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1
Tech
no
log
y I
nd
ep
en
den
t M
ult
i-L
evel
Lo
gic
Op
tim
izati
on
Pro
f. K
urt
Ke
utz
er
Pro
f. S
an
jit
Se
sh
ia
EE
CS
Un
ive
rsit
y o
f C
alifo
rnia
, B
erk
ele
y,
CA
Th
an
ks
to
R. R
ud
ell,
S. M
ali
k, R
. R
ute
nb
ar
K.
Keutz
er
& S
. S
eshia
2
Lo
gic
Op
tim
izati
on
Perf
orm
a v
ari
ety
of
tran
sfo
rmati
on
s a
nd
o
pti
miz
ati
on
s
Str
uctu
ral
gra
ph
tr
an
sfo
rmati
on
s
Bo
ole
an
tra
nsfo
rmati
on
s
Map
pin
g i
nto
a p
hysic
al
lib
rary
sm
all
er,
faste
rle
ss p
ow
er
log
ico
pti
miz
ati
on
netl
ist
netl
ist
Lib
rary
a b
s
q0 1
d
clk
a b
s
q0 1
d
clk
K.
Keutz
er
& S
. S
eshia
3
Wh
at
We’v
e S
een
: E
arl
y “
Syn
the
sis
”F
low
& 2
-Le
vel
Min
imiz
ati
on
FS
MS
yn
thesis
FS
M
log
ic
SO
P l
og
ico
pti
miz
ati
on
PL
A
ph
ysic
al
desig
n
layo
ut
F1 =
B +
D +
A C
+ A
C
I1 I2
O1
O2
K.
Keutz
er
& S
. S
eshia
4
Red
uce
to
Co
mb
inati
on
al
Op
tim
iza
tio
n
B
Fli
p-f
lop
s
Co
mb
inati
on
al
Lo
gic
input
arr
ival tim
es
input
drive
outp
ut
requ
ired
tim
es
outp
ut
loa
d
inputs
outp
uts
K.
Keutz
er
& S
. S
eshia
5
Co
mb
inati
on
al
Lo
gic
Op
tim
izati
on
Inp
ut:
Initia
l B
oole
an c
ircuit
Tim
ing c
hara
cte
rization for
the m
odule
-in
put arr
ival tim
es a
nd d
rive facto
rs
-outp
ut lo
adin
g facto
rs
Optim
ization g
oals
-outp
ut re
quired tim
es o
utp
ut lo
ad
Targ
et lib
rary
description
Ou
tpu
t:
Min
imum
-are
a n
et-
list of lib
rary
gate
s w
hic
h m
eets
tim
ing
constr
ain
ts
A v
ery
difficult o
ptim
ization p
roble
m !
K.
Keutz
er
& S
. S
eshia
6
Mo
dern
Ap
pro
ach
to
Lo
gic
Op
tim
izati
on
Div
ide l
og
ic o
pti
miz
ati
on
in
to t
wo
su
bp
rob
lem
s:
•T
echnolo
gy-independent optim
ization
-dete
rmin
e o
vera
ll lo
gic
str
uctu
re
-estim
ate
costs
(m
ostly)
independent of
technolo
gy
-sim
plif
ied c
ost m
odelin
g
•T
echnolo
gy-d
ependent optim
ization (
technolo
gy
mappin
g)
-bin
din
g o
nto
the g
ate
s in the lib
rary
-deta
iled technolo
gy-s
pecific
cost m
odel
Orc
hestr
ati
on
of
vari
ou
s o
pti
miz
ati
on
/tra
nsfo
rmati
on
te
ch
niq
ues f
or
each
su
bp
rob
lem
K.
Keutz
er
& S
. S
eshia
7
Lo
gic
Op
tim
izati
on
log
ico
pti
miz
ati
on
netl
ist
netl
ist
Lib
rary
tech
ind
ep
en
den
t
tech
dep
en
den
t
2-l
evel
Lo
gic
op
t
mu
ltil
evel
Lo
gic
op
t
Real
Lib
rary
Gen
eri
cL
ibra
ry
K.
