CSci 5403, Spring 2010
Hannah Jaber
Problem 4.2
Proposition.
BPL
⊆
P
where
BPL
is deﬁned as follows: the language
A
is in
BPL
iﬀ there
exists a logspace
PTM
M
such that
Pr[
M
(
x
;
r
) = (
x
∈
A
)]
≥
2
/
3
.
Proof.
With only logarithmic space,
M
can have at most
n
c
states, where
n
is the size of
the input. More states would require the machine would loop states and not halt. We can
therefore build an adjacency matrix
K
to descibe the possible conﬁgurations of
M
on input
A
. Iterate through all possible states
s
. If
s
j
is adjacent to
s
i
, the
i,j
entry in the matrix
is 1, otherwise it is 0.
s
j
is considered to be adjacent to state
s
i
iﬀ
M
can reach
s
j
from
s
i
in a single step. If
s
i
requests a random bit, there will be two adjacent states for the two
“output” edges. Random bit 1 will lead to one state, random bit 0 will lead to a diﬀerent
state.
Consider the matrix
K
x
. Entry
i,j
will list the number of paths from
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This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.
 Spring '08
 Sturtivant,C

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