subsemiautomaton

subsemiautomaton

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 former.

Definition

A semiautomaton

N

=

(

T

,

Γ

,

γ

)

is said to be a

subsemiautomaton

of a semiautomaton

M

=

(

S

,

Σ

,

δ

)

if

T

⊆

S

,

Γ

⊆

Σ

,

and

γ

⊆

δ

.

The last

inclusion

means the following:

γ

⁢

(

s

,

a

)

=

δ

⁢

(

s

,

a

)

for all

(

s

,

a

)

∈

T

×

Γ

. We write

N

≤

M

when

N

is a subsemiautomaton of

M

.

A subsemiautomaton

N

of

M

is said to be proper if

N

≠

M

, and is written

N

<

M

.

Examples

. Let

M

=

(

S

,

Σ

,

δ

)

be a semiautomaton.

•

M

can be represented by its

state diagram

, which is just a

directed graph

. Any

strongly connected component

of the state diagram

represents

a subsemiautomaton of

M

, characterized as a semiautomaton

(

S

′

,

Σ

,

δ

′

)

such that any state in

S

′

can be reached from any other state in

S

′

. In other words, for any

s

,

t

∈

S

′

, there is a word

u

over

Σ

such that either

t

∈

δ

′

⁢

(

s

,

u

)

, where

δ

′

is the

restriction

of

δ

to

S

′

×

Σ

. A semiautomaton whose state diagram is

strongly connected

is said to be

strongly connected

.

•

Suppose

M

is strongly connected. Then

M

has no proper subsemiautomaton whose input

alphabet

is equal to the input alphabet of

M

. In other words, if

N

=

(

T

,

Σ

,

γ

)

≤

M

, then

N

=

M

. However, proper subsemiautomata of

M

exist if we take

N

=

(

T

,

Γ

,

γ

)

for any

proper subset

Γ

of

Σ

, provided that

|

Σ

|

≥

2

. In this case,

γ

is just the restriction of

δ

to the set

T

×

Γ

.

•

On the other hand, if

Σ

is a singleton, and

M

is strongly connected, then no proper subsemiautomata of

M

exist. Notice that if the

transition function

δ

is

single valued

, then it is just a

permutation

on

S

of order

|

S

|

.

Specializations to Other Machines

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

machines

may too be considered as

algebras

. We record the definitions of subalgebras of these objects here.

Note: the following notations are used: 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)

.

A

=

(

A

′

,

I

,

F

)

is a

subautomaton

of

B

=

(

B

′

,

J

,

G

)

if

A

′

≤

B

′

,

I

⊆

J

,

and

F

⊆

G

.

Definition (state-output machine)

.

M

=

(

M

′

,

Δ

,

λ

)

is a

submachine

of

N

=

(

N

′

,

Ω

,

π

)

if

M

′

≤

N

′

,

Δ

⊆

Ω

,

and

λ

⁢

is the restriction of

⁢

π

⁢

to

⁢

S

×

Σ

,

where

S

and

Σ

are the state and input alphabets of

M

.

References

1

A. Ginzburg,

Algebraic Theory of Automata

, Academic Press (1968).

Title

subsemiautomaton

Canonical name

Subsemiautomaton

Date of creation

2013-03-22 19:01:03

Last modified on

2013-03-22 19:01:03

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 03D05

Classification

msc 68Q45

Classification

msc 68Q70

Defines

strongly connected semiautomaton

Defines

subsemiautomata

Defines

subautomaton

Defines

submachine

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