Lecture Notes in Computer Science 3472

274 Verena Wolf
model test processes boils down to characterization
⊑CH fully probabilistic τ-free, nonprob., classical extended trace
processes reactive (T np,re ) testing (⊑DH ) distribution
∼LS τ-free reactive prob. τ-free, nonprob. ordinary probabilistic
processes (T np ) bisimulation bisimulation (∼bs)
⊑CL fully probabilistic fully probabilistic classical probabilistic trace
processes (T fp
τ ) testing (⊑DH ) distribution
⊑ may
SE probabilistic probabilistic classical trace distr. preprocesses
(T pp
τ ) testing (⊑DH ) congruence (� ftp )
⊑ may
JY τ-free prob. τ-free, probabilistic ordinary probabilistic
processes (T pp ) simulation simulation
⊑BC action-labeled ”passive” classical extended trace
CTMCs (T pa
τ ) testing (⊑DH ) distribution
Table 9.1. Summary table: The first column lists the most important probabilistic
testing relations we have discussed in this chapter. In the second column the corresponding
class of processes is depict and the third column shows the set of applied test
processes. In the fourth column, the resulting relations are listed when the probabilistic
information is disregarded. The last columns lists the corresponding characterizations.
9.13 Summary
We have given an extensive survey of the testing theory for probabilistic systems
and presented the definitions of different preorders in a uniform style to ease the
task of establishing relationships between them. Moreover we saw that in most
cases the relations are closely connected with the classical testing relation of De
Nicola and Hennessy (see Figure 9.14, page 265) and we discussed characterizations
to better reflect the nature of the relations (see Figure 9.15, page 270).
Table 9.1 summarizes the main contents of this chapter.
Computational issues: Most authors do not not address computational issues,
but from the summary table we can see that some of the relations are
decidable. First, consider the characterization by extended traces. If the process
is finite, we can determine the (finite) set of extended traces with a non-zero
probability and compute the probability of each extended trace with the inductive
definitions of section 9.11.1. Christoff presents algorithms for verification
of his testing relations in [CC91]. Furthermore, we have a characterization by
probabilistic bisimulation that can be computed in polynomial time and space
by a partitioning technique [HT92]. To the best of our knowledge, for all other
relations no algorithms computing them exist.
Open problems:
• We have only pointed out the most obvious connections between the different
preorders presented here. Clarifying which relations are incomparable
and which are finer/coarser than others would be helpful to obtain a more
complete picture on probabilistic testing relations. For example, the rela-