Today Cook Levin Theorem

If L C HP non deterministic poly timme then

you can solve L in poly time if you have a

button that solves SAT or 3 SAT

Recall i SAT f f is a Boolean

formulathat is satisfiable

e gBodenformula e g X and 7 2

or Xg and n X or

f t E E g X G n 9 7 and or

net NT if

e s f X or 7133

f X 171 on 7 X l 3 3 E EsE

f is satisfiable if f f X Rn and there is

an assignment X TH X TH Xn Ttfthat makes f true

For reasons to bemade clear i

f is 3C IVF 3 conjuatuve normal form if

f blah and blah and blah and and blahswhere each blah B or B or Z each 3 is a

variable X Xn or TX TX in a

Idea SAT 3 SAT E N P I.e if fs

is a Boolean formula you can gess Tlf values farin

its variables write down non derministically

and you can verify whether f holds or not

Also 3COLOR c NP

we claim SAT 3 SAT is as difficult a question

as there is in HP

Pvs Np Is SAT in P

For this reason we say that SAT is NP complete

I SAT 2SAT 3 SAT 4 SAT

1COLOR 2COLORg3

COLOR 4COLOR

T TF Tfin P NP complete

Given a non det Turing machine M input w

let's simulate if w is accepted non det by Mi

T m computation

qInitial config i9o

step Inon deterministic

Step 2

step 3 d start

Hermean

tsetse

etc YXij

T ice wesymbol V

step gaF otherwise

when is the computation possible

yis ft ifstcnm.hn attmeiq fTifegttinneiIf otherwise State q

F otherwiseTime hs SAS steps

insteps

cell number E NS

Xijri cell www.fnnss rest e

j tinystep

X variables E ns.ns I r f s e n

y vars z vars E polynomial in h

can you reach qaco in Ms timelsteps

Yes If at step 1 i formula that express thatyou see c tape what

you shieldfrom step I step 2 everything is valid

AND step2 step3AND step step 4 u n l

ANDAND f stepns.t step ns n n n

ANDstepnsweare.in teqacc

EgcNfIZnsqaa T

7ns q.co