Game Development Reference
InDepth Information
H
= Knw(
G
)=
Q, q
I
,
Σ
,
Δ
H
, let Reach(
T
)with
T⊆
Q
be an observable
reachability objective in
H
, and
the equivalence relation between states
of
H
as defined above. Player 1 almostsurely wins from the set of states
W
≈
2
Σ
and Good :
Q
⊆
Q
if there exist functions Allow :
Q
→
→
Σ such that
for all
q
W
:
1 for all
q
≈
∈
∈
Allow(
q
), post
σ
(
q
)
q
and for all
σ
⊆
W
,
(
q, q
)
(
q,
Good(
q
)
,q
)
2 in the graph (
W, E
) with
E
=
{
∈
W
×
W

∈
Δ
H
}
,
all infinite paths visit a state in
T
,
3 Good(
q
)
∈
Allow(
q
).
Condition 1 ensures that the set
W
of winning states is never left. This
is necessary because if there was a positive probability to leave
W
, then
Player 1 would not win the game with probability 1. Condition 2 ensures
that from every state
q ∈ W
, the target
T
is entered with some positive
probability (remember that the action Good(
q
) is played with some positive
probability). Note that if all infinite paths in (
W, E
) eventually visit
T
, then
all finite paths of length
n
=
W
do so. Therefore, the probability to reach
T
within
n
rounds can be bounded by a constant
κ>
0, and thus after
every
n
rounds the target set
is reached with probability at least
κ
. Since
Condition 1 ensures the set
W
is never left, the probability that the target
set has not been visited after
m
T
κ
)
m
. Since the
game is played for infinitely many rounds, the probability to reach
·
n
rounds is at most (1
−
T
is
κ
)
m
= 1. By Condition 3, the actions that ensure progress
towards the target set can be safely played.
The algorithm to compute the set of states
W
lim
m→∞
1
−
(1
−
Q
from which Player 1
has an equivalencepreserving almostsurelywinning strategy for Reach(
⊆
T
)
is the limit of the following computations:
W
0
=
Q
W
i
+1
= PosReach(
W
i
) for all
i
≥
0
where the PosReach(
W
i
) operator is the limit of the sequence
X
j
defined by
X
0
=
T
X
j
+1
X
j
∪
Apre(
W
i
,X
j
) for all
j
=
≥
0
where
Apre(
W, X
)=
{
q
∈
W
∃
σ
Σ:
post
σ
(
q
)
∈
q
≈
q
: post
σ
(
q
)
⊆
X
and
∀
⊆
W
}
.
The operator Apre(
W, X
) computes the set of states
q
from which Player 1
can ensure that some state of
X
is visited in the next round with positive
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