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is a process whose only possible actions are ˄ and the elements of ʩ . Applying the
test T to the process P gives rise to the set of testing outcomes
( T , P ) defined in
(4.2), exactly one of which results from each resolution of the choices in T | Act P .
Each testing outcome is an ʩ -tuple of real numbers in the interval [0,1], that is, a
function o : ʩ
A
ʩ , gives the probability
that the resolution in question will reach an ˉ-success state , one in which the success
action ˉ is possible.
In Sect. 4.4 two reward testing preorders are obtained by associating each suc-
cess action ˉ
[0, 1], and its ˉ -component o ( ˉ ), for ˉ
ʩ a nonnegative reward. We refer to that approach of testing as
nonnegative-reward testing . If we also allow negative rewards, which intuitively can
be understood as costs, then we obtain an approach of testing called real-reward
testing . Technically, we simply let reward tuples h range over the set [
1, 1] ʩ .If
= ˉ ʩ h ( ˉ )
[0, 1] ʩ , we use the dot-product h
o
·
o
o ( ˉ ). It can apply to a set
[0, 1] ʩ
O
so that h
·
A
={
h
·
o
|
o
O
}
. Let A
[
1, 1]. We use the notation
A for the supremum of set A , and
A for the infimum.
Definition 6.30 (Real-Reward Testing Preorders)
rr may Q if for every ʩ -test T and real-reward tuple h
1, 1] ʩ ,
(i) P
[
h
h
· A
( T , P )
· A
( T , Q ).
(ii) P
rr must Q if for every ʩ -test T and real-reward tuple h
[
1, 1] ʩ ,
h · A
( T , P )
h · A
( T , Q ).
Note that for any test T and process P it is easy to see that
h ( T , P ) .
h
· A
( T , P )
= A
Therefore, the nonnegative-reward testing preorders presented in Definition 4.6 can
be equivalently formulated in the following way:
nr may Q if for every ʩ -test T and nonnegative-reward tuple h
[0, 1] ʩ ,
(i) P
h
h
· A
( T , P )
· A
( T , Q ).
nr must Q if for every ʩ -test T and nonnegative-reward tuple h
[0, 1] ʩ ,
(ii) P
h
· A
( T , P )
h
· A
( T , Q ).
Although the two nonnegative-reward testing preorders are in general incomparable,
the two real-reward testing preorders are simply the inverse relations of each other.
rr may Q if and only if
Theorem 6.21
For any processes P and Q, it holds that P
ʩ
Q
rr must P .
Proof
[0, 1] ʩ
We first notice that for any nonempty set A
and any reward tuple
1, 1] ʩ ,
h
[
A ,
h
·
A
=−
(
h )
·
(6.29)
where
ʩ .We
consider the “if” direction; the “only if” direction is similar. Let T be any ʩ -test
h is the negation of h , that is, (
h )( ˉ )
=−
( h ( ˉ )) for any ˉ
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