Lofgren, Musbach
CSci 5403— Homework 4
Problem 1
Suppose
L
∈
PPSPACE
. Then there exists a polynomial
p
:
N
→
N
and a
p
(
n
)space PTM
M
that runs for
exactly 2
p
(
n
)
steps on every input (it counts its steps to ensure this), has a unique accepting configuration
a
(it clears its tape before accepting), and is such that
x
∈
L
⇐⇒
Pr[
M
(
x
)]
≥
1
/
2.
We will construct
an algorithm that decides
L
in polynomial space, proving that
L
∈
PSPACE
.
By
δ
(
x, b
), we mean the
configuration that comes after configuration
x
with random bit
b
.
By [
δ
(
u, b
) =
v
], we mean the Iverson
bracket: 1 if
δ
(
u, b
) =
v
, and 0 otherwise. Let
C
be the set of configurations of
M
, so

C

=
θ
(2
p
(
n
)
). We
define a recursive function
CountPaths
(
u, v, i
) which counts the number of computation paths from
u
to
v
of length 2
i
:
CountPaths
(
u, v, i
) =
(
[
δ
(
u,
0) =
v
] + [
δ
(
u,
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 Spring '08
 Sturtivant,C
 Addition, Recursion

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