simplified automaton

simplified automaton

Word acceptance is the primary

function

of an

automaton

. When a word is fed into an automaton at an initial state, it is accepted exactly when a final state is reached upon reading the last symbol of the word. So when a state does not contribute in anyway to word acceptance, it may be tossed out without changing the accepting power of the automaton.

Definition

. Let

A

=

(

S

,

Σ

,

δ

,

I

,

F

)

be an automaton. A state

s

∈

S

is said to be

reachable

from a state

t

∈

S

, if for some word

u

over

Σ

,

t

∈

δ

⁢

(

s

,

u

)

. A state is said to be

accessible

if it is reachable from a start state. A state is

live

or

coaccessible

if a final state can be reached from it. If a state is both accessible and live, then it is called

useful

.

Clearly, every start state is accessible, and every final state is live. The following is the

state diagram

of an automaton that illustrates the

concepts

defined above.

States

σ

,

p

,

r

,

s

are accessible, while states

σ

,

p

,

q

are live. So the useful states are

σ

and

p

.

The automaton

A

is said to be

accessible

if every state in

S

is accessible, and is

live

if every state is live.

A

is called

simplified

or

trim

if it is both accessible and live (every state is useful).

Proposition 1

.

An automaton is

equivalent

(

http://planetmath.org/EquivalentAutomata

) to a simplified automaton iff it accepts at least one word.

Proof.

Suppose an automaton

A

is equivalent to a simplified automaton

A

′

. Then the set of start states of

A

′

is non-empty, say containing a state

q

. So

q

is live, which means there is a word

u

such that

δ

⁢

(

p

,

u

)

contains a final state, or that

u

∈

L

⁢

(

A

′

)

=

L

⁢

(

A

)

.

Conversely

, suppose an automaton

A

=

(

S

,

Σ

,

δ

,

I

,

F

)

accepts a word

u

. This means that there is a start state

q

and a final state

r

such that

r

∈

δ

⁢

(

q

,

u

)

, so that both

q

and

r

are useful. Now, let

S

′

be the subset (of

S

) all useful states in

A

, and let

I

′

=

I

∩

S

′

,

F

′

=

F

∩

S

′

. Then neither

I

′

nor

F

′

is empty. Finally, let

δ

′

be the

restriction

of

δ

on the set

S

′

×

Σ

, such that

δ

′

⁢

(

s

,

a

)

:=

δ

⁢

(

s

,

a

)

∩

S

′

.

Then the five-tuple

A

′

:=

(

S

′

,

Σ

,

δ

′

,

I

′

,

F

′

)

is an automaton. Obviously,

L

⁢

(

A

′

)

⊆

L

⁢

(

A

)

. On the other hand, the word

u

above is accepted by

A

′

. Since

u

is arbitrary, we see that

L

⁢

(

A

′

)

=

L

⁢

(

A

)

as a result.
∎

Remark

. If we allow the start state set of an automaton to be empty, then every automaton can be simplified, whether or not it accepts any words.

In the example above, the automaton is equivalent to the following simplified automaton:

However, the construction in the proof shows that, given a deterministic automaton

A

, the simplified automaton

A

′

may not be determinstic. In the example above,

A

′

is not

deterministic

because

δ

⁢

(

σ

,

b

)

=

∅

. With respect to deterministic automata, the best we can do is the following:

Proposition 2

.

A deterministic automaton is equivalent (

http://planetmath.org/EquivalentAutomata

) to a deterministic accessible automaton.

The proof of this is very similar to the one given above, except

S

′

,

I

′

, and

F

′

now consist of all accessible states, and

δ

′

is just a restriction on the domain of

δ

, since if a state

s

is accessible, so is any state in

δ

⁢

(

s

,

a

)

for any

a

∈

Σ

.

Again, referring the example above,

A

is deterministic, and it is equivalent to the following deterministic accessible automaton:

The

language

accepted by the automaton is represented by the

regular expression

a

⁢

(

a

⁢

b

)

*

.

Title

simplified automaton

Canonical name

SimplifiedAutomaton

Date of creation

2013-03-22 19:01:51

Last modified on

2013-03-22 19:01:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D05

Classification

msc 68Q45

Synonym

trim automaton

Synonym

coaccessible

Related topic

ReducedAutomaton

Defines

reachable

Defines

accessible

Defines

live

Defines

useful

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