Keutz
er
& S
. S
eshia
8
Ou
tlin
e
•M
oti
vati
on
fo
r M
ult
ilevel
Ckts
•O
verv
iew
of
Mu
ltil
evel O
pti
miz
ati
on
•D
eta
ils o
n M
ult
ilevel O
pti
miz
ati
on
Tech
niq
ues
K.
Keutz
er
& S
. S
eshia
9
Wh
y M
ult
ile
ve
l C
om
bin
ati
on
al
Cir
cu
its?
•T
here
are
man
y f
un
cti
on
s t
hat
are
to
o
“exp
en
siv
e”
to i
mp
lem
en
t in
tw
o-l
evel fo
rm
–T
ry 1
6-b
it a
dd
er
⇒ ⇒⇒⇒
32 in
pu
t lin
es a
nd
216
pro
du
ct
term
s!
•D
ela
y v
s.
Are
a t
rad
eo
ff
–2-l
evel ckt:
tin
y d
ela
y, la
rge a
rea (
man
y g
ate
s &
lite
rals
)
–m
ult
i-le
vel:
big
ger
dela
y, le
ss a
rea
K.
Keutz
er
& S
. S
eshia
11
Ou
tlin
e
•M
oti
vati
on
fo
r M
ult
ilevel
Ckts
•O
verv
iew
of
Mu
ltil
evel O
pti
miz
ati
on
•D
eta
ils o
n M
ult
ilevel O
pti
miz
ati
on
Tech
niq
ues
K.
Keutz
er
& S
. S
eshia
12
Rep
res
en
tati
on
: B
oo
lean
Ne
two
rk
a b c
x y
a b c
x y
ab
c +
x
K.
Keutz
er
& S
. S
eshia
13
Bo
ole
an
Netw
ork
, E
xp
lain
ed
It’s
a g
rap
h:
•P
rim
ary
in
pu
ts (
vari
ab
les)
•P
rim
ary
ou
tpu
ts
•In
term
ed
iate
no
des (
in S
OP
fo
rm in
term
s o
f it
s
inp
uts
)
Gen
eri
c lib
rary
–te
ch
no
log
y i
nd
ep
en
den
t
•H
as s
tan
dard
fu
ncti
on
s –
ND
2, N
D4,
AO
I22
•Q
uality
of
netw
ork
: are
a,
dela
y,
…
–m
easu
red
in
term
s o
f #
(lit
era
ls),
dep
th, …
K.
Keutz
er
& S
. S
eshia
14
Tech
.-In
dep
en
den
t M
ult
i-L
evel
Op
tim
izati
on
: O
pera
tio
ns
Invo
lves p
erf
orm
ing
th
e f
ollo
win
g o
pera
tio
ns
“it
era
tively
”u
nti
l “g
oo
d e
no
ug
h”
resu
lt is o
bta
ined
:
1.
Sim
plifi
cati
on
•M
inim
izin
g t
wo
-level lo
gic
fu
ncti
on
(S
OP
fo
r a s
ing
le n
od
e)
2.
Eli
min
ati
on
•S
ub
sti
tuti
ng
on
e e
xp
ressio
n in
to a
no
ther.
3.
De
co
mp
os
itio
n
•E
xp
ressin
g a
sin
gle
SO
P w
ith
2 o
r m
ore
sim
ple
r fo
rms
4.
Extr
acti
on
•F
ind
ing
& p
ullin
g o
ut
su
bexp
ressio
ns
co
mm
on
to
man
y
no
des
5.
Su
bsti
tuti
on
•L
ike e
xtr
acti
on
, b
ut
no
des in
th
e n
etw
ork
are
re-u
sed
K.
