Information Technology Reference
In-Depth Information
c) f satisfies condition C3 . For any R R , we can see that the partial sum
r R lim n ₒ∞ f ( r , n ) is bounded because
lim
n ₒ∞
f ( r , n )
=
lim
n ₒ∞
ʔ ( r )
·
f n ( r )
ʔ ( r )
ʔ ( r )
=
1 .
r R
r R
r R
r R
Because of Lemma 4.3 and Proposition 2.1, the least fixed point of
C
can be
V = n ∈N C
n (
written as
), where
( r )
=
0 for all r
R .
4.4
Reward Testing
In this section we introduce an alternative testing approach based on the probabilistic
testing discussed in Sect. 4.2 . The idea is to associate each success action ˉ
ʩ
a reward, and performing a success action means accumulating some reward. The
outcomes of this reward testing are expected rewards.
4.4.1
A Geometric Property
We have seen in Proposition 4.1 that the comparison of two sets with respect to the
Hoare and Smyth preorders can be simplified if they are closed subsets of [0, 1], as it
suffices to consider their maximum and minimum elements. This simplification does
not apply if we want to compare two subsets of [0, 1] n , even if they are closed, because
maximum and minimum elements might not exist for sets of vectors. However, we
can convert a set of vectors into a set of scalars by taking the expected reward
entailed by each vector with respect to a reward vector. Interestingly, the comparison
of two sets of vectors is related to the comparison of suprema and infima of two
sets of scalars, provided some closure conditions are imposed. Therefore, to some
extent we generalise Proposition 4.1 from the comparison of sets of scalars to the
comparison of sets of vectors. Mathematically, the result can be viewed as an analytic
property in geometry, which could be of independent interest.
Suppose that in vector-based testing we use at most n success actions taken from
the set ʩ
[0, 1] ʩ
={
ˉ 1 , ... , ˉ n }
with n> 1. A testing outcome o
can be viewed
as the n -dimensional vector
where o ( ˉ i ) is the probability of suc-
cessfully reaching success action ˉ i . Similarly, a reward vector h
o ( ˉ 1 ), ... , o ( ˉ n )
[0, 1] ʩ
can be
regarded as the vector
where h ( ˉ i ) is the reward given to ˉ i .We
sometimes take the dot product of a particular vector h
h ( ˉ 1 ), ... , h ( ˉ n )
[0, 1] n and a set of vectors
O
[0, 1] n , resulting in a set of scalars given by h
·
O :
={
h
·
o
|
o
O
}
.
Definition 4.5 A subset O of the n -dimensional Euclidean space is p-closed (for
probabilistically closed) iff
Search WWH ::




Custom Search