Equivalent Languages and Simulation
Simulations
Assumo two NFAs A = 1Q , g ., F, 5) and B (P , p ., E, Y) ↳ = he iff he a ha and he the
,
Definition We that relation simulation between automata A and B ↳ I he iff he n complement ((2) =
⑪
16 :
say a -
from Q to P is a
( (a) 0
*
if and only if ↳d La iff he n
- =
(E -4)
*
iff =
·
q ~p
. .
Lach Lan
·
if gup for some qeQ and peP then
ifa <q" then there exists pleP such that p
<p' and
q'up
qF implies pEE
=
Lxample
Al A B C D Give a simulation from A2 to A
a
b
, 2
> > > a ·
(2 A) ,
-f *
(y B) accepted by A2
a
·
, any words
will also be accepted by A1
·
(2 C) ,
#
Az X Y Z
a b
> 2 >
Give a simulation from Al to Az
There is no simulation
from As to A2
Further Examplo : DFA
Al A B D Simulation from As to A2 Simulation from A2 to A
b
T
a
(W , Al
a
(A w] start states start state
·
- 7 7 ·
,
*
I * A
& ·
(B x),
·
(X , B)
b
a ·
(C w),
·
(w c) ,
·
(D , Y)
·
(y D)
,
C
*
·
(D , 2)
·
(2 D)
,
B
Az w
# 2536
-
I A
b a b
-
...
W
2
Further Examplo : NFA
a
As a
*
Simulation from As to A2 Simulation from As to An
①
b
D
#
D ·
(A , A) Start state There is no simulation
A b B C
·
(B, Y)
·
(C 2)
,
Az ·
(A 2)
"
,
.... (C y)
·
D
,
N Y I
Conclusions
Two DFAs equivalent if simulation exists
are and only if a
·
Two NFAs equivalent if simulation exists. (it cannot exist they could still be equivalent )
·
are a and .
Simulations
Assumo two NFAs A = 1Q , g ., F, 5) and B (P , p ., E, Y) ↳ = he iff he a ha and he the
,
Definition We that relation simulation between automata A and B ↳ I he iff he n complement ((2) =
⑪
16 :
say a -
from Q to P is a
( (a) 0
*
if and only if ↳d La iff he n
- =
(E -4)
*
iff =
·
q ~p
. .
Lach Lan
·
if gup for some qeQ and peP then
ifa <q" then there exists pleP such that p
<p' and
q'up
qF implies pEE
=
Lxample
Al A B C D Give a simulation from A2 to A
a
b
, 2
> > > a ·
(2 A) ,
-f *
(y B) accepted by A2
a
·
, any words
will also be accepted by A1
·
(2 C) ,
#
Az X Y Z
a b
> 2 >
Give a simulation from Al to Az
There is no simulation
from As to A2
Further Examplo : DFA
Al A B D Simulation from As to A2 Simulation from A2 to A
b
T
a
(W , Al
a
(A w] start states start state
·
- 7 7 ·
,
*
I * A
& ·
(B x),
·
(X , B)
b
a ·
(C w),
·
(w c) ,
·
(D , Y)
·
(y D)
,
C
*
·
(D , 2)
·
(2 D)
,
B
Az w
# 2536
-
I A
b a b
-
...
W
2
Further Examplo : NFA
a
As a
*
Simulation from As to A2 Simulation from As to An
①
b
D
#
D ·
(A , A) Start state There is no simulation
A b B C
·
(B, Y)
·
(C 2)
,
Az ·
(A 2)
"
,
.... (C y)
·
D
,
N Y I
Conclusions
Two DFAs equivalent if simulation exists
are and only if a
·
Two NFAs equivalent if simulation exists. (it cannot exist they could still be equivalent )
·
are a and .
