Ted Kaminski
CSci 5403  Computational Complexity
Homework 2 Problem 4 Solution
Corrected problem statement
Prove that if
NP
⊆
S
c
∈
N
dtime
(
n
c
log
n
) =
dtime
(
n
O
(log
n
)
), then for each
i
there exists a constant
k
such that Σ
i
⊆
dtime
(
n
O
((log
n
)
k
)
).
Conclude that under this hypothesis,
PH
⊆
S
k
∈
N
dtime
(
n
(log
n
)
k
) =
dtime
(
n
log
O
(1)
n
).
Proof of the second part
We show that
PH
⊆
S
k
∈
N
dtime
(
n
(log
n
)
k
) =
dtime
(
n
log
O
(1)
n
) under the above hypothesis.
Recall the
definition of
PH
:
PH
=
[
k
∈
N
Σ
k
Our hypothesis is that Σ
i
⊆
dtime
(
n
O
((log
n
)
k
)
), from which we can conclude:
[
k
∈
N
Σ
k
⊆
[
k
∈
N
dtime
(
n
O
((log
n
)
k
)
)
And to be absolutely precise, we can show that
S
k
∈
N
dtime
(
n
O
((log
n
)
k
)
) =
S
k
∈
N
dtime
(
n
(log
n
)
k
) because
dtime
(
n
O
((log
n
)
k
)
)
⊆
dtime
(
n
(log
n
)
k
0
) for some
k
0
> k
. Thus demonstrating that
PH
⊆
[
k
∈
N
dtime
(
n
(log
n
)
k
)
Proof of the first part
We show by induction that assuming
NP
⊆
S
c
∈
N
dtime
(
n
c
log
n
) =
dtime
(
n
O
(log
n
)
), we can conclude that
for every
i
, there is a constant
k
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 Spring '08
 Sturtivant,C
 Computational complexity theory, Σi

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