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# Lecture 27 207 Lecture 27 207 Notes | EduRev

## : Lecture 27 207 Lecture 27 207 Notes | EduRev

``` Page 1

Lecture 27

207

Page 2

Lecture 27

207

Beyond NP

Many complexity classes are worse, e.g. time 2
2
n
, 2
2
2
n
, …

Others seem to be “worse” in a different sense, e.g., not in
NP, but still exponential time.  E.g., let

Lp = “assignment y satis?es formula x”, ? P

Then :

SAT = { x | ?y ?x,y??L
P
}

UNSAT = { x | ?y ?x,y??L
P
}

QBF
k
= { x | ?y
1
?y
2
?y
3
…
k
y
k
?x,y
1
…y
k
??L
P
}

QBF
8
= { x | ?y
1
?y
2
?y
3
…

?x,y
1
…   ??L
P
}

208

Q

Page 3

Lecture 27

207

Beyond NP

Many complexity classes are worse, e.g. time 2
2
n
, 2
2
2
n
, …

Others seem to be “worse” in a different sense, e.g., not in
NP, but still exponential time.  E.g., let

Lp = “assignment y satis?es formula x”, ? P

Then :

SAT = { x | ?y ?x,y??L
P
}

UNSAT = { x | ?y ?x,y??L
P
}

QBF
k
= { x | ?y
1
?y
2
?y
3
…
k
y
k
?x,y
1
…y
k
??L
P
}

QBF
8
= { x | ?y
1
?y
2
?y
3
…

?x,y
1
…   ??L
P
}

208

Q

S
P
2
:  { x | ?y?z ?x,y,z??L
P
}
?
?
P
0
:
P
The “Polynomial Hierarchy”

?
P
1
: P time
given SAT

S
P
1
(NP):
{ x | ?y ?x,y??L
P
}
SAT, Clique, VC, HC, Knap,…
?
P
1

(co-NP):
{ x | ?y ?x,y??L
P
}
UNSAT,…
?
P
2

:  { x | ?y?z ?x,y,z??L
P
}
Potential Utility: It is often easy to give such a quantifier-based
characterization of a language; doing so suggests (but doesn’t prove) whether it
is in P, NP, etc. and suggests candidates for reducing to it.
Page 4

Lecture 27

207

Beyond NP

Many complexity classes are worse, e.g. time 2
2
n
, 2
2
2
n
, …

Others seem to be “worse” in a different sense, e.g., not in
NP, but still exponential time.  E.g., let

Lp = “assignment y satis?es formula x”, ? P

Then :

SAT = { x | ?y ?x,y??L
P
}

UNSAT = { x | ?y ?x,y??L
P
}

QBF
k
= { x | ?y
1
?y
2
?y
3
…
k
y
k
?x,y
1
…y
k
??L
P
}

QBF
8
= { x | ?y
1
?y
2
?y
3
…

?x,y
1
…   ??L
P
}

208

Q

S
P
2
:  { x | ?y?z ?x,y,z??L
P
}
?
?
P
0
:
P
The “Polynomial Hierarchy”

?
P
1
: P time
given SAT

S
P
1
(NP):
{ x | ?y ?x,y??L
P
}
SAT, Clique, VC, HC, Knap,…
?
P
1

(co-NP):
{ x | ?y ?x,y??L
P
}
UNSAT,…
?
P
2

:  { x | ?y?z ?x,y,z??L
P
}
Potential Utility: It is often easy to give such a quantifier-based
characterization of a language; doing so suggests (but doesn’t prove) whether it
is in P, NP, etc. and suggests candidates for reducing to it.
Examples

QBF
k
in S
P
k

Given graph G, integers j & k, is there a set U of = j vertices
in G such that every k-clique contains a vertex in U?

Given graph G, integers j & k, is there a set U of = j vertices
in G such removal of any k edges leaves a Hamilton path
in U?

210

Page 5

Lecture 27

207

Beyond NP

Many complexity classes are worse, e.g. time 2
2
n
, 2
2
2
n
, …

Others seem to be “worse” in a different sense, e.g., not in
NP, but still exponential time.  E.g., let

Lp = “assignment y satis?es formula x”, ? P

Then :

SAT = { x | ?y ?x,y??L
P
}

UNSAT = { x | ?y ?x,y??L
P
}

QBF
k
= { x | ?y
1
?y
2
?y
3
…
k
y
k
?x,y
1
…y
k
??L
P
}

QBF
8
= { x | ?y
1
?y
2
?y
3
…

?x,y
1
…   ??L
P
}

208

Q

S
P
2
:  { x | ?y?z ?x,y,z??L
P
}
?
?
P
0
:
P
The “Polynomial Hierarchy”

?
P
1
: P time
given SAT

S
P
1
(NP):
{ x | ?y ?x,y??L
P
}
SAT, Clique, VC, HC, Knap,…
?
P
1

(co-NP):
{ x | ?y ?x,y??L
P
}
UNSAT,…
?
P
2

:  { x | ?y?z ?x,y,z??L
P
}
Potential Utility: It is often easy to give such a quantifier-based
characterization of a language; doing so suggests (but doesn’t prove) whether it
is in P, NP, etc. and suggests candidates for reducing to it.
Examples

QBF
k
in S
P
k

Given graph G, integers j & k, is there a set U of = j vertices
in G such that every k-clique contains a vertex in U?

Given graph G, integers j & k, is there a set U of = j vertices
in G such removal of any k edges leaves a Hamilton path
in U?

210

Space Complexity

DTM M has space complexity S(n) if it halts on all inputs, and
never visits more than S(n) tape cells on any input of
length n.

NTM …on any input of length n on any computation path.

DSPACE(S(n)) = { L | L acc by some DTM in space O(S(n)) }

NSPACE(S(n)) = { L | L acc by some NTM in space O(S(n)) }

211

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