two-level logic minimization prof. srinivas devadas mit ...keutzer/classes/... · the quine -...
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1
Tw
o-L
evel
Lo
gic
Min
imiz
ati
on
Pro
f. S
rin
iva
s D
ev
ad
as
MIT
Pro
f. K
urt
Ke
utz
er
Pro
f. R
ich
ard
Ne
wto
n
Pro
f. S
an
jit
Se
sh
ia
Un
ive
rsit
y o
f C
ali
forn
ia
Be
rke
ley,
CA
2
To
pic
s
•M
oti
vati
on
•B
oo
lean
fu
ncti
on
s &
no
tati
on
•E
xact
2-l
evel lo
gic
min
imiz
ati
on
–Q
uin
e-M
cC
luske
y
•H
eu
risti
c 2
-level m
inim
izati
on
–M
INI, E
sp
resso
3
Sch
em
ati
c E
ntr
y E
ra
Giv
en
:
•G
ate
-level sch
em
ati
c e
ntr
y e
dit
or
•G
ate
-level sim
ula
tor
(we h
aven
’t t
alk
ed
ab
ou
t th
is)
•G
ate
level sta
tic-t
imin
g a
naly
zer
•N
etl
ist� ���
Layo
ut
flo
w
•W
e c
an
(an
d d
id)
bu
ild
larg
e-s
cale
in
teg
rate
d (
35,0
00 g
ate
) cir
cu
its
•E
DA
ven
do
rs p
rovid
ed
fro
nt-
en
d t
oo
ls a
nd
AS
IC v
en
do
r (e
.g.
LS
I L
og
ic)
pro
vid
ed
back-e
nd
flo
w
•B
ut
…It
ma
y b
e m
uch
mo
re n
atu
ral, a
nd
pro
du
cti
ve, to
d
escri
be c
om
ple
x c
on
tro
l lo
gic
by B
oo
lean
eq
uati
on
s t
han
by
a s
ch
em
ati
c n
etl
ist
of
gate
s
4
Fo
r exam
ple
: tr
aff
ic lig
ht
co
ntr
oller
5
As a
Sta
te t
ran
sit
ion
dia
gra
m
6
Syn
thesiz
e L
og
ic t
o Im
ple
men
t eq
uati
on
s
B
Fli
p-f
lop
s
Co
mb
inati
on
al
Lo
gic
inputs
outp
uts
Desc
rib
e u
sin
g B
oo
lean
eq
uati
on
s
7
Ph
ys
ica
lly I
mp
lem
en
t: A
ND
-OR
an
d N
OR
-NO
R P
LA
s
I1 I2
O1
O2
I1 I2
O1
O2
Lo
gic
in
cre
as
es w
ith
th
e n
um
ber
of
pro
du
ct
term
s
8
Earl
y “
Syn
thesis
”F
low
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
9
Key T
ech
no
log
y:
SO
P L
og
ic M
inim
izati
on
Can
reali
ze a
n a
rbit
rary
lo
gic
fu
ncti
on
in
su
m-o
f-p
rod
ucts
or
two
-level fo
rm
F1 =
A B
+ A
B D
+ A
B C
D
+ A
B C
D +
A B
+ A
B D
F1 =
B +
D +
A C
+ A
C
Of
gre
at
inte
rest
to f
ind
a m
inim
um
su
m-o
f-p
rod
ucts
rep
res
en
tati
on
10
Defi
nit
ion
s -
1
Ba
sic
defi
nit
ion
s:
Let
B =
{0, 1}
an
dY
= {
0, 1, 2}
Inp
ut
vari
ab
les:
X1, X
2…
Xn
Ou
tpu
t vari
ab
les:
Y1,
Y2
…Y
m
A l
og
ic f
un
cti
on
ff
(or
Bo
ole
an
fu
ncti
on
, sw
itch
ing
fu
ncti
on
) in
nin
pu
ts a
nd
m
ou
tpu
ts i
s t
he m
ap
ff:
Bn
Ym
do
n’t
care
–aka
“X
”
11
Defi
nit
ion
s -
2
If b
∈ ∈∈∈B
nis
map
ped
to
a 2
th
en
fu
ncti
on
is
inco
mp
lete
ly s
pe
cif
ied
, els
e c
om
ple
tely
sp
ecif
ied
OF
F-S
ET
iB
n, th
e s
et
of
all in
pu
t valu
es
for
wh
ich
ff i(x
) =
