⊆ obs(o, q) for any test o ∈O. The observable testing
equivalence is a relation �⊆ Q × Q, withp � q iff p ⊑ q and q ⊑ p. ⊓⊔
If we restrict the definition of O (and thus the definition of the function obs),
we obtain a different preorder, and thus a different equivalence. In other words, if
we change the set of possible tests that can be applied to processes (the testing
scenario), then we obtain a different classification of processes.
We will present in what follows various preorder relations under various testing
scenarios. These preorders correspond to sets of changes imposed on O and
obs, and we shall keep comparing various scenarios with the testing scenario
presented in Section 5.2.2. As it turns out, the changes we impose on O are in
all but one case restrictions (i.e., simplification of the possible tests).
We will in most cases present an equivalent modal characterization corresponding
to these restrictions. Such a modal characterization (containing a set
of testing formulae and a satisfaction operator) will in essence model exactly the
same thing, but we are able to offer some results that are best shown using the
modal characterization rather than other techniques.
When we say that a preorder ⊑α makes more distinction than another preorder
⊑β we mean that there exist processes that are distinguishable under ⊑α
but not under ⊑β. This does not imply that ⊑α and ⊑β are comparable, i.e., it
could be possible that ⊑α makes more distinction than ⊑β and that ⊑β makes
more distinction than ⊑α. Whenever ⊑α makes more distinction than ⊑β but
not the other way around we say that ⊑α is coarser than ⊑β, orthat⊑β is finer
than ⊑α.
5.3 Trace Preorders
We thus begin our discussion on preorder and equivalence relations with what
we believe to be the simplest assumption: we compare two processes by their
trace, i.e., by the sequence of actions they perform. In this section we follow
roughly [vG01, dN87].
We consider that the divergence predicate ↑B of the underlying transition
system is empty (no process diverges). The need for such a strong assumption
will become clear later, when we discover that trace preorders do not cope well
with divergence.
The trace preorder is based on the following testing scenario: Weviewa
process as a black box that contains only one interface to the real world. This
interface is a window displaying at any given moment the action that is currently
carried out by the process. The process chooses its execution path autonomously,
according to the given transition system. As soon as no action is carried out, the
display becomes empty. The observer records a sequence of actions (a trace), or
a sequence of actions followed by an empty window (a complete trace). Internal
moves are ignored (indeed, by their definition they are not observable). We
regard two processes as equivalent if we observe the same complete trace using
our construction for both processes.