bisimilar automata

bisimilar automata

Let

M

=

(

S

M

,

Σ

,

δ

M

,

I

M

,

F

M

)

and

N

=

(

S

N

,

Σ

,

δ

N

,

I

N

,

F

N

)

be two NDFA’s (non-deterministic finite

automata

). Let

≈

⊆

S

M

×

S

N

be a

binary relation

between the states of the automata

M

and

N

. We may extend

≈

to a binary relation between subsets of the states of the automata, as follows: for any

P

⊆

S

M

and

Q

⊆

S

N

, set

C

⁢

(

P

)

:=

{

q

∈

S

N

∣

p

≈

q

⁢

for some

⁢

p

∈

P

}

⁢

and

⁢

C

⁢

(

Q

)

:=

{

p

∈

S

M

∣

p

≈

q

⁢

for some

⁢

q

∈

Q

}

.

Then, using the same notation, define

≈

⊆

P

(

S

M

)

×

P

(

S

N

)

by

P

≈

Q

⁢

iff

⁢

P

⊆

C

⁢

(

Q

)

⁢

and

⁢

Q

⊆

C

⁢

(

P

)

.

Definition

. We say that

M

is

bisimilar

to

N

if there is a binary relation

≈

⊆

S

M

×

S

N

such that

1.

I

M

≈

I

N

,

2.

if

p

≈

q

, then

δ

M

⁢

(

p

,

a

)

≈

δ

N

⁢

(

q

,

a

)

for any

a

∈

Σ

,

3.

if

p

≈

q

, then

p

∈

F

M

iff

q

∈

F

N

.

In other words,

M

is bisimilar to

N

as automata precisely when

M

is bisimilar to

N

as LTS, and

satisfy

conditions

1

and

3

above.

Any NDFA

M

=

(

S

,

Σ

,

δ

,

I

,

F

)

is bisimilar to itself, for the

identity relation

is clearly a bisimulation. Next, if

M

is bisimilar to

N

with bisimulation

≈

,

N

is bisimilar to

M

with the converse relation

≈

-

1

. Finally, if

M

is bisimilar to

N

with bisimulation

≈

1

and

N

is bisimilar to

P

with bisimulation

≈

2

,

M

is bisimilar to

N

with bisimulation

≈

1

⁣

∘

⁣

≈

2

. Therefore, bisimilarity is an

equivalence relation

on the class of NDFA’s.

Another

property

of bisimulations on NDFA’s is that the they are preserved by taking unions: an arbitrary non-empty union of bisimulations is again a bisimulation. From this property, it is not hard to show that if

A

≈

B

, then

δ

⁢

(

A

,

x

)

≈

δ

⁢

(

B

,

x

)

for any word over

Σ

. As a result, bismilar NDFA’s accept the same set of words.

By taking the union of all bisimulations on a given NDFA

M

=

(

S

,

Σ

,

δ

,

I

,

F

)

, we get a bisimulation that is also an equivalence relation on the set of states of

M

. For each

p

∈

S

, let

[

p

]

be the

equivalence class

containing

p

, and for any subset

A

⊆

S

, let

[

A

]

:=

{

[

p

]

∣

p

∈

A

}

. Then we get an NDFA

[

M

]

:=

(

[

S

]

,

Σ

,

[

Δ

]

,

[

I

]

,

[

F

]

)

, with

[

Δ

]

⁢

(

[

p

]

,

a

)

:=

[

δ

⁢

(

p

,

a

)

]

for any

a

∈

Σ

. It can be shown that

[

M

]

is

minimal

in the sense that

[

[

M

]

]

is isomorphic to

[

M

]

, and that

M

is bisimilar to

[

M

]

. In

addition

, if

M

has no

inaccessible states

, then

M

is bisimilar to a unique minimal automaton, in the sense that, if

N

is any minimal automaton bisimlar to

M

, then

N

is isomorphic to

[

M

]

.

Title

bisimilar automata

Canonical

name

BisimilarAutomata

Date of creation

2013-03-22 19:23:17

Last modified on

2013-03-22 19:23:17

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q85

Classification

msc 68Q10

Synonym

strongly bisimilar automata

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