Lecture Notes in Computer Science 3472

or (must-testing):
9 Testing Theory for Probabilistic Systems 249
If all paths in P�T are successful then all paths in Q�T are successful.
For fully probabilistic processes we do not distinguish between may- and musttesting.
There are two approaches to relate P, Q ∈ FPP:
and
”The probability of all successful paths in P�T is at most the probability
of all successful paths in Q�T .”
The probability of any trace α in P�T is at most the probability of α in
Q�T .
Intuitively speaking, an implementation (Q) should be at least as successful as
the specification (P).
9.7.1 Testing with Non-probabilistic Test Processes
In this section, we present the approach of Christoff [Chr90]. He presents a testing
theory for fully probabilistic processes that is based on reactive, non-probabilistic
test processes without internal actions. Two fully probabilistic processes P and
Q are related with regard to their trace distributions while running in parallel
with a non-probabilistic test process. Note that T np,re is a proper subset of T np ,
the set of all non-probabilistic test processes (compare Definition 9.7, page 245).
Definition 9.14. [Chr90] Let P, Q ∈ FPP.
• P ⊑ tr CH
Q iff ∀ T ∈Tnp,re seq , α ∈ Act ∗ :
Pr trace
trace
P�T (α) � PrQ�T (α).
• P ⊑ wte
CH Q iff ∀ T ∈Tnp,re , α ∈ Act ∗ :
�
a∈Act
�
trace PrP�T (α a) � a∈Act
• P ⊑ ste
CH Q iff ∀ T ∈Tnp,re , α ∈ Act ∗ :
Pr trace
trace
P�T (α) � PrQ�T (α).
Recall that T np,re
seq
than or equal to ⊑wte CH
trace PrQ�T (α a).
⊓⊔
is coarser
can be characterized by
contains only ”sequential” test processes, so ⊑ tr CH
and ⊑ste
CH . It turns out that ⊑tr CH
trace inclusion (hence, the superscript ”tr”), i.e.
P ⊑ tr CH Q iff {β ∈ Act∗ | Pr trace
P
(β) > 0} ⊆{β ∈ Act ∗ | Pr trace
Q (β) > 0}.
The probabilistic information gets lost due to the restrictions of the test pro-
compares the sum of the trace probabilities whereas the
cesses. Since ⊑ wte
CH