0
DC
-SE
Ti
Bn, th
e s
et
of
all in
pu
t valu
es
for
wh
ich
ff i(x
) =
2
⊆
⊆
Fo
r each
ou
tpu
t w
e d
efi
ne:
ON
-SE
Ti
Bn,
the s
et
of
all in
pu
tvalu
es f
or
wh
ich
ff i(x
) =
1
⊆
12
Th
e B
oo
lean
n-C
ub
e, B
n
13
Lit
era
ls
Gre
en
–O
N-s
et
Red
–O
FF
-set
14
Bo
ole
an
Fo
rmu
las -
-S
yn
tax
15
“S
em
an
tic”
Descri
pti
on
of
Bo
ole
an
F
un
cti
on
EX
AM
PL
E:
Tru
th t
ab
le f
orm
of
an
in
co
mp
lete
ly s
pe
cif
ied
fu
ncti
on
ff:
B3
Y2
Y1:
ON
-SE
T1 =
{
000, 001, 100, 101, 110}
OF
F-S
ET
1=
{
010, 011}
DC
-SE
T1
= {
111}
X1 X
2X
3Y
1Y
2
0 0
0 1
1
0 0
1 1
0
0 1
0 0
1
0 1
1 0
1
1 0
0 1
0
1 0
1 1
2
1 1
0 1
1
1 1
1 2
1
16
Cu
be R
ep
resen
tati
on
F1 =
A
B +
A B
D +
A B
C D
+ A
B C
D +
A B
+ A
B D
F1 =
B +
D +
A C
+ A
C
min
imu
m r
ep
resen
tati
on
0 0
--
1
0 1
-1
10 1
0 0
1
1 1
1 0
1
1 0
--
11 1
-1 1
-0 -
-1
--
-1
10 -
0 -
11 -
1 -
1
Inp
uts
Ou
tpu
ts
17
Op
era
tio
ns o
n L
og
ic F
un
cti
on
s
(1)
Co
mp
lem
en
t: f
f
inte
rch
an
ge O
N a
nd
OF
F-S
ET
S
(2)
Pro
du
ct
(or
inte
rsecti
on
or
log
ical
AN
D)
h =
f • •••
go
r h
= f
∩ ∩∩∩g
(3)
Su
m (
or
un
ion
or
log
ical O
R):
h =
f +
go
r h
= f
∪ ∪∪∪g
(4)
Dif
fere
nce
h
= f
-g
= f
∩ ∩∩∩g
18
Pri
me Im
plican
ts
A c
ub
e p
is a
n im
plican
t o
ff
if it
do
es n
ot
inte
rsect
the O
FF
-SE
T o
f f
p f
ON
∪ ∪∪∪f D
C(o
r p
∩ ∩∩∩f O
FF
= 0
)
A p
rim
e im
plican
t o
f f
is a
n im
plican
t p
su
ch
th
at
(1)
No
oth
er
imp
lican
t q
is s
uch
th
at
q⊃ ⊃⊃⊃
pin
th
e s
en
se t
hat
qco
vers
all
vert
ices o
f p
(2)
fD
C⊃ ⊃⊃⊃
p
A m
inte
rmis
a f
ull
y s
pecif
ied
im
plican
te.g
., 011, 111
(no
t 01-)
⊆
19
Exam
ple
s o
f Im
plican
ts/P
rim
es
X1 X
2X
3Y
1
0 0
0 1
0 0
1 1
0 1
0 0
0 1
1 0
1 0
0 1
1 0
1 1
1 1
0 1
1 1
1 2
000, 00-
are
im
plican
ts, b
ut
no
t p
rim
es (
-0-
)
1-1
0-0
20
Pri
me a
nd
Irr
ed
un
dan
t C
overs
A c
over
is a
set
of
cu
bes C
su
ch
th
at
C
⊇ ⊇⊇⊇f O
Nan
dC
f
ON
∪ ∪∪∪f D
C
All o
f th
e O
N-s
et
is c
overe
d b
y C
C is c
on
tain
ed
in
th
e O
N-s
et
an
d D
on
’t C
are
Set
A p
rim
e c
over
is a
co
ver
wh
ose c
ub
es a
re a
ll p
rim
e
imp
lican
ts
An
irr
ed
un
dan
t co
ver
is a
co
ver
Csu
ch
th
at
rem
ovin
g a
ny c
ub
e f
rom
Cre
su
lts in
a s
et
of
cu
bes t
hat
no
lo
ng
er
co
vers
th
e f
un
cti
on
⊆
21
Min
imu
m c
overs
A m
inim
um
co
ver
is a
co
ver
of
min
imu
m
card
inali
ty
Th
eo
rem
: A
min
imu
m c
over
can
alw
ays b
e
fou
nd
by r
estr
icti
ng
th
e s
earc
h t
o p
rim
e
an
d i
rred
un
dan
t co
vers
.
