1
CS151
Complexity Theory
Lecture 13
May 10, 2011
May 10, 2011
2
Alternating quantifiers
Pleasing viewpoint:
P
NP
coNP
Δ
3
Σ
2
Π
2
Δ
2
“
9
”
“
8
”
“
98
”
“
89
”
Σ
3
Π
3
PSPACE
PH
“
989
”
“
898
”
“
9898989
…”
Σ
i
Π
i
“
989
…”
“
898
…”
const. # of
alternations
poly(n)
alternations
May 10, 2011
3
PH collapse
Theorem
: if
Σ
i
=
Π
i
then for all j > i
Σ
j
=
Π
j
=
Δ
j
=
Σ
i
“the polynomial hierarchy
collapses
to the i-
th level”
•
Proof:
–
sufficient to show
Σ
i
=
Σ
i+1
–
then
Σ
i+1
=
Σ
i
=
Π
i
=
Π
i+1
; apply theorem again
P
NP
coNP
Σ
3
Π
3
Δ
3
PH
Σ
2
Π
2
Δ
2
May 10, 2011
4
PH collapse
–
recall: L
Σ
i+1
iff expressible as
L = { x |
9
y (x, y)
R }
where R
Π
i
–
since
Π
i
=
Σ
i
,
R expressible as
R = { (x,y) |
9
z (
(x, y),
z)
R’ }
where R’
Π
i-1
–
together:
L = { x |
9
(y, z)
(x, (y, z))
R’}
–
conclude L
Σ
i
May 10, 2011
5
Karp-Lipton
•
we know that
P
=
NP
implies SAT has
polynomial-size circuits.
•
suppose SAT has poly-size circuits
–
any consequences?
–
might hope: SAT
P/poly
PH
collapses to
P
, same as if SAT
P
Theorem
(KL): if SAT has poly-size circuits
then
PH
collapses to the
second
level.
May 10, 2011
6
BPP
PH
•
Recall: don’t know
BPP
different from
EXP
Theorem
(S,L,GZ):
BPP
(
Π
2
Σ
2
)
•
don’t know
Π
2
Σ
2
different from
EXP
but
believe much weaker

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