p-morphism

p-morphism

Let

ℱ

1

=

(

W

1

,

R

1

)

and

ℱ

2

=

(

W

2

,

R

2

)

be

Kripke frames

. A

p-morphism

from

ℱ

1

to

ℱ

2

is a

function

f

:

W

1

→

W

2

such that

•

if

u

⁢

R

1

⁢

w

, then

f

⁢

(

u

)

⁢

R

2

⁢

f

⁢

(

w

)

,

•

if

s

⁢

R

2

⁢

t

and

s

=

f

⁢

(

u

)

for some

u

∈

W

1

, then

u

⁢

R

1

⁢

w

and

t

=

f

⁢

(

w

)

for some

w

∈

W

1

,

We write

f

:

ℱ

1

→

ℱ

2

to denote that

f

is a p-morphism from

ℱ

1

to

ℱ

2

.

Let

M

1

=

(

ℱ

1

,

V

1

)

and

M

2

=

(

ℱ

2

,

V

2

)

be Kripke models of modal

propositional logic

PL

M

. A

p-morphism

from

M

1

to

M

2

is a p-morphism

f

:

ℱ

1

→

ℱ

2

such that

•

M

1

⊧

w

p

iff

M

2

⊧

f

⁢

(

w

)

p

for any

propositional variable

p

.

Proposition 1

.

For any wff

A

,

M

1

⊧

w

A

iff

M

2

⊧

f

⁢

(

w

)

A

.

Proof.

Induct on the number

n

of

logical connectives

in

A

. When

n

=

0

,

A

is either

⟂

or a propositional variable. The case when

A

is

⟂

is

obvious

, and the other case is

definition

. Next, suppose

A

is

B

→

C

, then

M

1

⊧

w

A

iff

M

1

⊧̸

w

B

or

M

1

⊧

w

C

iff

M

2

⊧̸

f

⁢

(

w

)

B

or

M

2

⊧

f

⁢

(

w

)

C

iff

M

2

⊧

f

⁢

(

w

)

A

. Finally, suppose

A

is

□

⁢

B

, and

M

1

⊧

w

A

. To show

M

2

⊧

f

⁢

(

w

)

A

, let

t

be such that

f

⁢

(

w

)

⁢

R

2

⁢

t

. Then there is a

u

such that

t

=

f

⁢

(

u

)

and

w

⁢

R

1

⁢

u

, so that

M

1

⊧

u

B

. By

induction

,

M

2

⊧

f

⁢

(

u

)

B

, or

M

2

⊧

t

B

. Hence

M

2

⊧

f

⁢

(

w

)

A

.

Conversely

, suppose

M

2

⊧

f

⁢

(

w

)

A

. To show

M

1

⊧

w

A

, let

u

be such that

w

⁢

R

1

⁢

u

. So

f

⁢

(

w

)

⁢

R

2

⁢

f

⁢

(

u

)

, and therefore

M

2

⊧

f

⁢

(

u

)

B

. By induction,

M

1

⊧

u

B

, whence

M

1

⊧

w

A

.
∎

Proposition 2

.

If a p-morphism

f

:

F

1

→

F

2

is

one-to-one

, then

F

2

⊧

A

implies

F

1

⊧

A

for any wff

A

.

Proof.

Suppose

ℱ

2

⊧

A

. Let

M

=

(

W

1

,

R

1

,

V

1

)

be any model based on

ℱ

1

and

w

any world in

W

1

. We want to show that

M

⊧

w

A

.

Define a Kripke model

M

′

:=

(

W

2

,

R

2

,

V

2

)

as follows: for any propositional variable

p

, let

V

2

⁢

(

p

)

:=

{

s

∈

W

2

∣

f

-

1

⁢

(

s

)

∩

V

1

⁢

(

p

)

≠

∅

}

. Then

M

⊧

w

p

iff

w

∈

V

1

⁢

(

p

)

iff

f

-

1

⁢

(

f

⁢

(

w

)

)

=

{

w

}

⊆

V

1

⁢

(

p

)

(since

f

is one-to-one) iff

f

-

1

⁢

(

f

⁢

(

w

)

)

∩

V

1

⁢

(

p

)

≠

∅

iff

f

⁢

(

w

)

∈

V

2

⁢

(

p

)

iff

M

′

⊧

f

⁢

(

w

)

p

. This shows that

f

is a p-morphism from

M

to

M

′

.

Now, let

w

∈

W

1

. Then

M

′

⊧

f

⁢

(

w

)

A

by

assumption

. By the last

proposition

,

M

⊧

w

A

.
∎

Proposition 3

.

