bisimulation

bisimulation

Given two

labelled state transition systems

(LTS)

M

=

(

S

1

,

Σ

,

→

1

)

,

N

=

(

S

2

,

Σ

,

→

2

)

, a

binary relation

≈

⊆

S

1

×

S

2

is called a

simulation

if whenever

p

≈

q

and

p

→

α

1

p

′

, then there is a

q

′

such that

p

′

≈

q

′

and

q

→

α

2

q

′

. An LSTS

M

=

(

S

1

,

Σ

,

→

1

)

is

similar

to an LTS

N

=

(

S

2

,

Σ

,

→

2

)

if there is a simulation

≈

S

1

×

S

2

.

For example, any LTS is similar to itself, as the

identity relation

on the set of states is a simulation. In

addition

,

1.

if

M

is similar to

N

and

N

is similar to

P

, then

M

is similar to

P

:

Proof.

Let

M

=

(

S

1

,

Σ

,

→

1

)

,

N

=

(

S

2

,

Σ

,

→

2

)

, and

P

=

(

S

3

,

Σ

,

→

3

)

be LSTS, and suppose

p

≈

1

∘

≈

2

q

with

p

→

α

1

p

′

, where

≈

1

and

≈

2

are simulations. Then there is an

r

such that

p

≈

1

r

and

r

≈

2

q

. Since

≈

1

is a simulation, there is an

r

′

such that

r

→

α

2

r

′

. But then since

≈

2

is a simulation, there is a

q

′

such that

q

→

α

3

q

′

. As a result,

≈

1

⁣

∘

⁣

≈

2

is a simulation.
∎

2.

a union of simulations is a simulation.

Proof.

Let

≈

be the union of simulations

≈

i

, where

i

∈

I

, and suppose

p

≈

q

, with

p

→

α

p

′

. Then

p

≈

i

q

for some

i

. Since

≈

i

is a simulation, there is a state

q

′

such that

p

′

≈

i

q

′

and

q

→

α

q

′

. So

p

′

≈

q

′

and therefore

≈

is a simulation.
∎

A binary relation

≈

between

S

1

and

S

2

is a

bisimulation

if both

≈

and its

converse

≈

-

1

are simulations. A bisimulation is also called a

strong bisimulation

, in contrast with

weak bisimulation

. When there is a bisimulation between the state sets of two LTS, we say that the two systems are

bisimilar

, or

strongly bisimilar

. By abuse of notation, we write

M

≈

N

to denote that

M

is bisimilar to

N

.

An

equivalent

formulation of bisimulation is given by extending the binary relation on the sets to a binary relation on the corresponding

power sets

. Here’s how: let

≈

⊆

S

1

×

S

2

. For any

A

⊆

S

1

and

B

⊆

S

2

, define

C

⁢

(

A

)

:=

{

b

∈

S

2

∣

a

≈

b

⁢

for some

⁢

a

∈

A

}

⁢

and

⁢

C

⁢

(

B

)

:=

{

a

∈

S

1

∣

a

≈

b

⁢

for some

⁢

b

∈

B

}

.

Then the binary relation

≈

can be extended to a binary relation from

P

⁢

(

S

1

)

to

P

⁢

(

S

2

)

, still denoted by

≈

, as

A

≈

B

iff

A

⊆

C

⁢

(

B

)

⁢

and

⁢

B

⊆

C

⁢

(

A

)

,

for any

A

⊆

S

1

and

B

⊆

S

2

. In other words,

A

≈

B

iff for any

a

∈

A

, there is a

b

∈

B

such that

a

≈

b

and for any

b

∈

B

, there is an

a

∈

A

such that

a

≈

b

. Now, for any

p

∈

S

1

and

α

∈

Σ

, let

δ

1

⁢

(

p

,

a

)

=

{

q

∈

S

1

∣

p

→

α

1

q

}

.

We can similar define

function

δ

2

:

S

2

×

Σ

→

P

⁢

(

S

2

)

. So a binary relation

≈

between

S

1

and

S

2

is a bisimilation iff for any

(

p

,

q

)

∈

S

1

×

S

2

such that

p

≈

q

, we have

δ

1

⁢

(

p

,

a

)

≈

δ

2

⁢

(

q

,

a

)

for any

a

∈

Σ

.

