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Intr
od
uct
ion
to
Co
mp
uta
bil
ity
Th
eo
ry
Dis
cuss
ion
1:
Co
nv
ers
ion
of
A D
FA t
o a
Re
gu
lar
Ex
pre
ssio
n
1
A D
FA t
o a
Re
gu
lar
Ex
pre
ssio
nP
rof.
Am
os
Isra
eli
Giv
en
a G
NFA
G,
an
y s
tate
of
G,
no
t in
clu
din
g
an
d
,
ca
n b
e r
ipp
ed
off
G,
wh
ile
pre
serv
ing
.
Th
is i
s d
em
on
stra
ted
in
th
e n
ext
slid
e b
y
Rip
pin
g a
sta
te f
rom
a G
NFA
(re
m.)
()
GL
sta
rtq
accep
tq
Th
is i
s d
em
on
stra
ted
in
th
e n
ext
slid
e b
y
con
sid
eri
ng
a g
en
era
l st
ate
, d
en
ote
d b
y
,
an
d a
n a
rbit
rary
pa
ir o
f st
ate
s,
a
nd
:
2
rip
q
iq
jq
Re
mo
vin
g a
sta
te f
rom
a G
NFA
iq
jq
4R
()(
)(
)4
3
*
21
RR
RR
∪
iq
jq
Be
fore
Rip
pin
gA
fte
r R
ipp
ing
3
1R
ij
q
rip
q
3R
2R
iq
jq
No
te:
Th
is s
ho
uld
be
do
ne
fo
r e
ve
ry p
air
of
inco
min
g a
nd
ou
tgo
ing
tra
nsi
tio
ns.
Ell
ab
ora
tio
n
Ass
um
e t
he
fo
llo
win
g s
itu
ati
on
:
In o
rde
r to
rip
,
all
pa
irs
of
inco
min
g a
nd
ou
tgo
ing
tra
nsi
tio
ns
sho
uld
be
co
nsi
de
red
1t2t
3tri
pq
qtr
an
siti
on
s sh
ou
ld b
e c
on
sid
ere
d
in t
he
wa
y s
ho
we
d o
n t
he
pre
vio
us
slid
e n
am
ely
co
nsi
de
r
on
e a
fte
r th
e o
the
r. A
fte
r th
at
c
an
be
rip
pe
d w
hil
e p
rese
rvin
g
.
4
5t4t
()(
)(
)(
)(
)
()
53
43
52
42
51
41
,
,,
,,
,,
,,
,,
tt
tt
tt
tt
tt
tt
rip
q
rip
q
()
GL
Th
e C
on
ve
rsio
n A
lgo
rith
m -
Ou
tlin
e
Th
e c
on
ve
rsio
n a
lgo
rith
m h
as
3 s
tag
es:
1.
Co
nve
rtin
g a
DFA
D w
ith
k s
tate
s to
an
eq
uiv
ale
nt
GN
FA G
wit
h
sta
tes
.
2.
Re
pe
ate
dly
rip
pin
g a
n a
rbit
rari
ly c
ho
sen
sta
te
2+
k
2.
Re
pe
ate
dly
rip
pin
g a
n a
rbit
rari
ly c
ho
sen
sta
te
of
G w
hil
e p
rese
rvin
g i
ts f
un
ctio
na
lity
un
til
rem
ain
ing
wit
h a
2 s
tate
s e
qu
iva
len
t G
NFA
wit
h t
wo
sta
tes.
3.
Re
turn
th
e R
E l
ab
eli
ng
re
ma
inin
g t
ran
siti
on
.
5
Exe
rcis
e
Ap
ply
th
e a
lgo
rith
m t
o o
bta
in t
he
re
gu
lar
exp
ress
ion
eq
uiv
ale
nt
to D
:
0
aq
bq
1
cq
Wh
at
is t
he
eq
uiv
ale
nt
Re
gu
lar
exp
ress
ion
?
6
()*
*1
01
01
∪+a
qb
qc
q
10
1,0
Sta
ge
1:
Co
nv
ert
Dto
a G
NFA
1.0
Sta
rt w
ith
D
aq
cq
7
0
aq
bq
1
1
0
1,0
Sta
ge
1:
Co
nv
ert
Dto
a G
NFA
1.1
Ad
d 2
ne
w s
tate
s
aq
cq
8
0
aq
bq
1
1
0
1,0
acc
ept
qst
art
q
Sta
ge
1:
Co
nv
ert
Dto
a G
NFA
1.2
Ma
ke
the
in
itia
l st
ate
an
d
th
e f
ina
l
sta
te.
aq
cq
sta
rtq
acc
ept
q
9
0
aq
bq
1
1
0
1,0
acc
ept
qst
art
qεε
Sta
ge
1:
Co
nv
ert
Dto
a G
NFA
1.3
Re
pla
ce m
ult
i la
be
l tr
an
siti
on
s b
y t
he
ir
un
ion
.
aq
cq
10
0
aq
bq
1
1
0
10∪
acc
ept
qst
art
qεε
Sta
ge
1:
Co
nv
ert
Dto
a G
NFA
1.4
Ad
d a
ll m
issi
ng
tra
nsi
tio
ns
an
d l
ab
el
the
m
.
