CPSC 421/501

- End of new material :

A portion of 89.3 of [Sip]

"

How to solve P vs . HP"

really How people are trying to solve P vs HP. . .

-

• 1 Presentation today ! Pumping Lemma

- I'll post the topics for Hou 26

Deal ④→ thee:bit of time

[ Think: final 17that 7-p.m )t-

-

§ 9.3 ! You have a Boolean function

f -- fcx , , - - . ,xn ) : { F. T }

"

→ { f. T}

often I In f I

{ o , I } → { c ,I}

Sipser :"

parity"

f ( Xuxa ,xg,Xy) = X,④ A.④ Xz④X34

where ④ = XOR = exclusive or

Min formula Size , Min Circuit Size-

Overview !*# Boden functors on h -variables 2k

"

)

f s f ( x, , - - , Xnt = TH-

zn TIF

⇒"

Most"

Baden functions on n - variables

have minimum circuit size I 2%3h- Cook - Levin approach ⇒

SAT,3SAT

,. .. any

L language

if LEP,then there are polynomial

size circuits to @ompute !

given input w , Iwish,

there is

a circuit of size poly Cm) to determine

whether or not w c- L.

Gwen FCK,- → Xy) F X ,@X*Xz④Xy

f ( x. .kz) - X

,④ Xz

"

parity functions"

f- GO,Ta l

f ( x, ,×,)

= fl if Xiso , Xin

Xi. I , Xzso

0 otherwise

= ( X ..

+ Xz) mod 2

( HII,

'

IIin contexts thy Cpsc 506,1

fix, ,Xz , Xz ,x.D= X

, Xz E Xz ④ Xu,

= (X ,exit xzxx.ie ) mad 2

Compute :

ANDf-as formula

, say or

T

X, ④ Xz = ( X

,

AND 2X,) or

(TX,AND Xz )

OR- n

ANDAND

Ii il f :I I n

X, Xz X

, Xz at(bottom

E.g .

formulaOR

-- i

/ OR

DAD l l

n :ON AND

'

7

/ OR

AND ← y yl ln

.

n

f,

f-z f ! f2

' ' ' '

Smallest formula for fcxi,-

→ Xn)

is smallest tree size to compute\

fon

# leaves

← →AND AND

= #

literals← b ← y

X,7 Xz TX

, Xz

(X , n 7×2) v ( ax , rxz )

r i i r

size formula = 4

-Circuit to I

f)t

compute f ←A" D

J N V

X,TX

, Xz - Xz . -

- Xn

Circuit ( formula t . compete f :

(X. ④ Xz) ⑤ ( Xz ④ Xy)

A

fond#AND y.at#D

X,- X

, Xz axe Xz TX , Xu, My

formula

,#Into

✓A

#AND

pX,TX

, Ky nxzX,TX

,H2 TX

,

- formula-

can't reuse

intermediate nodes

Formula.-⇒ Tree ; Circuit → Directed

Acyclic CryL

Another example :

Say that you have LEP

Cook - Levin Thm :given W

,

Odp)⇐"

"

:{"

"* "" """

# variables

¥ii¥÷¥f÷:1-H ' pdyln) tape calls I

,. -

,Htt

Xij, s fT if step i , cell j has symtdkF otherwise

←# vars :

(Ntl) -CNN) . try

Xijk Yij ZigTinta - ~ } Ideatape head state your

location time Variable,

TM man- deterministic -7 SAT,3 SAT

- instance

TM deterministic !

j.k.is/ixi,j,kYiel,jZixi,svarying-

step ix. I

= some Boolean functions )L

step i

-Xijk , HY -

ij ,

2-i. s

a',I

step i -2

⇒XH, j ,k i Y N

, j i7His

↳xp -link ,

- -

C. x..

"l

l

l

×0.j.la most Erath

of n variables§input

require §%⇒ given input , 1W

,lwlsn ,

you have circuit based on 01182 )functions Xij .k , Yij ,

2-isto see

whether a- not w c- L

If you can prove that given

graph size r ,

(I ) possible edges

n -- (5)-

COLOUR i ix.→ TIF

these functions require more

then pdylr) spdyln ) circuit gates=) p f- NP Es 9.3 of Grp]