2
May 5, 2011
7
Useful characterization
• Recall: L
NP
iff expressible as
L = { x |
9
y, |y| ≤ |x|
k
, (x, y)
R }
where R
P
.
• Corollary: L
coNP
iff expressible as
L = { x |
8
y, |y| ≤ |x|
k
, (x, y)
R }
where R
P
.
May 5, 2011
8
Useful characterization
Theorem
: L
Σ
i
iff expressible as
L = { x |
9
y, |y| ≤ |x|
k
, (x, y)
R }
where
R
Π
i-1
.
• Corollary: L
Π
i
iff expressible as
L = { x |
8
y, |y| ≤ |x|
k
, (x, y)
R }
where
R
Σ
i-1
.
May 5, 2011
9
Useful characterization
• Proof of Theorem:
– induction on i
– base case (i =1) on previous slide
(
)
– we know
Σ
i
= NP
Σ
i-1
= NP
Π
i-1
– guess y, ask oracle if
(x, y)
R
Theorem
: L
Σ
i
iff expressible as
L = { x |
9
y, |y| ≤ |x|
k
, (x, y)
R },
where
R
Π
i-1
.
May 5, 2011
10
Useful characterization
(
)
– given
L
Σ
i
= NP
Σ
i-1
decided by ONTM M
running in time n
k
– try:
R = { (x, y) : y describes valid path of M’s
computation leading to q
accept
}
– but how to recognize valid computation path
when it depends on result of oracle queries?
Theorem
: L
Σ
i
iff expressible as
L = { x |
9
y, |y| ≤ |x|
k
, (x, y)
R },
where
R
Π
i-1
.
May 5, 2011
11
Useful characterization
– try:
R = { (x, y) : y describes valid path of M’s
computation leading to q
accept
}
– valid path
= step-by-step description including
correct
yes/no answer for each A-oracle query z
j
(A
Σ
i-1
)
– verify “no” queries in
Π
i-1
:
e.g: z
1
A
z
3
A
…
z
8
A
– for each “yes” query z