Giv
en
an
y c
over
C(a
) if
red
un
dan
t, n
ot
min
imu
m
(b)
if
an
y c
ub
e q
is n
ot
pri
me, re
pla
ce q
wit
h p
rim
e p
⊃ ⊃⊃⊃q
an
d c
on
tin
ue u
nti
l all
cu
bes p
rim
e;
it is a
min
imu
m p
rim
e c
over
22
Exam
ple
Co
vers
X1 X
2X
3Y
1
0 0
0 1
0 0
1 1
0 1
0 0
0 1
1 0
1 0
0 1
1 0
1 1
1 1
0 1
1 1
1 2
0 0
-1 0
-is
a c
over.
Is
it
pri
me?
1 1
-Is
it
irre
du
nd
an
t?
Wh
at
is a
min
imu
m p
rim
e a
nd
ir
red
un
dan
t co
ver
for
the f
un
cti
on
?
23
Exam
ple
Co
vers
X1 X
2X
3Y
1
0 0
0 1
0 0
1 1
0 1
0 0
0 1
1 0
1 0
0 1
1 0
1 1
1 1
0 1
1 1
1 2
0 0
-1 0
-is
a c
over.
Is
it
pri
me?
1 1
-Is
it
irre
du
nd
an
t?
-0 -
1 1
-is
a c
over.
Is it
pri
me?
Is it
irre
du
nd
an
t?Is
it
min
imu
m?
Wh
at
is a
min
imu
m p
rim
e a
nd
ir
red
un
dan
t co
ver
for
the f
un
cti
on
?
24
Th
e Q
uin
e-
McC
luskey
Meth
od
Ste
p 1
:L
ist
all m
inte
rms
in O
N-S
ET
an
d D
C-S
ET
Ste
p 2
:U
se a
pre
scri
bed
seq
uen
ce o
f ste
ps t
o f
ind
all t
he p
rim
e
imp
lican
tso
f th
e f
un
cti
on
Ste
p 3
:C
on
str
uct
the p
rim
e im
plican
tta
ble
Ste
p 4
:F
ind
a m
inim
um
set
of
pri
me
imp
lican
tsth
at
co
ver
all t
he
min
term
s
25
Exam
ple
0 5 7 8 9 10
11
14
15
0000
0101
0111
1000
1001
1010
1011
1110
1111
0,8
5,7
7,1
5
8,9
8,1
0
9,1
1
10,1
1
10,1
4
11,1
5
14,1
5
-000 E
01-1
D
-111 C
100-
10-0
10-1
101-
1-1
0
1-1
1
111-
8,9
,10,1
110--
B
10,1
1,1
4,1
51-1
-A
A B
C
D
E
are
pri
me im
plican
ts
26
Pri
me Im
plican
t T
ab
le
AB
CD
E XX
XX
XX
XX
XX
XX X
X
0 5 7 8 9 10
11
14
15
Min
term
s(O
N-S
ET
on
ly)
X’s
in
dic
ate
min
term
sco
vere
d b
y P
Is
27
Essen
tial P
rim
e Im
plican
ts
Ro
w w
ith
a s
ing
le X
iden
tifi
es a
n e
ssen
tial
pri
me
imp
lican
t (E
PI)
Ess
en
tial P
I’s
E, D
, B
, A
⇒ ⇒⇒⇒
Fo
rm m
inim
um
co
ver
AB
CD
E XX
XX
XX
XX
XX
XX X
X
0 5 7 8 9 10
11
14
15
28
Do
min
