Solutions 3
Question 1
Show that
{
P

P
is a position in the game where
A
has a winning strategy
}
is in PSPACE.
A reminder of the deﬁnitions: a language is in AP if it is decided by an alternating Turing
machine
N
(a nondeterministic TM whose states
Q
are divided into two parts
Q
and
and
Q
or
) and all accepting computations use only polynomial time. An eventually accepting
conﬁguration (e.a.c.) of
N
on input
x
is deﬁned as follows:
•
all accepting leaves in the computation tree
N
(
x
) are e.a.c;
•
a conﬁguration with state in
Q
and
is e.a.c. iﬀ all its successor conﬁgurations are e.a.c;
•
A conﬁguration with state in
Q
or
is e.a.c. iﬀ at least one of its successor conﬁgurations
is e.a.c.
N
accepts
x
iﬀ the initial conﬁguration of
N
(
x
) is e.a.c.
Thus to answer the question we must describe a machine
N
that decides the language
given in the question; let’s call it
L
. We do this by describing the computation tree
N
(
x
).
Each vertex in the tree will correspond to a position in the game. The root corresponds to
the position given in the input, and we deﬁne further vertices recursively:
•
a vertex representing a position
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 Fall '10
 Veneris
 Algorithms, Graph Theory, Data Structures, Position, Computational complexity theory, PSPACE, computation tree

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