CSci 5403, Spring 2010
Homework 2
due: Feb 16, 2010
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.
(b) The Hamiltonian Path Game is a game played on a graph by two players.
An in
stance of the game is specified by a (
G, v
) pair and means that Player I starts at
node
v
.
At each turn of the game, the current node is marked, and the current
player may choose to move to any unmarked node adjacent to the current node; the
next turn is made by the other player.
If at any turn there are unmarked nodes
but all neighbors of the current node are marked, Player II wins.
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 Spring '08
 Sturtivant,C
 Computational complexity theory, NPcomplete, current node, quantiﬁed Boolean formula, Qn xn, formula game instance

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