CSci 5403, Spring 2010
Stefan NelsonLindall
Homework 2
Example Solution
1. Games.
Prove that the following problems are
PSPACE
complete:
(a) A Stochastic Satisfiability instance is a quantified boolean formula, only the universal
∀
quantifier is replaced by the random
R
quantifier (meaning, “for a random x”). An
instance
∃
x
1
R
x
2
∃
x
3
. . .
Q
n
x
n
φ
is in
ssat
if
∃
x
1
R
x
2
∃
x
3
. . .
Q
n
x
n
Pr[
φ
(
x
1
, . . . , x
n
)]
≥
1
2
.
Show that
ssat
is
PSPACE
complete.
In some sense,
ssat
is a “game against nature,” e.g. we can think of the
∃
player in a
formula game instance playing against an opponent who is not explicitly adversarial,
but simply moves randomly at each stage. It might seem like this should be an easier
game to win, but your reduction shows this is not always true.
Pf:
ssat
is
PSPACE
hard
Reduction from
fg
. Map the formula game instance
f
=
Q
1
x
1
. . .
Q
i
x
i
φ
(
x
1
, . . . x
i
) to
the
ssat
problem
f
=
R
x
0
Q
1
x
1
...
Q
i
x
i
(
x
0
∧
φ
(
x
1
, . . . x
i
))
where
Q
k
=
∃
if
Q
k
is
∃
R
if
Q
k
is
∀
.
x
0
will be assigned 1 with probability
1
2
, and Pr[
x
0
∧
φ
] = Pr[
x
0
]
·
Pr[
φ
].
If
f
∈
fg
, then for any possible assignment of random variables the resulting formula
is satisfiable, so Pr[
φ
] = 1, Pr[
x
0
∧
φ
] =
1
2
, and hence
f
∈
ssat
.
If
f
∈
fg
, then there exists some assignment of random variables such that the resulting
formula is not satisfiable. This makes Pr[
φ
]
<
1, so Pr[
x
0
∧
φ
]
<
1
2
, and hence
f
∈
ssat
.
Thus,
fg
≤
P
ssat
.
ssat
∈
PSPACE
Iterate through each possible assignment of variables with the
R
quantifier, in each
checking each possible assignment of variables with the
∃
quantifier to determine
whether or not
φ
can be satisfied.
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 Spring '08
 Sturtivant,C
 Graph Theory, Vertex, Randomness, Glossary of graph theory

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