semiautomaton homomorphism

semiautomaton homomorphism

Just like groups and rings, a

semiautomaton

can be viewed as an

algebraic structure

. As such, one may define algebraic

constructs

such as

subalgebras

and

homomorphisms

. In this entry, we will briefly discuss the later.

Definition

A

homomorphism

from a semiautomaton

M

=

(

S

,

Σ

,

δ

)

to a semiautomaton

N

=

(

T

,

Γ

,

γ

)

is a pair of maps

h

:=

(

f

:

S

→

T

,

g

:

Σ

→

Γ

)

such that

f

⁢

(

δ

⁢

(

s

,

a

)

)

=

γ

⁢

(

f

⁢

(

s

)

,

g

⁢

(

a

)

)

The

image

of the homomorphism

h

is the

subsemiautomaton

h

⁢

(

M

)

:=

(

f

⁢

(

S

)

,

g

⁢

(

Σ

)

,

γ

′

)

of

N

, where

γ

′

is the

restriction

of

γ

to

f

⁢

(

S

)

×

g

⁢

(

Σ

)

, also known as the

homomorphic image

of

M

under

h

.

In general, a semiautomaton

N

is said to be a homomorphic image of a semiautomaton

M

, if there is a homomorphism

h

such that

h

⁢

(

M

)

=

N

.

When comparing two semiautomata

M

,

N

, it is customary to identify the two corresponding input

alphabets

Σ

,

Γ

, especially when one wants to study the interaction between the states of

M

and the states of

N

. This can be done by taking

Δ

:=

Σ

∪

Γ

, and extend the corresponding

transition functions

so that they are

identities

on the extended portions of the input alphabets. Call the extended semiautomata

M

′

,

N

′

.

If

h

:

M

→

N

is a homomorphism, then

h

′

:

M

′

→

N

′

extends

h

in a natural way:

f

′

=

f

, and

g

′

:

Δ

→

Δ

given by

g

′

⁢

(

a

)

=

g

⁢

(

a

)

if

a

∈

Σ

, and

g

′

⁢

(

a

)

=

a

if

a

∈

Γ

-

Σ

. Routine verification shows that

h

′

is indeed a homomorphism: if

a

∈

Σ

,

f

′

⁢

(

δ

′

⁢

(

s

,

a

)

)

=

f

⁢

(

δ

′

⁢

(

s

,

a

)

)

=

f

⁢

(

δ

⁢

(

s

,

a

)

)

=

γ

⁢

(

f

⁢

(

s

)

,

g

⁢

(

a

)

)

=

γ

′

⁢

(

f

⁢

(

s

)

,

g

⁢

(

a

)

)

=

γ

′

⁢

(

f

′

⁢

(

s

)

,

g

′

⁢

(

a

)

)

,

and if

a

∈

Γ

-

Σ

, then

f

′

⁢

(

δ

′

⁢

(

s

,

a

)

)

=

f

⁢

(

δ

′

⁢

(

s

,

a

)

)

=

f

⁢

(

s

)

=

γ

′

⁢

(

f

⁢

(

s

)

,

a

)

=

γ

′

⁢

(

f

′

⁢

(

s

)

,

a

)

=

γ

′

⁢

(

f

′

⁢

(

s

)

,

g

′

⁢

(

a

)

)

.

A homomorphism

h

=

(

f

,

g

)

:

M

→

N

is an

isomorphism

if both

f

and

g

are

bijections

. As discussed above, if we identify the input alphabets of

M

and

N

, and treat

g

as the

identity map

on the input alphabet, then

(

S

,

Σ

,

δ

)

and

(

T

,

Σ

,

γ

)

are isomorphic if there is a bijection

f

:

S

→

T

such that

f

⁢

(

δ

⁢

(

s

,

a

)

)

=

γ

⁢

(

f

⁢

(

s

)

,

a

)

.

Specializations to Other Machines

Computing devices derived from semiautomata such as automata and state-output

machines

may also be considered as

algebras

. We record the definitions of homomorphisms with respect to these objects here.

We adopt the following notations: given an automaton

A

=

(

S

,

Σ

,

δ

,

I

,

F

)

and a state-output machine

M

=

(

S

,

Σ

,

Δ

,

δ

,

λ

)

, let

A

′

and

M

′

be the associated semiautomaton

(

S

,

Σ

,

δ

)

. So

A

and

M

may be written

(

A

′

,

I

,

F

)

and

(

M

′

,

Δ

,

λ

)

respectively.

Definition (automaton)

. Given automata

A

=

(

A

′

,

I

,

F

)

and

B

=

(

B

′

,

J

,

G

)

. Then

h

=

(

f

,

g

)

:

A

→

B

is a

homomorphism

if

•

h

:

A

′

→

B

′

is a semiautomaton homomorphism, such that

•

f

⁢

(

I

)

⊆

J

, and

f

⁢

(

F

)

⊆

G

.

A homomorphism

h

:

A

→

B

is an isomorphism if

h

:

A

′

→

B

′

is an isomorphism such that

f

⁢

(

I

)

=

J

and

f

⁢

(

F

)

=

G

.

Definition (state-output machine)

. Given state-output machines

M

=

(

M

′

,

Δ

,

λ

)

and

N

=

(

N

′

,

Ω

,

π

)

. Then

ϕ

=

(

h

,

k

)

:

M

→

N

is a homomorphism if

•

h

:

M

′

→

N

′

is a semiautomaton homomorphism, and

•

k

:

Δ

→

Ω

, such that

k

⁢

(

λ

⁢

(

s

,

a

)

)

=

π

⁢

(

f

⁢

(

s

)

,

g

⁢

(

a

)

)

,

for all

(

s

,

a

)

∈

S

×

Σ

, where

S

and

Σ

are the state and input alphabets of

M

.

A homomorphism

ϕ

:

A

→

B

is an isomorphism if

h

:

M

′

→

N

′

is an isomorphism and

k

is a bijection.

References

1

A. Ginzburg,

Algebraic Theory of Automata

, Academic Press (1968).

Title

semiautomaton homomorphism

Canonical name

SemiautomatonHomomorphism

Date of creation

2013-03-22 19:01:07

Last modified on

2013-03-22 19:01:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 20M35

Classification

msc 03D05

Classification

msc 68Q70

Defines

homomorphism

Defines

homomorphic image

Defines

isomorphism

Defines

automaton homomorphism

Defines

machine homomorphism

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