Game Development Reference
InDepth Information
u
a
u
1
4
1
4
v
r
3
4
3
4
d
≡
P
>
0
G
(
(
P
>
0
F
a
)), and let
σ
be a HD strategy which in every
Let
ϕ
¬
a
∧
wv
selects either
v
0
or not, respectively. Here #
d
(
w
) and #
u
(
w
) denote the number of occurrences
of
d
and
u
in
w
, respectively. Obviously, the strategy
σ
can be implemented
by a onecounter automaton. The play
G
(
σ
v
initiated in
v
closely resembles a
oneway infinite random walk where the probability of going right is
→
or
v
→
r
, depending on whether #
d
(
w
)
−
#
u
(
w
)
≤
3
4
and
4
. More precisely, the play
G
(
σ
v
corresponds
to the unfolding of the following infinitestate Markov chain (the initial state
is grey):
1
the probability of going left is
1
4
1
4
1
4
u
u
3
4
3
4
3
4
u
a
v
v
r
v
r
d
d
Run
(
G
(
σ
v
)
A standard calculation shows that the probability of all
w
∈
1
initiated in
v
such that
w
visits a state satisfying
a
is equal to
3
. Note that
Run
(
G
(
σ
v
) initiated in
v
which does
not
visit a state satisfying
a
we have that
w
(
i
)
for every
w
∈
=
ν
(
P
>
0
F
a
) for every
i

¬
a
∧
≥
0. Since the probability
2
of all such runs is
3
, we obtain that the formula
ϕ
is valid in the state
v
of
G
(
σ
v
. On the other hand, there is no finitememory
ϕ
winning strategy
σ
in
v
, because then the play
G
(
σ
v
corresponds to an unfolding of a finitestate
Markov chain, and the formula
ϕ
does not have a finitestate model (see,
e.g., Brazdil et al. [2008]).
The memory requirements of
ϕ
winning strategies for various fragments of
qualitative branchingtime logics were analysed by Brazdil and Forejt [2007]
and Forejt [2009]. The decidability/complexity of the existence of HR (or
HD)
ϕ
winning strategies in turnbased stochastic games with qualitative
branchingtime objectives is still open.
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