Keutz
er
& S
. S
eshia
15
Exam
ple
(du
e t
o G
. D
e M
ich
eli)
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
er
= p
+ a
’s =
r +
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
’c+
qc’
+ q
c
w x y z
#li
tera
ls =
33,
dep
th =
3
K.
Keutz
er
& S
. S
eshia
16
Exam
ple
: E
lim
inati
on
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
er
= p
+ a
’s =
r +
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
’c+
qc’
+ q
c
w x y z
#li
tera
ls =
33,
dep
th =
3
K.
Keutz
er
& S
. S
eshia
17
Exam
ple
: E
lim
inate
no
de r
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
’c+
qc’
+ q
c
w x y z
#li
tera
ls =
32,
dep
th =
2
K.
Keutz
er
& S
. S
eshia
18
Exam
ple
: S
imp
lifi
ca
tio
n
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
’c+
qc’
+ q
c
w x y z
#li
tera
ls =
32,
dep
th =
2
K.
Keutz
er
& S
. S
eshia
19
Exam
ple
: S
imp
lify
ing
no
de u
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
28,
dep
th =
2
K.
Keutz
er
& S
. S
eshia
20
Exam
ple
: D
eco
mp
osit
ion
a b c d e
v =
a’d
+ b
d+
c’d
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
28,
dep
th =
2
K.
Keutz
er
& S
. S
eshia
21
Exam
ple
: D
eco
mp
osin
g n
od
e v
a b c d e
v =
jd
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
27,
dep
th =
2
j =
a’+
b +
c’
K.
Keutz
er
& S
. S
eshia
22
Exam
ple
: E
xtr
acti
on
a b c d e
v =
jd
+ a
e’
p =
ce
+ d
es =
p +
a’+
b’
t =
ac +
ad
+ b
c+
bd
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
27,
dep
th =
2
j =
a’+
b +
c’
K.
Keutz
er
& S
. S
eshia
23
Exam
ple
: E
xtr
acti
ng
fro
m p
an
d t
a b c d e
v =
jd
+ a
e’
p =
ke
s =
p +
a’+
b’
t =
ka +
kb
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
23,
dep
th =
3
j =
a’+
b +
c’
k =
c +
d
K.
Keutz
er
& S
. S
eshia
24
Exam
ple
: W
ha
t n
ex
t?
a b c d e
v =
jd
+ a
e’
p =
ke
s =
p +
a’+
b’
t =
ka +
kb
+ e
q =
a +
bu
= q
+ c
w x y z
#li
tera
ls =
23,
dep
th =
3
j =
a’+
b +
c’
k =
c +
d
K.
Keutz
er
& S
. S
eshia
25
Wh
ich
Op
era
tio
ns D
o W
e K
no
w H
ow
to
Do
?
Invo
lves p
erf
orm
ing
th
e f
ollo
win
g o
pera
tio
ns
“it
era
tively
”u
nti
l “g
oo
d e
no
ug
h”
resu
lt is o
bta
ined
:
1.
Sim
plifi
cati
on
•M
inim
izin
g t
wo
-level lo
gic
fu
ncti
on
(S
OP
fo
r a s
ing
le n
od
e)
2.
Eli
min
ati
on
•S
ub
sti
tuti
ng
on
e e
xp
ressio
n in
to a
no
ther.
3.
De
co
mp
os
itio
n
•E
xp
ressin
g a
sin
gle
SO
P w
ith
2 o
r m
ore
sim
ple
r fo
rms
4.
Extr
acti
on
•F
ind
ing
& p
ullin
g o
ut
su
bexp
ressio
ns
co
mm
on
to
man
y
no
des
5.
Su
bsti
tuti
on
•L
ike e
xtr
acti
on
, b
ut
no
des in
th
e n
etw
ork
are
re-u
sed
K.
Keutz
er
& S
. S
eshia
26
Ou
tlin
e
•M
oti
vati
on
fo
r M
ult
ilevel
Ckts
•O
verv
iew
of
Mu
ltil
evel O
pti
miz
ati
on
•D
eta
ils o
n M
ult
ilevel O
pti
miz
ati
on
Tech
niq
ues
K.