ati
ng
Ro
ws
In g
en
era
l E
PIs
do
no
t fo
rm a
co
ver
At
Ste
p 4
, w
e n
eed
to
sele
ct
PIs
to
ad
d t
o
the E
PIs
so
as t
o f
orm
a m
inim
um
co
ver
AB
CD
FG
XX
XX
XX
XX
XX
XX
XX
X
1 8 9 24
25
27
Ro
w 9
do
min
ate
s 8
Ro
w 2
5 d
om
inate
s 2
4
Can
rem
ove 8
sin
ce c
overi
ng
9 im
plies c
overi
ng
of
8
29
Do
min
ati
ng
Co
lum
ns
Fd
om
inate
s
D
Can
rem
ove
Dsin
ce
Fco
vers
all m
inte
rms
Dco
vers
Can
th
is h
ap
pen
in
th
e o
rig
inal ta
ble
?
Ma
y h
ap
pen
aft
er
rem
oval o
f P
Is
AB
CD
FG
XX
XX
XX
XX
XX
XX
XX
X
1 8 9 24
25
27
30
Ste
p 4
Issu
es
Rem
oval
of
do
min
ati
ng
co
lum
ns o
r d
om
inate
d r
ow
s m
ay in
tro
du
ce c
olu
mn
s
wit
h s
ing
le X
’s.
–N
eed
to
ite
rate
A c
over
ma
y s
till n
ot
be f
orm
ed
aft
er
all
ess
en
tial ele
men
ts a
nd
do
min
an
ce
rela
tio
ns h
ave b
een
rem
oved
–N
eed
to
bra
nch
over
po
ssib
le
so
luti
on
s
31
Recu
rsiv
e B
ran
ch
ing
(S
tep
4)
(a)
Sele
ct
EP
Is,
rem
ove d
om
inate
d c
olu
mn
s
an
d d
om
inati
ng
ro
ws ite
rati
vely
till ta
ble
d
oes n
ot
ch
an
ge
(b)
If t
he s
ize o
f th
e s
ele
cte
d s
et
(+ lo
wer
bo
un
d)
exceed
s o
r eq
uals
best
so
luti
on
so
far,
retu
rn f
rom
th
is level
of
recu
rsio
n. If
no
ele
men
ts l
eft
to
be
co
vere
d,
decla
re s
ele
cte
d s
et
as t
he
best
so
luti
on
reco
rded
.
(c)
Sele
ct
(heu
risti
call
y)
a b
ran
ch
ing
co
lum
n.
32
Recu
rsiv
e B
ran
ch
ing
(S
tep
4)
-2
(d)
Giv
en
th
e s
ele
cte
d c
olu
mn
, re
cu
r
–O
n t
he s
ub
-tab
le r
esu
ltin
g f
rom
d
ele
tin
g t
he c
olu
mn
an
d a
ll r
ow
s
co
vere
d b
y t
his
co
lum
n. A
dd
th
is
co
lum
n t
o t
he s
ele
cte
d s
et.
–O
n t
he s
ub
-tab
le r
esu
ltin
g f
rom
d
ele
tin
g t
he c
olu
mn
wit
ho
ut
ad
din
g it
to t
he s
ele
cte
d s
et.
33
Exam
ple
-a1
No
ess
en
tial
pri
mes, d
om
inate
d r
ow
s o
r co
lum
ns.
