equivalent automata

equivalent automata

Two

automata

are said to be

equivalent

if they accept the same

language

. Explicitly, if

A

1

=

(

S

1

,

Σ

1

,

δ

1

,

I

1

,

F

1

)

and

A

2

=

(

S

2

,

Σ

2

,

δ

2

,

I

2

,

F

2

)

are two automata, then

A

1

is equivalent to

A

2

if

L

⁢

(

A

1

)

=

L

⁢

(

A

2

)

. We write

A

1

∼

A

2

when they are equivalent. It is clear that

∼

is an

equivalence relation

on the class of automata.

First, note that if

A

1

∼

A

2

, then every symbol

α

∈

Σ

1

in a word

a

∈

L

⁢

(

A

1

)

is a symbol

α

in

Σ

2

. In other words, every symbol in a word accepted by

A

1

(or

A

2

) belongs to

Σ

:=

Σ

1

∩

Σ

2

. As a result,

L

⁢

(

A

1

)

=

L

⁢

(

A

2

)

⊆

Σ

*

. If

B

i

is an automaton obtained from

A

i

by replacing the

alphabet

Σ

i

with

Σ

, where

i

=

1

,

2

, then

B

i

∼

A

i

. This shows that we may, without loss of generality, assume outright, in the definition of equivalence of

A

1

and

A

2

, that they have the same underlying alphabet.

The most striking aspect of equivalence of automata is the following:

Proposition 1

.

Every non-deterministic automaton is equivalent to a

deterministic

one.

Proof.

Suppose

A

=

(

S

1

,

Σ

,

δ

1

,

I

1

,

F

1

)

be a non-deterministic automaton. We seek a deterministic automaton

B

=

(

S

1

,

Σ

,

δ

2

,

I

2

,

F

2

)

such that

A

∼

B

. Recall that the

difference

between

A

and

B

lie in the transition functions:

δ

1

is a

function

from

S

1

×

Σ

to

P

⁢

(

S

1

)

, whereas

δ

2

is a function from

S

2

×

Σ

to

S

2

, and the fact that

I

2

is required to be a singleton. The key to finding

B

is to realize that

δ

1

can be converted into a function from

P

⁢

(

S

1

)

×

Σ

to

P

⁢

(

S

1

)

.

Now, define

S

2

:=

P

⁢

(

S

1

)

,

I

2

:=

I

1

. For

T

⊆

S

1

and

α

∈

Σ

, let

δ

2

⁢

(

T

,

α

)

:=

⋃

s

∈

T

δ

1

⁢

(

s

,

α

)

.

As usual, we extend

δ

2

so it is defined on all of

S

2

×

Σ

*

. We want to show that

δ

2

⁢

(

{

s

}

,

a

)

=

δ

1

⁢

(

s

,

a

)

for any

s

∈

S

1

and any

a

∈

Σ

*

. This can be done by

induction

on the length of

a

:

•

if

a

=

λ

, then

δ

2

⁢

(

{

s

}

,

λ

)

=

{

s

}

=

δ

1

⁢

(

s

,

λ

)

by definition;

•

if

a

∈

Σ

, then

δ

2

⁢

(

{

s

}

,

a

)

=

⋃

s

∈

{

s

}

δ

1

⁢

(

s

,

a

)

=

δ

1

⁢

(

s

,

a

)

, again by definition;

•

if

a

=

b

⁢

α

, where

b

∈

Σ

*

and

α

∈

Σ

, then by the induction step,

δ

2

⁢

(

{

s

}

,

b

)

=

δ

1

⁢

(

s

,

b

)

, so that

δ

2

⁢

(

{

s

}

,

a

)

=

δ

2

⁢

(

{

s

}

,

b

⁢

α

)

=

δ

2

⁢

(

δ

2

⁢

(

{

s

}

,

b

)

,

α

)

=

δ

2

⁢

(

δ

1

⁢

(

s

,

b

)

,

α

)

=

⋃

t

∈

δ

1

⁢

(

s

,

b

)

δ

1

⁢

(

t

,

α

)

=

δ

1

⁢

(

δ

1

⁢

(

s

,

b

)

,

α

)

=

δ

1

⁢

(

s

,

b

⁢

α

)

=

δ

1

⁢

(

s

,

a

)

.

Suppose

a

is accepted by

A

, so that

δ

1

⁢

(

s

,

a

)

∩

F

1

≠

∅

for some

s

∈

I

1

. Then

δ

2

⁢

(

I

2

,

a

)

=

⋃

s

∈

I

2

δ

1

⁢

(

s

,

a

)

=

⋃

s

∈

I

1

δ

1

⁢

(

s

,

a

)

,

(1)

which has non-empty

intersection

with

F

1

. So, we want

F

2

to consists of every element of

S

2

that has non-empty intersection with

F

1

. Formally, we define

F

2

:=

{

F

⊆

S

1

∣

F

∩

F

1

≠

∅

}

. So what we have just shown is that

L

⁢

(

A

)

⊆

L

⁢

(

B

)

.

On the other hand, if

a

is accepted by

B

, then (1) above says that

⋃

s

∈

I

1

δ

1

⁢

(

s

,

a

)

∈

F

2

, or

(

⋃

s

∈

I

1

δ

1

⁢

(

s

,

a

)

)

∩

F

1

≠

∅

, or

δ

1

⁢

(

s

,

a

)

∩

F

1

≠

∅

for some

s

∈

I

1

, which means

a

is accepted by

A

, proving the

proposition

.
∎

Title

equivalent automata

Canonical name

EquivalentAutomata

Date of creation

2013-03-22 18:03:21

Last modified on

2013-03-22 18:03:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D05

Classification

msc 68Q45

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