Keutz
er
& S
. S
eshia
27
Need
fo
r F
ac
tori
ng
/Div
isio
n
Facto
red
vers
us D
isju
ncti
ve f
orm
s
su
m-o
f-p
rod
ucts
or
dis
jun
cti
ve f
orm
facto
red
fo
rm
mu
lti-
level o
r co
mp
lex g
ate
Wh
at
we n
eed
is a
wa
y t
o d
o “
div
isio
n”
f=
ac
+a
d+
bc
+b
d+
ae
f=
a+
b(
)c
+d
()
+a
e
K.
Keutz
er
& S
. S
eshia
28
Div
iso
rs a
nd
Deco
mp
osit
ion
Giv
en
Bo
ole
an
fu
nc
tio
n F
, w
e w
an
t to
wri
te i
t a
s
F
=
D .
Q
+
R
wh
ere
D –
Div
iso
r, Q
–Q
uo
tie
nt,
R –
Re
ma
ind
er
De
co
mp
os
itio
n:
Se
arc
hin
g f
or
div
iso
rs w
hic
h a
re
co
mm
on
to
ma
ny f
un
cti
on
s i
n t
he
ne
two
rk
–id
en
tify
div
iso
rs w
hic
h a
re c
om
mo
n t
o s
evera
l fu
ncti
on
s
–in
tro
du
ce c
om
mo
n d
ivis
or
as a
new
no
de
–re
-exp
ress e
xis
tin
g n
od
es u
sin
g t
he n
ew
div
iso
r
K.
Keutz
er
& S
. S
eshia
29
Bo
ole
an
Div
isio
n
Giv
en
Bo
ole
an
fu
nc
tio
n F
, w
e w
an
t to
wri
te it
as
F =
D
. Q
+
R
•D
is a
facto
r o
f F
iff
F . D
’=
0
–O
N-S
ET
(D)
co
nta
ins O
N-S
ET
(F)
•If
F. D
!=
0, t
hen
D i
s a
div
iso
r o
f F
•H
ow
man
y p
ossib
le f
acto
rs D
can
th
ere
be f
or
a
giv
en
F?
K.
Keutz
er
& S
. S
eshia
30
Alg
eb
raic
Mo
del
Real N
um
bers
a.b
= b
.a
a+
b=
b+
a
a.(
b.c
) =
(a.b
).c
a+
(b+
c)
= (
a+
b)+
c
a.(
b+
c)
= a
.b+
a.c
a.1
= a
a
.0 =
0 a+
0 =
a
Bo
ole
an
Alg
eb
ra
a.b
= b
.a
a+
b=
b+
a
a.(
b.c
) =
(a.b
).c
a+
(b+
c)
= (
a+
b)+
c
a.(
b+
c)
= a
.b+
a.c
a.1
= a
a
.0 =
0 a+
0 =
a
a+
(b.c
) =
(a+
b).
(a+
c)
a+
a’=
1 a.a
’=
0 a.a
= a
a
+a=
a
a+
1 =
1 a
+ab
= a
a.(
a+
b)
= a
…
Idea:
Perf
orm
div
isio
n u
sin
g o
nly
th
e r
ule
s (
axio
ms)
of
real
nu
mb
ers
, n
ot
all o
f B
oo
lean
alg
eb
ra
K.
Keutz
er
& S
. S
eshia
31
Alg
eb
raic
Div
isio
n
•A
lit
era
l an
d its
co
mp
lem
en
t are
tre
ate
d a
s
un
rela
ted
–E
ach
lit
era
l as a
fre
sh
vari
ab
le
–E
.g. f =
ab
+ a
’x+
b’y
as f
= a
b+
dx
+ e
y
•T
reat
SO
P e
xp
ressio
n a
s a
po
lyn
om
ial
–D
ivis
ion
/facto
rin
g t
hen
beco
mes p
oly
no
mia
l d
ivis
ion
/facto
rin
g
•B
oo
lean
id
en
titi
es a
re i
gn
ore
d
–E
xcep
t in
pre
-pro
cessin
g
–S
imp
le lo
cal sim
plifi
cati
on
s lik
e a
+ a
b� ���
a p
erf
orm
ed
K.