Sele
ct
pri
me A
AB
CD
EF
GH
XX
XX X
X XX
XX
XX
XX
XX
0 1 5 7 8 10
14
15
34
Exam
ple
-a2
BC
DE
FG
HX
X XX
XX
XX
XX
XX
5 7 810
14
15
Bis
do
min
ate
d b
y C
His
do
min
ate
d b
y G
Rem
ove B
, H
Sele
cte
d s
et
= {
A}
35
Exam
ple
-a3
C D
EF
GX X
XX
XX
XX
XX
5 7 810
14
15
C, G
essen
tial to
this
tab
le
Sele
cte
d s
et
= {
A, C
, G
}
D E
FX
XX
X14
15
Sele
cte
d s
et
= {
A, C
, G
, E
}
36
Exam
ple
-b
1
Sele
cte
d s
et
= {
}B
CD
EF
GH X
X XX X
XX
XX
XX
XX
X
0 1 5 7 810
14
15
Ess
en
tial p
rim
es
in t
his
tab
leare
B,
H
Sele
cte
d s
et
= {
B, H
}
CD
EF
GX
XX
XX
XX
X
710
14
15
Sele
cte
d s
et
= {
B, H
, D
, F
}
37
Eff
icie
nt
low
er
bo
un
din
g a
t S
tep
4(b
) to
term
inate
u
np
rofi
tab
le s
earc
hes h
igh
in
th
e r
ecu
rsio
n
Siz
e o
f sele
cte
d s
et
+ L
ow
er
bo
un
d e
qu
als
or
exceed
s b
est
so
luti
on
alr
ead
y k
no
wn
, q
uit
level
of
recu
rsio
n
Esp
resso
-Exact
(1987)
Inclu
de
AD
iscard
A
Inclu
de
B
Inclu
de C
Ob
tain
co
ver
wit
h c
ost
10
Dis
card
B
Co
mp
ute
lo
wer
bo
un
d o
f 9
38
Lo
wer
Bo
un
din
g
Lo
wer
bo
un
d:
Maxim
al
ind
ep
en
den
t set
of
row
s a
ll o
f w
hic
h a
re
pair
wis
ed
isjo
int
Maxim
al
ind
ep
en
den
t set
= {
1, 4, 8}
or
{0, 6, 10}
Need
to
sele
ct
at
least
on
e P
I/co
lum
n t
o c
ove
r ea
ch
ro
w.
NO
TE
: F
ind
ing
maxim
um
ind
ep
en
den
t set
is i
tself
NP
-hard
AB
CD
EF
XX
XX
XX
XX
XX
XX
X
0 1 4 6 810
12
39
Co
mp
lexit
y o
f Q
-M b
ased
Meth
od
s
(1)
Th
ere
exis
t fu
ncti
on
s f
or
wh
ich
th
e
nu
mb
er
of
pri
me im
plican
tsis
O(3
n)
(nis
nu
mb
er
of
inp
uts
)
(2)
Giv
en
a P
I ta
ble
, re
cu
rsiv
e b
ran
ch
ing
co
uld
re
qu
ire O
(2m
)ti
me (
mis
th
e n
um
ber
of
PIs
)
Cu
rren
t lo
gic
min
imiz
ers
ab
le t
o f
ind
exact
so
luti
on
s f
or
fun
cti
on
s w
ith
20-2
5 in
pu
t vari
ab
les
⇒ ⇒⇒⇒N
eed
heu
risti
c m
eth
od
s f
or
larg
er
fun
cti
on
s
40
Heu
risti
c L
og
ic M
inim
izati
on
Pre
sen
tly,
there
ap
pears
to
be a
lim
it o
f ~
20-2
5 i
np
ut
va
riab
les i
n p
rob
lem
s t
hat
can
be h
an
dle
d b
y e
xa
ct
min
imiz
ers
Eas
y f
or
co
mp
lex c
on
tro
l lo
gic
to
excee
d 2
0-
25 i
np
ut
va
riab
les
HIS
TO
RY
50’s
Karn
au
gh
Map
≤ ≤≤≤5 v
ari
ab
les
60’s
Q-M
meth
od
< 1
0 v
ari
ab
les
70’s
Sta
rner,
Die
tme
yer
< 1
5 v
ari
ab
les
1974
MIN
Ih
eu
risti
c
1980-8
4E
SP
RE
SS
Oap
pro
ach
es
1986
McB
oo
le<
25 v
ari
ab
les
1987
ES
PR
ES
SO
-EX
AC
T <
25 v
ari
ab
les
41
Als
o, M
ult
iple
Ou
tpu
t F
un
cti
on
s
Tru
th t
ab
le is A
ND
-OR
rep
resen
tati
on
AN
D O
Ra b
c
f
g
0 1
− −−−1 0
0 1
1
1 1
1 0
1
0 1
Wh
at
do
es v
ecto
r 0
1 1
pro
du
ce?
