Lecture Notes in Computer Science 3472

Definition 9.30. Let P, Q ∈ PP.
9 Testing Theory for Probabilistic Systems 269
• The trace distribution preorder � td is given by
P � td Q iff tdistr(P) ⊆ tdistr(Q).
• The finite trace distribution preorder � ftd is given by
P � ftd Q iff ftdistr(P) ⊆ ftdistr(Q).
• The trace distribution precongruence � tp (finite trace distribution
precongruence � ftp , respectively) is the coarsest precongruence with respect
to || 6 that is contained in � td (� ftd , respectively).
⊓⊔
Segala shows that � td and � ftd coincide [Seg96]. This is also stated by Stoelinga
and Vaandrager as ”Approximation Induction Principle” [SV03]. Furthermore
we have that � ftd and � td characterize ⊑ may
SE .
Theorem 9.31. [Seg96] Let P, Q be probabilistic processes.
P ⊑ may
SE QiffP�tp QiffP� ftp Q.
⊓⊔
Segala also provides a characterization by failure distributions for ⊑must SE [Seg96].
Failures are similar to traces but end in a set of actions that cannot be performed
by the last state. The details of this characterization are omitted here because
it is similar to the case of trace distributions.
Stoelinga and Vaandrager present an intuitive ”testing scenario” (also known as
button pushing experiment) and proved that the resulting relation is equivalent
to the trace distribution preorder [SV03]. Note that in a sense this also motivates
Segala’s may-preorder due to theorem 9.31.
We have also a characterization of ⊑JY by structures called ”chains of a process”
that are similar to traces [JY02]. A very interesting result is the characterization
by probabilistic simulation as briefly presented in the following section.
of ⊑ may
JY
9.11.3 Probabilistic Simulation
The idea of ordinary simulation is to prove that an implementation Q refines
an abstract specification P in such a way that required properties are fulfilled
[Jon91]. So Q is simulated by P if ”every step in Q can be simulated by a step
in P” but not necessarily vice versa. In the probabilistic setting, simulation relations
have been defined amongst others by Jonsson [JGL91] and Segala [SL94].
In 2002, Jonsson and Yi proposed an alternative definition of probabilistic simulation
which coincides with their probabilistic may-testing preorder [JY02].
6 ArelationR is a precongruence with respect to || if P R Q implies (P || ˆ P) R (Q ||
ˆP) for an arbitrary probabilistic process ˆ P.Notethat� denotes the parallel composition
operator from Definition 9.8 here.