NP theory
Pasi Fränti
25.10.2016
Turing machine vs. RAM
© GEOFF DRAPER
Palindrome example
…
…
SOS
ABBA
SAIPPUAKAUPPIAS
INNOSTUNUT SONNI
SAIPPUAKIVIKAUPPIAS
Turing machine solving palindrome
q0 b b
0 b
1 b
q1 0 b
1 b
b b
q2 b 0
b 1
b b
q3 0 0
1 1
B b
q4
q1
q1
q1
q1
q2
q2
q2
q3
q3
q3
q4
(b,S)
(0,S)
(1,S)
(0,R)
(1,R)
(b,S)
(b,S)
(b,S)
(b,L)
(0,L)
(1,L)
(b,S)
(b,S) Empty
(b,S)
(b,S)
(0,R) Copy
(1,R)
(b,L)
(0,L) Move
(1,L)
(b,R)
(0,R) Compare
(1,R)
(b,S)
Initial
b,b
q0
0,b
1,b
b,b
Rewind
q2
Copy
q1
0,b
1,b
b,b
b,0
b,1
Compare
b,b
q3
0,0
1,1
q4
Final
Accept!
Three programs to calculate NN
PASCAL
Assembler
product := n;
FOR i:=1 TO (n-1) DO
product := product * n;
ALGOL
Input
READ r1;
IF r1≤0 THEN WRITE 0
Loop
ELSE
variable
r2  r1;
r3  r1-1;
WHILE r3>0 DO
r2 r2*r1;
r3  r3-1;
Output
OD;
WRITE r2;
READ
LOAD
JGTZ
WRITE
JUMP
pos:
LOAD
STORE
LOAD
SUB
STORE
while: LOAD
JGTZ
JUMP
cont: LOAD
MULT
STORE
LOAD
SUB
STORE
JUMP
endw: WRITE
endif: HALT
1
1
pos
=0
endif
1
2
1
=1
3
3
cont
endw
2
1
2
3
=1
3
while
2
READ r1
IF r1≤0
r2  r1
r3  r1-1
WHILE r3>0
r2  r2*r1
r3  r3-1
WRITE r2
RAM and Turing compared
RAM
Sequence:
Turing
…
r2 r2*r1;
r3  r3-1;
…
Condition:
IF r1≤0 THEN
Loop:
WHILE r3>0 DO
q1
q2
q3
a
q4
q5
a
b
b
Simulating RAM by Turing
One step:
T(N) steps
RAM
Turing
O(1)
T(N)
S(N)
T(N)∙S(N) ≤ T(N)2
S(N)
… Ь Ь
x
x
x
R/W head
x
x
Rewind
x
Ь Ь …
Next needed
Empty space for notes
P and NP classes
P
Problems that can be solved
in polynomial time
by deterministic Turing machine
P
P and NP classes
P
Problems that can be solved
in polynomial time
by deterministic Turing machine
NP Problems that can be solved
in polynomial time
by non-deterministic Turing machine
NP
P
Known problems in P
NP
P
SEARCH
SORT
MM
MST
SP
SELECTION
Known problems in NP
NP
COLORING
TSP
P
SEARCH
SORT
MM
EXACT
COVER
MST
KP
SP
SAT
SELECTION
Anything outside of NP ?
NP
EXACT
COVER
COLORING
TSP
KP
P
SEARCH
SORT
MM
GO
MST
SP
SELECTION
SAT
CHESS
HANOI
Non-deterministic algorithm
Coloring problem
Coloring(E,V,k)
FOR i1 TO n DO
Color[i]  Choose(1,k);
Select
combination
FOR All e(i,j)V DO
IF Color[i]=Color[j] FAIL;
Check
validity
SUCCESS;
Non-deterministic algorithm
Traveling salesman problem
TSP(E,V,k)
Sum  0;
Select
combination
FOR i1 TO n-1 DO
Node[i]  Choose(1,n);
Sum  Sum + w(Node[i-1],Node[i]);
IF Sum<k THEN SUCCESS;
ELSE FAIL;
Check
validity
Reduction of problems
L1 ≤p L2
Search ≤p Sort
p
I1
O1
p
Isearch
Osearch
I2
O2
min
A2
Osort
Sort
Transitivity:
L1 ≤p L2 and L2 ≤p L3

L1 ≤p L3
I1
O1
p
p
I2
O2
p
p
I3
O3
A3
Empty space for notes
NP-hard
Problem L is NP-hard iff all problems in NP
reduces to L in polynomial time:
Lx ≤p L  LxNP
L3
NP
≤p
L2
P
L1
≤p
NP-hard
≤p
L4
L
≤p
NP complete
Problem L is NP-complete iff
 L is NP-hard
 LNP
NP-complete
NP
P
NP-hard
NP-complete
Another view
Problem L is NP-complete iff
 L is NP-hard
 LNP
NP
complete
Difficulty
increases
P
Intersection of P and NP-hard
Case 1: PNP-hard = 
NP
NP-hard
P
Case 2: L: LP  L NP-hard
NP
P
NP-hard
Open question: P=NP ?
NP
P

P=NP
NP-hard
Empty space for notes
Turing is still going on strong…
http://www.youtube.com/watch?v=cYw2ewoO6c4