φ
11
0
aq
bq
1
cq
1
0
sta
rtq
εε
acc
ept
q
10∪
Sta
ge
2:
Rip
a s
tate
2.0
Sta
rt w
ith
G.
12
0
aq
bq
1
cq
1
0
sta
rtq
εε
acc
ept
q
10∪
Sta
ge
2:
Rip
a s
tate
2.1
Ch
oo
se a
n a
rbit
rary
sta
te t
o b
e r
ipp
ed
.
13
0
aq
bq
1
cq
1
0
sta
rtq
εε
acc
ept
q
10∪
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
in
com
ing
an
d
ou
tgo
ing
tra
nsi
tio
ns.
(No
te:
Th
is s
tag
e d
oe
s n
ot
ap
pe
ar
in t
he
bo
ok
).
φ
bo
ok
).
2.3
Re
pla
ce e
ach
pa
ir o
f in
com
ing
an
d o
utg
oin
g
tra
nsi
tio
ns
usi
ng
th
e p
roce
du
re w
e s
ho
we
d
be
fore
.
14
Sta
ge
2:
Rip
a s
tate
Re
min
de
r: i
fth
e i
nco
min
g t
ran
siti
on
fro
m
t
o
is l
ab
ele
d
, th
e s
elf
-lo
op
of
,
,
th
e
tra
nsi
tio
n f
rom
to
w
ith
, a
nd
th
e
tra
nsi
tio
n f
rom
to
i
s la
be
led
wit
h
1R
2R
rip
qri
pq
iq
rip
qj
q3
R
4R
iq
jq
tra
nsi
tio
n f
rom
to
i
s la
be
led
wit
h
the
n t
he
ne
w l
ab
el
fro
m
t
o
is
lab
ele
d
.
Als
o n
ote
: fo
r a
ny
re
gu
lar
exp
ress
ion
R,
15
()(
)(
)4
3
*
21
RR
RR
∪i
qj
q
4R
φφ
φ=
=R
Ro
o
iq
jq
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
in
com
ing
tra
nsi
tio
ns.
φ
16
0
aq
bq
1
cq
1
0
10∪
sta
rtq
εε
acc
ept
qφ
φ
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
ou
tgo
ing
tra
nsi
tio
ns.
φ
17
0
aq
bq
1
cq
1
0
10∪
sta
rtq
εε
acc
ept
qφ
φ
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
ou
tgo
ing
tra
nsi
tio
ns.
φ
18
0
aq
bq
1
cq
1
0
10∪
sta
rtq
εε
acc
ept
qφ
φ
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
tra
nsi
tio
ns
ne
w l
ab
el
is
.
φ1
01
00
*+
=∪
φ
10+
19
0
aq
bq
1
cq
1
0
sta
rtq
εε
acc
ept
q
φ
10∪
Sta
ge
2:
Rip
a s
tate
2.1
No
w a
ll i
nco
min
g t
ran
siti
on
s a
re r
em
ove
d.
φ1
0+
20
aq
bq
1
cq
1
0
sta
rtq
εε
acc
ept
q
φ
10∪
Sta
ge
2:
Rip
a s
tate
2.2
Re
mo
ve
ou
tgo
ing
tra
nsi
tio
ns.
φ1
0+
21
aq
bq
cq
1
sta
rtq
ε
acc
ept
q
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.3
ch
oo
se a
ne
w
.
φ1
0+
rip
q
22
aq
cq
1
sta
rtq
ε
acc
ept
q
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
in
com
ing
tra
nsi
tio
ns.
φ1
0+
φ
23
aq
cq
1
sta
rtq
ε
acc
ept
q
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
ou
tgo
ing
tra
nsi
tio
ns.
10+
φ
24
aq
cq
1
sta
rtq
ε
acc
ept
q
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.1
Re
mo
ve
all
-la
be
led
ou
tgo
ing
tra
nsi
tio
ns.
10+
φ
25
aq
cq
1
sta
rtq
ε
acc
ept
q
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.2
Re
mo
ve
tra
nsi
tio
ns
ne
w l
ab
el
is
.
.
10+
()*
10
10
∪+
26
aq
cq
1
sta
rtq
ε
acc
ept
q
φ
()
()*
*1
01
01
01
0∪
=∪
∪+
+ε
φ
ε1
0∪
Sta
ge
2:
Rip
a s
tate
2.3
Re
mo
ve
a
nd
all
its
tra
nsi
tio
ns.
rip
q
27
aq
1
sta
rtq
ε
acc
ept
q
φ
()
()*
*1
01
01
01
0∪
=∪
∪+
+ε
φ
Sta
ge
2:
Rip
la
st s
tate
2.3
Ch
oo
se t
he
la
st r
em
ain
ing
sta
te t
o b
e r
ipp
ed
.
28
aq
1
sta
rtq
ε
acc
ept
q
φ
()
()*
*1
01
01
01
0∪
=∪
∪+
+ε
φ
Sta
ge
2:
Rip
a s
tate
2.3
ch
oo
se a
ne
w
a
nd
re
pe
at
pro
ced
ure
.ri
pq
29
sta
rtq
acc
ept
q
()
()*
**
*1
01
01
10
10
1∪
=∪
∪+
+ε
φ
Sta
ge
3:
Re
turn
th
e r
em
ain
ing
RE
. 30
sta
rtq
acc
ept
q
()*
*1
00
11
∪+