Keutz
er
& S
. S
eshia
32
Alg
eb
raic
facto
rizati
on
pro
du
ces
Bo
ole
an
facto
rizati
on
pro
du
ces
Alg
eb
raic
vs
. B
oo
lea
n f
ac
tori
zati
on
f=
ab
+a
c+
ba
+b
c+
ca
+c
b
f=
ab
+c
()
+a
b+
c(
)+
bc
+c
b
f=
a+
b+
c(
)a
+b
+c
()
K.
Keutz
er
& S
. S
eshia
33
Alg
eb
raic
Div
isio
n E
xam
ple
F =
ac +
ad
+ b
c+
bd
+ e
Wan
t to
get
Q, R
, w
here
F
= D
Q +
R fo
r
1.
D =
a +
b
2.
D =
a
K.
Keutz
er
& S
. S
eshia
34
Alg
eb
raic
Div
isio
n A
lgo
rith
m
•W
hat
we w
an
t:
–G
iven
F, D
, fi
nd
Q, R
–F
, D
exp
ressed
as s
ets
of
cu
bes (
sam
e f
or
Q, R
)
•A
pp
roach
:
–F
or
each
cu
be C
in
D {
let
B =
{cu
bes in
F c
on
tain
ed
in
C}
if (
B is e
mp
ty)
retu
rn Q
= {
}, R
= F
let
B =
{cu
bes in
B w
ith
vari
ab
les in
C r
em
oved
}
if (
C is t
he f
irst
cu
be in
D w
e’r
e lo
okin
g a
t)
let
Q =
B;
els
e Q
= Q
∩ ∩∩∩B
;
} R =
F \
(Q x
D);
Co
mp
lexit
y?
K.
Keutz
er
& S
. S
eshia
35
Takin
g S
tock
•W
hat
we k
no
w:
–H
ow
to
perf
orm
Alg
eb
raic
div
isio
n g
iven
a d
ivis
or
D
•W
hat
we d
on
’t
–H
ow
to
pic
k D
?
•R
ecall
wh
at
we w
an
ted
to
do
:
–G
iven
2 f
un
cti
on
s F
an
d G
, fi
nd
a c
om
mo
n d
ivis
or
D a
nd
fa
cto
rize t
hem
as
F =
D Q
1 +
R1
G =
D
Q2 +
R2
K.
Keutz
er
& S
. S
eshia
36
New
Te
rmin
olo
gy:
Kern
els
•A
kern
el
of
a B
oo
lean
exp
ressio
n F
is a
cu
be-f
ree e
xp
ressio
nth
at
resu
lts w
hen
yo
u
div
ide F
by a
sin
gle
cu
be
–T
hat
“sin
gle
cu
be”
is c
alled
a c
o-k
ern
el
•C
ub
e-f
ree e
xp
ressio
n:
Can
no
t fa
cto
r o
ut
a
sin
gle
cu
be t
hat
leaves b
eh
ind
no
re
main
der
•E
xam
ple
s:
Wh
ich
are
cu
be-f
ree?
–F
= a
–F
= a
+ b
–F
= a
bc
+ a
bd
K.
Keutz
er
& S
. S
eshia
37
Kern
els
: E
xa
mp
les
F =
ae
+ b
e +
cd
e+
ab
K(f
)
Kern
el C
o-k
ern
el
{a,b
,cd
} e
{e,b
} ?
? b
{ae,b
e,c
de,a
b}
?
K.