ON
-SE
T o
f f
=
{0
1 − −−−
, 0 1
1}
= {0
1 − −−−
}
ON
-SE
T o
f g
= {0
1 1
, 1 0
1}
42
Mu
ltip
le-O
utp
ut
Fu
ncti
on
Pri
mes
Sam
e d
efi
nit
ion
as in
sin
gle
-ou
tpu
t case
–C
ub
e w
ith
mo
st
min
term
sth
at
will in
ters
ect
OF
F-S
ET
if
yo
u a
dd
an
y m
ore
min
term
sto
th
em
f g
CU
BE
TY
PE
.
0 0
0 0
1 0
0 0
0 0
1
00 0
0 1
1 0
0 0
0 − −−−
1 0
1 0
0 1
1 0
1 0
0 1
1
00 0
0 0
0 1
1 0
0 1
1
10 0
1 0
0 1
0 0
0 − −−−
1 1
1 0
0 1
0 1
43
MIN
I
S.J
. H
on
g, R
.G. C
ain
, D
.L. O
sta
pko
-1974
Fin
al so
luti
on
is o
bta
ined
fro
m in
itia
l so
luti
on
by
itera
tive im
pro
vem
en
t ra
ther
than
by g
en
era
tin
g
an
d c
overi
ng
pri
me im
plican
ts
Th
ree b
asic
mo
dif
icati
on
s a
re p
erf
orm
ed
–R
ed
ucti
on
of
imp
lican
tsw
hil
e
main
tain
ing
co
vera
ge
–R
esh
ap
ing
im
plican
tsin
pair
s
–E
xp
an
sio
n o
f im
pli
can
ts(a
nd
re
mo
val
of
co
vere
d im
plican
ts)
44
MIN
I A
lgo
rith
m
MIN
I (F
, D
C)
{F
is O
N-S
ET
DC
is D
on
’t C
are
Set
1.
F =
U -
(F ∨ ∨∨∨
DC
) U
is u
niv
ers
e c
ub
e
2.
(Co
ver)
f=
Exp
an
df
ag
ain
st
Fp
= C
om
pu
te s
olu
tio
n s
ize
3.
f =
Red
uce
each
cu
be o
f f
ag
ain
st
oth
er
cu
bes o
f F
∨ ∨∨∨D
C
4.
Resh
ap
ef
5.
f=
Exp
an
df
ag
ain
st
Fn
= c
om
pu
te s
olu
tio
n s
ize
6.
If n
< p
g
o t
o 3
, els
e, exit
}
45
Exam
ple
: E
xp
an
sio
n
bc
a
46
Exp
an
sio
n E
xam
ple
Ste
p 2
in
MIN
I:
Exp
an
d f
a
gain
st
F
ff e
xp
an
ded
F
10
01
01
01
10
10
11
01
10
10
00
10
10
10
01
− −−−1
00
10
− −−−1
11
01
10
11
01
− −−−1
11
10
− −−−0
10
10
− −−−1
− −−−0
01
− −−−0
− −−−1
10
10
01
01
0− −−−
10
10
11
01
10
10
− −−−0
10
10
10
01
− −−−1
00
10
− −−−− −−−
11
11
− −−−1
− −−−0
01
− −−−0
− −−−1
10
11
10
10
01
01
10
00
10
01
00
01
01
00
00
10
− −−−1
01
01
− −−−0
00
01
Ord
er
sm
all
cu
bes f
irst
47
Red
ucti
on
Red
uce t
he s
ize (
in t
he s
en
se o
f th
e n
um
ber
of
min
term
s/v
ert
ices t
hat
it c
overs
) o
f cu
bes in
f
wit
ho
ut
aff
ecti
ng
co
vera
ge
Th
e s
maller
the s
ize o
f th
e c
ub
e, th
e m
ore
lik
ely
it
will b
e c
overe
d b
y a
n e
xp
an
ded
cu
be
48
Red
ucti
on
Ex
am
ple
s
1 − −−−
− −−−1
− −−−1 − −−−
