Post on 22-Jan-2022
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 .
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