Keutz
er
& S
. S
eshia
38
Wh
y a
re K
ern
els
Us
efu
l?
Go
al o
f m
ult
i-le
vel lo
gic
op
tim
izer
is t
o f
ind
co
mm
on
d
ivis
ors
of
two
(o
r m
ore
) fu
ncti
on
s f
an
d
g
Th
eo
rem
: [B
rayto
n&
McM
ullen
]
fan
d
gh
ave a
no
n-t
rivia
l (m
ult
iple
-cu
be)
co
mm
on
d
ivis
or
dif
an
d o
nly
if
there
exis
t kern
els
kf
∈ ∈∈∈K
( f
),
kg
∈ ∈∈∈K
( g )
su
ch
th
at
kf
∩ ∩∩∩k
gis
no
n-t
rivia
l, i.
e., n
ot
a c
ub
e
∴ ∴∴∴can
use k
ern
els
of
fan
d
gto
lo
cate
co
mm
on
d
ivis
ors
K.
Keutz
er
& S
. S
eshia
39
Th
eo
rem
, P
ut
An
oth
er
Way
•F
= D
1 . K
1 +
R1
•G
= D
2 . K
2 +
R2
•K
1 =
(X
+ Y
+ …
) +
stu
ff1
•K
2 =
(X
+ Y
+ …
) +
stu
ff2
•T
hen
,
–F
= (
X +
Y +
…)
D1
+ s
tuff
3
–G
= (
X +
Y +
…)
D2
+ s
tuff
4
•S
o, if
we f
ind
kern
els
an
d in
ters
ect
them
, th
e
inte
rsecti
on
giv
es u
s o
ur
co
mm
on
div
iso
r
K.
Keutz
er
& S
. S
eshia
40
Kern
el
Inte
rse
cti
on
: E
xam
ple
F =
ae
+ b
e +
cd
e+
ab
K(f
)
Kern
el C
o-k
ern
el
{a,b
,cd
} e
{e,b
} a
{e,a
} b
{ae,b
e,c
de,a
b}
1
G =
ad
+ a
e+
bd
+ b
e +
bc
K(g
)
Kern
el C
o-k
ern
el
{a,b
} d
or
e
{d,e
} a
or
b
{d,e
,c}
b
{ad
,ae,b
d,b
e,b
c}
1
K.
Keutz
er
& S
. S
eshia
41
Ho
w d
o w
e f
ind
Kern
els
?
Overv
iew
: G
iven
a f
un
cti
on
F
1.
Pic
k a
vari
ab
le x
ap
peari
ng
in
F, an
d u
se it
as a
d
ivis
or
2.
Fin
d t
he c
orr
esp
on
din
g k
ern
el K
if
on
e e
xis
ts (a
t le
ast
2 c
ub
es i
n F
co
nta
in x
)
•If
no
t, g
o b
ack t
o (
1)
an
d p
ick a
no
ther
vari
ab
le
3.
Us
e K
in
pla
ce o
f F
an
d r
ecu
rse
to f
ind
kern
els
of
K •F
= x
K +
R an
d K
= y
M +
S � ���
F =
xy
M +
…
•A
dd
kern
els
of
K t
o t
ho
se o
f F
4.
Go
back t
o (
1)
an
d p
ick a
no
ther
vari
ab
le t
o k
eep
fi
nd
ing
kern
els
K.
Keutz
er
& S
. S
eshia
42
Fin
din
g K
ern
els
: E
xa
mp
le
F =
ab
c+
ab
d+
bcd
bc
+ b
dac +
ad
+ c
dab
+ b
dab
+ b
c
Can
we d
o b
ett
er?
K.
Keutz
er
& S
. S
eshia
43
Fin
din
g K
ern
els
: E
xa
mp
le
F =
ab
c+
ab
d+
bcd
bc
+ b
dab
c+
ad
+ c
dab
c&
bcd
bo
th
co
nta
in c
.In
ters
ecti
on
is
bc.