1− −−−
− −−−1 1
f 1 0
0 1
− −−−1 − −−−
1− −−−
− −−−1 1
f red
uced
Red
ucin
g c
overs
:
10
01
01
0− −−−
10
10
11
01
10
10
− −−−0
10
10
10
01
− −−−1
00
10
− −−−− −−−
11
11
− −−−1
− −−−0
01
− −−−0
− −−−1
10
− −−−− −−−
11
11
− −−−0
− −−−1
10
− −−−1
− −−−0
01
10
− −−−0
10
− −−−1
00
10
0− −−−
10
10
11
01
10
10
10
01
10
01
01
Reo
rder,
p
ut
larg
er
cu
bes f
irst
− −−−− −−−
11
11
− −−−0
01
10
− −−−1
− −−−0
01
10
− −−−0
10
− −−−1
00
10
0− −−−
10
10
11
01
10
10
10
01
10
01
01
(− −−−0 1
1 1
0)
49
Resh
ap
ing
Att
em
pt
to c
han
ge t
he s
hap
e o
f th
e c
ub
es w
ith
ou
t ch
an
gin
g c
overa
ge o
r n
um
ber
Resh
ap
ing
tra
nsfo
rms a
pair
of
cu
bes in
to a
no
ther
pair
su
ch
th
at
co
vera
ge is u
naff
ecte
d (
pert
urb
s
so
luti
on
so
next
exp
an
d d
oes t
hin
gs d
iffe
ren
tly)
50
Resh
ap
ing
Exam
ple
− −−−− −−−
11
11
− −−−0
01
10
− −−−1
− −−−0
01
10
− −−−0
10
− −−−1
00
10
0− −−−
10
10
11
01
10
10
10
01
10
01
01
f
− −−−− −−−
11
11
− −−−1
− −−−0
01
10
− −−−0
10
− −−−0
01
10
− −−−1
00
10
0− −−−
10
10
11
01
10
10
10
01
10
01
01
f resh
ap
ed
− −−−− −−−
11
11
− −−−1
− −−−0
01
10
− −−−0
10
− −−−0
01
10
− −−−1
00
10
0− −−−
10
10
11
01
10
10
10
01
10
01
01
1 2 3 4 5 6 7 8 9
− −−−− −−−
11
11
− −−−1
10
01
− −−−1
00
11
10
00
10
10
10
11
00
01
10
10
01
11
0− −−−
10
10
11
01
10
1
(2,5
)
(3,8
)
(4,9
) 6 7
f ord
ere
d
51
A C
om
ple
te E
xam
ple
00
01
11
10
cd
ab
00
01
11
10
00
01
11
10
00
01
11
10
4 1 8
9 1 7
9 3 1
10 4 1 8
1
55
2
11
1
55
6
fg
init
ial f
− −−−− −−−
11
11
10
01
01
11
01
10
− −−−0
01
10
− −−−1
− −−−0
01
10
10
01
01
10
10
− −−−0
10
10
− −−−1
00
10
10
00
10
1 2 3 4 5 6 7 8 9 10
a
b
c d
f
g
− −−−− −−−
11
11
10
01
01
11
01
10
− −−−0
− −−−1
10
− −−−1
− −−−0
01
10
10
01
0− −−−
10
10
− −−−1
00
10
10
− −−−0
10
exp
an
d
52
Exam
ple
-2
00
01
11
10
cd
ab
00
01
11
10
00
01
11
10
00
01
11
10
4 1,4 7
9 1 7
9 3 1
10 4
1,4 10
1
55
2
11
1
55
6
exp
an
ded
f
− −−−− −−−
11
11
10
01
01
11
01
10
− −−−0
− −−−1
10
− −−−1
− −−−0
01
10
10
01
0− −−−
10
10
− −−−1
00
10
10
− −−−0
10
1 2 3 4 5 6 7 9 10
a
b
c d
f
g
− −−−− −−−
11
11
10
01
01
11
01
10