If a p-morphism

f

:

F

1

→

F

2

is onto, then

F

1

⊧

A

implies

F

2

⊧

A

for any wff

A

.

Proof.

Suppose

ℱ

1

⊧

A

. Let

M

=

(

W

2

,

R

2

,

V

2

)

be any model based on

ℱ

2

and

s

any world in

W

2

. We want to show that

M

⊧

s

A

.

Define a Kripke model

M

′

:=

(

W

1

,

R

1

,

V

1

)

as follows: for any propositional variable

p

, let

V

1

⁢

(

p

)

:=

{

w

∈

W

1

∣

f

⁢

(

w

)

∈

V

2

⁢

(

p

)

}

. Then

w

∈

V

1

⁢

(

p

)

iff

f

⁢

(

w

)

∈

V

2

⁢

(

p

)

, so

f

is a p-morphism from

M

′

to

M

, and by assumption

M

′

⊧

A

for any wff

A

.

Now, let

w

∈

W

1

be a world such that

f

⁢

(

w

)

=

s

. Since

M

′

⊧

A

,

M

′

⊧

w

A

in particular, and therefore

M

⊧

f

⁢

(

w

)

A

or

M

⊧

s

A

by the last proposition.
∎

Corollary 1

.

If

f

:

F

1

→

F

2

is

bijective

, then

F

1

⊧

A

iff

F

2

⊧

A

for any wff

A

.

A frame

ℱ

′

is said to be a p-morphic image of a frame

ℱ

if there is an onto p-morphism

f

:

ℱ

→

ℱ

′

. Let

𝒞

be the class of all frames validating a wff. Then by the third proposition above,

𝒞

is

closed under

p-morphic images: if a frame is in

𝒞

, so is any of its p-morphic images. Using this

property

, we can show the following: if

𝒞

is the class of all frames validating a wff

A

, then

𝒞

can not be

•

the class of all

irreflexive

frames

•

the class of all asymetric frames

•

the class of all anti-symmetric frames

Proof.

Let

ℱ

1

=

(

ℕ

,

<

)

and

ℱ

2

=

(

{

0

}

,

R

)

, where

0

⁢

R

⁢

0

. Notice that

ℱ

1

is in both the class of irreflexive frames and the class of asymetric frames, but

ℱ

2

is in neither. Let

f

:

ℕ

→

{

0

}

be the obvious

surjection

. Clearly,

m

<

n

implies

f

⁢

(

m

)

⁢

R

⁢

f

⁢

(

n

)

. Also, if

f

⁢

(

m

)

⁢

R

⁢

0

, then

f

⁢

(

m

)

⁢

R

⁢

f

⁢

(

m

+

1

)

. So

f

is a p-morphism. Suppose

𝒞

is either the class of all irreflexive frames or the class of all asymetric frames. If

A

is validated by

𝒞

,

A

is validated by

ℱ

1

in particular (since

ℱ

1

is in

𝒞

), so that

A

is validated by

ℱ

2

as well, which means

ℱ

2

is

𝒞

too, a

contradiction

. Therefore, no such an

A

exists.

Next, let

ℱ

3

=

(

ℕ

,

S

)

, where

n

⁢

S

⁢

(

n

+

1

)

for all

n

∈

ℕ

and

ℱ

4

=

(

{

0

,

1

}

,

R

)

, where

R

=

{

(

0

,

1

)

,

(

1

,

0

)

}

. Let

𝒞

be the class of all anti-symmetric frames. Then

ℱ

3

is in

𝒞

but

ℱ

4

is not. Let

f

:

ℱ

3

→

ℱ

4

be given by

f

⁢

(

n

)

=

0

if

n

is even and

f

⁢

(

n

)

=

1

if

n

is odd. If

a

⁢

S

⁢

b

, then

f

⁢

(

a

)

and

f

⁢

(

b

)

differ by

1

, so

f

⁢

(

a

)

⁢

R

⁢

f

⁢

(

b

)

. On the other hand, if

f

⁢

(

a

)

⁢

R

⁢

x

, then

x

is either

0

or

1

, depending on whether

a

odd or even. Pick

b

=

a

+

1

, so

a

⁢

S

⁢

b

and

f

⁢

(

b

)

=

x

. This shows that

f

is a p-morphism. By the same

argument

as in the last paragraph, no wff

A

is validated by precisely the members of

𝒞

.
∎

Title

p-morphism

Canonical

name

Pmorphism

Date of creation

2013-03-22 19:34:54

Last modified on

2013-03-22 19:34:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Related topic

Bisimulation

\@unrecurse

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