Re
cu
rse
on
F
/bc
= a
+d
ab
+ b
c
Take i
nte
rsecti
on
of
all c
ub
es c
on
tain
ing
a v
ari
ab
le
K.
Keutz
er
& S
. S
eshia
44
Kern
el
Fin
din
g A
lgo
rith
m
Fin
dK
ern
els
(F)
{
K =
{ }
;
for
(each
vari
ab
le x
in
F)
{
if (
F h
as a
t le
ast
2 c
ub
es c
on
tain
ing
x)
{
let
S =
{cu
bes in
f c
on
tain
ing
x};
let
c =
cu
be r
esu
ltin
g f
rom
in
ters
ecti
on
of
all c
ub
es in
S
K =
K ∪ ∪∪∪
Fin
dK
ern
els
(F/c
); //re
cu
rsio
n
}
} K =
K ∪ ∪∪∪
F ;
retu
rn K
;
}
K.
Keutz
er
& S
. S
eshia
45
Giv
en
a f
un
cti
on
f
to b
e s
tro
ng
div
ided
by
g
Ad
d a
n e
xtr
a i
np
ut
to
fco
rresp
on
din
g t
o
g,
nam
ely
G
an
d o
bta
in f
un
cti
on
h
as f
ollo
ws
Min
imiz
e
hu
sin
g t
wo
-level
min
imiz
er
Str
on
g (
or
Bo
ole
an
) D
ivis
ion
hO
N=
f ON
− −−−h
DC
hO
FF
= ===f O
N+ +++
hD
C
hD
C=
Gg
+G
gIn
pu
ts t
o f
th
at
can
no
t o
ccu
r
Get:
h
= Q
G +
R
K.
Keutz
er
& S
. S
eshia
46
Ad
dit
ion
al
Rea
din
g
1.
Re
ad
Ch
ap
ter
7 u
pto
Sec.
7.6
2.
R.
Ru
dell, ‘L
og
ic S
yn
thesis
fo
r V
LS
I D
esig
n’,
Ph
D T
hesis
, U
C B
erk
ele
y,
198
9.
3.
R.
Bra
yto
n, G
. H
ach
tel, A
. S
an
gio
van
ni-
Vin
cen
telli, ‘M
ult
ileve
l L
og
ic S
yn
thesis
’,
Pro
ce
ed
ing
s o
f th
e IE
EE
, F
eb
’90.
OP
TIO
NA
L
K.
Keutz
er
& S
. S
eshia
47
Lo
gic
op
tim
iza
tio
n -
su
mm
ary
Cu
rren
t fo
rmu
lati
on
of
log
ic s
yn
thesis
an
d o
pti
miz
ati
on
is t
he
mo
st
co
mm
on
tech
niq
ues f
or
desig
nin
g in
teg
rate
d c
ircu
its
tod
ay
Has b
een
th
e m
ost
su
ccessfu
l d
esig
n p
ara
dig
m 1
989 -
pre
sen
t
Alm
ost
all d
igit
al cir
cu
its a
re t
ou
ch
ed
by lo
gic
syn
thesis
•M
icro
pro
cesso
rs (
co
ntr
ol p
ort
ion
s/r
an
do
m g
lue lo
gic
~ 2
0%
)
•A
pp
licati
on
sp
ecif
ic s
tan
dard
part
s (A
SS
Ps)-
20 -
90
%
•A
pp
licati
on
sp
ecif
ic in
teg
rate
d c
ircu
its (
AS
ICs)
-40 -
100
%
Real lo
gic
op
tim
izati
on
syste
ms o
rch
estr
ate
op
tim
izati
on
s
•T
ech
no
log
y in
dep
en
den
t
•T
ech
no
log
y d
ep
en
den
t
NE
XT
LE
CT
UR
E
•A
pp
licati
on
sp
ecif
ic (
e.g
. d
ata
path
ori
en
ted
)
