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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
ₒ
[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
ↆ
so that
h
·
A
={
h
·
o
|
o
∈
O
}
. Let
A
ↆ
[
−
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
ↆ
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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