Let

M

=

(

S

1

,

Σ

,

→

1

)

,

N

=

(

S

2

,

Σ

,

→

2

)

, and

P

=

(

S

3

,

Σ

,

→

3

)

be LTS. The following are some basic properties of bisimulation:

1.

The identity relation

=

is a bisimilation on any LTS, since

=

is a simulation and

=

-

1

is just

=

.

2.

If

M

is bisimilar to

N

via

≈

, then

N

is bisimilar to

M

via

≈

-

1

, since both

≈

-

1

and

(

≈

-

1

)

-

1

=

≈

are simulations.

3.

If

M

≈

1

and

N

≈

2

P

, then

M

≈

1

∘

≈

2

P

, since

≈

1

⁣

∘

⁣

≈

2

and

(

≈

1

∘

≈

2

)

-

1

=

≈

2

1

-

∘

≈

1

-

1

are both simulations according to the

argument

above.

4.

A union of bisimilations is a bisimilation.

Proof.

Let

≈

be the union of bisimulations

≈

i

, where

i

∈

I

. Then

≈

is a simulation by the argument above. Now, suppose

p

≈

-

1

q

and

p

→

α

p

′

, then

q

≈

p

. Then

q

≈

i

p

for some

i

∈

I

. So

p

≈

i

-

1

q

. Since

≈

i

is a bisimulation, so is

≈

i

-

1

, and therefore for some state

q

′

,

p

′

≈

i

-

1

q

′

and

q

→

α

q

′

. This means that

p

′

≈

-

1

q

′

, implying that

≈

-

1

is a simulation. Hence

≈

is a bisimulation.
∎

5.

The union of all bisimilations on an LTS is a bisimulation that is also an

equivalence relation

.

Proof.

For an LTS

M

, let

≈

M

be the union of all bisimulations on

M

. Then

≈

M

is a bisimulation by the previous result. Since

=

is a bisimulation on

M

,

≈

M

is

reflexive

. If

p

≈

M

q

,

p

≈

q

for some bisimulation

≈

on

M

. Then

≈

-

1

is a bisimulation and therefore

q

≈

-

1

p

implies that

q

≈

M

p

, so that

≈

M

is

symmetric

. Finally, if

p

≈

M

q

and

q

≈

M

r

, then

p

≈

1

q

and

q

≈

2

r

for some bisimulations

≈

1

and

≈

2

. So

≈

1

⁣

∘

⁣

≈

2

is a bisimulation. Since

p

≈

1

∘

≈

2

r

,

p

≈

M

r

and therefore

≈

M

is

transitive

.
∎

For an LTS

M

=

(

S

,

Σ

,

→

)

, let

≈

M

be the maximal bisimulation on

M

as defined above. Since

≈

M

is an equivalence relation, we can form an

equivalence class

[

p

]

for each state

p

∈

S

. Let

[

S

]

be the set of all such equivalence classes:

[

S

]

:=

{

[

p

]

∣

p

∈

S

}

. Define

[

→

]

on

S

×

Σ

×

S

by

[

p

]

[

→

]

α

[

q

]

iff

p

→

α

q

.

This is a well-defined

ternary relation

, for if

p

≈

M

p

′

and

q

≈

M

q

′

, we have

p

′

→

α

q

′

. Now,

[

M

]

:=

(

[

S

]

,

Σ

,

[

→

]

)

is an LSTS, and

M

is bisimilar to it, with bisimulation given by the

relation

{

(

p

,

[

p

]

)

∣

p

∈

S

}

.

Title

bisimulation

Canonical name

Bisimulation

Date of creation

2013-03-22 19:23:14

Last modified on

2013-03-22 19:23:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

44

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Classification

msc 68Q85

Classification

msc 68Q10

Synonym

strong bisimulation

Synonym

strongly bisimilar

Related topic

PMorphism

Defines

simulation

Defines

bisimilar

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