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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