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s
0
˄
˄
1/2
1/2
s
1
˄
˄
˄
˄
1/
2
1
/2
1/2
1/
2
1
/2
1/2
s
s
s
2
3
4
a
˄
a
˄
ˉ
ˉ
(a)
(b)
Fig. 4.4
Testing the process
Q
2
(1) For each test
T
, process
P
and reward vector
h
, calculate a set of outcomes
A
h
(
T
,
P
), which is a subset of [0, 1].
(2) For each pair of processes
P
,
Q
, compare the corresponding sets of outcomes,
A
h
(
T
,
Q
) for every test
T
and reward vector
h
, in terms of their
suprema and infima.
h
(
T
,
P
) and
A
But our methods for comparing sets of outcomes do not necessarily require us to
calculate the entire set of outcomes. For example, Proposition
4.1
says that for
closed sets, it suffices to compare extremal outcomes. Here we propose an alternative
approach to testing based on calculating directly the extremal values of possible
outcomes.
Note that the results in Sects.
4.5
and
4.6
are not used in the rest of this topic,
except for a notion of extreme policy (Definition
4.8
) that is referred to in Sect. 6.5.1.
So the reader may skip these two sections in the first reading, and return back when
necessary.
The function
h
used to associate a reward with a resolution is only defined in
(
4.10
) above for deterministic pLTSs. Here we consider generalisations to an arbitrary
finitely branching pLTS
C
S
,
ʩ
˄
,
ₒ
.
h
min
Now consider the function
C
:(
S
ₒ
[0, 1])
ₒ
(
S
ₒ
[0, 1]) defined by:
⊧
⊨
ˉ
−ₒ
h
(
ˉ
)
if
s
h
min
(
f
)(
s
)
ˉ
˄
C
=
0
if
s
−ₒ
and
s
−ₒ
⊩
˄
−ₒ
˄
−ₒ
ˉ
min
{
Exp
ʔ
(
f
)
|
s
ʔ
}
if s
−ₒ
and
s
.
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