NP theory

Pasi Fränti22.10.2013

Turing machine vs. RAM

© GEOFF DRAPER

Palindrome example

……

SOS

ABBA

SAIPPUAKAUPPIAS

INNOSTUNUT SONNI

SAIPPUAKIVIKAUPPIAS

Turing machine solving palindrome

Accept!

q1

q2

q4

q0

q3

Final

Initial

0,b1,b

b,b

1,b0,b

Copy

Rewind

Compare

b,b

b,b

b,0b,1

1,10,0

b,b

q0 b b q4 (b,S) (b,S) Empty

0 b q1 (0,S) (b,S)

1 b q1 (1,S) (b,S)

q1 0 b q1 (0,R)(0,R

)Copy

1 b q1 (1,R)(1,R

)

b b q2 (b,S) (b,L)

q2 b 0 q2 (b,S) (0,L) Move

b 1 q2 (b,S) (1,L)

b b q3 (b,L)(b,R

)

q3 0 0 q3 (0,L)(0,R

)Compar

e

1 1 q3 (1,L)(1,R

)

B b q4 (b,S) (b,S)

product := n;FOR i:=1 TO (n-1) DO

product := product * n;

Assembler

READ r1;IF r1≤0 THEN WRITE 0ELSE

r2 r1;r3 r1-1;WHILE r3>0 DOr2 r2*r1;r3 r3-1;OD;

WRITE r2;

PASCAL

ALGOL

READ 1LOAD 1JGTZ posWRITE =0JUMP endif

pos: LOAD 1STORE 2LOAD 1SUB =1STORE 3

while: LOAD 3JGTZ contJUMP endw

cont: LOAD 2MULT 1STORE 2LOAD 3SUB =1STORE 3JUMP while

endw: WRITE 2endif: HALT

Three programs to calculate NN

Loopvariable

Output

READ r1

IF r1≤0

r2 r1

r3 r3-1

r2 r2*r1

r3 r3-1

WHILE r3>0

WRITE r2

Input

RAM and Turing compared

RAM Turing

Sequence:

Loop:

Condition:

WHILE r3>0 DO

IF r1≤0 THEN

…r2 r2*r1;r3 r3-1;…

q1

q2

q3

q4a b

q5

a

b

RAM TuringOne step: O(1) S(N)T(N) steps T(N) T(N)∙S(N) ≤ T(N)2

Simulating RAM by Turing

x x x x x Ь ЬxЬЬ ……

R/W head Next needed

S(N)

Rewind

Empty space for notes

P and NP classes

P Problems that can be solvedin polynomial timeby deterministic Turing machine

P

P and NP classes

P Problems that can be solvedin polynomial timeby deterministic Turing machine

NP Problems that can be solvedin polynomial timeby non-deterministic Turing machine

NP

P

Known problems in P

NP

P

SORT

MM

SEARCH

SP

SELECTION

MST

Known problems in NP

NP

P

COLORING

TSP

EXACTCOVER

KP

SATSORT

MM

SEARCH

SP

SELECTION

MST

Anything outside of NP ?

NP

P CHESS

GO

COLORING

TSP

EXACTCOVER

KP

SAT

HANOISORT

MM

SEARCH

SP

SELECTION

MST

Coloring(E,V,k)

FOR i1 TO n DOColor[i] Choose(1,k);

FOR All e(i,j)V DOIF Color[i]=Color[j] FAIL;

SUCCESS;

Select combination

Check validity

Non-deterministic algorithmColoring problem

TSP(E,V,k)

Sum 0;

FOR i1 TO n-1 DONode[i] Choose(1,n);Sum Sum + w(Node[i-1],Node[i]);

IF Sum<k THEN SUCCESS;ELSE FAIL;

Select combination

Check validity

Non-deterministic algorithmTraveling salesman problem

Reduction of problems

L1 ≤p L2

Search ≤p Sort

Transitivity:

L1 ≤p L2 and L2 ≤p L3

L1 ≤p L3

A2

I1 I2p

pO1 O2

SortIsearch

Osearch Osortmin

A3

I1 I2p

pO1 O2

I3p

p O3

Empty space for notes

NP-hard

NP

P

NP-hard

L1

LL2

L4

L3 ≤p

≤p

≤p≤p

Problem L is NP-hard iff all problems in NP reduces to L in polynomial time:

Lx ≤p L LxNP

NP complete

NP

P

Problem L is NP-complete iff L is NP-hard LNP

NP-hard

NP-complete

NP-completeAnother view

P

NPcomplete

Difficultyincreases

Problem L is NP-complete iff L is NP-hard LNP

NP

P

Intersection of P and NP-hard

Case 1: PNP-hard =

NP-hard

NP

P

Case 2: L: LP L NP-hard

NP-hard

P=NP

Open question: P=NP ?

NP

PNP-hard

Empty space for notes

Turing is still going on strong…

http://www.youtube.com/watch?v=cYw2ewoO6c4