− −−−0
01
10
− −−−1
− −−−0
01
10
10
01
0− −−−
10
01
− −−−1
00
10
10
− −−−0
10
red
uce
53
Exam
ple
-3
00
01
11
10
cd
ab
00
01
11
10
00
01
11
10
00
01
11
10
4 1 7
9 1 7
9 3 1
10 4 1 10
1
55
2
11
1
55
6
red
uced
f
1 2 3 4 5 6 7 9 10
a
b
c d
f
g
− −−−− −−−
11
11
10
01
01
11
01
10
− −−−0
01
10
− −−−1
− −−−0
01
10
10
01
0− −−−
10
10
− −−−1
00
10
10
− −−−0
10
− −−−− −−−
11
11
10
01
11
11
01
10
00
01
10
− −−−1
10
01
10
10
11
0− −−−
10
10
− −−−1
00
11
10
00
10
2,4
5,9
6,1
0
2,4
5,9
6,1
0
resh
ap
e
54
Exam
ple
-4
ab
00
01 11
10
2
00
01 11
10
cd
00
01
11
10
00
01
11
10
4 1 7
9 1 7
9 3 1
10 1 6
1
99
2
11
1
55
6
resh
ap
ed
f
1 2 3 4 5 6 7 9 10
a
b
c d
f
g
− −−−− −−−
11
11
10
01
01
11
01
10
00
01
10
− −−−1
10
01
10
10
11
0− −−−
10
10
− −−−1
00
11
10
00
10
− −−−− −−−
11
11
10
− −−−1
11
1− −−−
0− −−−
10
− −−−0
− −−−1
10
− −−−1
− −−−0
01
10
1− −−−
11
0− −−−
10
10
− −−−1
00
11
exp
an
d
55
Exam
ple
-5
cd
4
1,4
,7
7
9 1,7 7
3,9 3 1
3
1,2
,4,6
6
1
5,9
5,9
2
11
1,2
,6
55
6
2,3
,4
ab
fin
al
exp
an
ded
f
− −−−− −−−
11
11
10
− −−−1
11
1− −−−
0− −−−
10
− −−−0
− −−−1
10
− −−−1
− −−−0
01
10
1− −−−
11
0− −−−
10
10
− −−−1
00
11
1 2 3 4 5 6 7 9
a
b
c d
f
g
00
01
11
10
00
01
11
10
00
01
11
10
00
01
11
10
fin
al F
56
Esp
resso
Alg
ori
thm
ES
PR
ES
SO
(F
, D
C)
{F
is O
N-S
ET
DC
is D
on
’t C
are
Set
1.
F =
U -
(F ∨ ∨∨∨
DC
) U
is u
niv
ers
e c
ub
e
2.
n =
|F
|
3.
F =
Red
uce (
F, D
C);
4.
F =
Exp
an
d (
F, F
);
5.
F =
Irr
ed
un
dan
t (F
, D
C);
6.
If |F
|<
n g
oto
2, els
e, p
ost-
pro
cess &
exit
}
57
Su
mm
ary
of
2-l
evel
2-l
evel o
pti
miz
ati
on
is v
ery
eff
ecti
ve a
nd
matu
re.
Exp
resso
(develo
ped
at
Berk
ele
y)
is t
he “
last
wo
rd”
on
th
e s
ub
ject
2-l
evel o
pti
miz
ati
on
is d
irectl
y u
sefu
l fo
r P
LA
’s/P
LD
’s–
these w
ere
wid
ely
used
to
im
ple
men
t co
mp
lex c
on
tro
l lo
gic
in
th
e e
arl
y 8
0’s
–
they a
re r
are
ly u
sed
th
ese d
ays
2-l
evel o
pti
miz
ati
on
fo
rms t
he t
heo
reti
cal
fou
nd
ati
on
fo
r m
ult
ilevel lo
gic
op
tim
izati
on
2-l
evel o
pti
miz
ati
on
is u
sefu
l as a
su
bro
uti
ne in
m
ult
ilevel o
pti
miz
ati
on