product of automata

product of automata

One way to manufacture an

automaton

out of existing automata is by taking

products

.

Products of Two Automata

Let

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

)

be two automata. We define the product

A

of

A

1

and

A

2

, written

A

1

×

A

2

, as the quituple

(

S

,

Σ

,

δ

,

I

,

F

)

:=

(

S

1

×

S

2

,

Σ

1

×

Σ

2

,

δ

1

×

δ

2

,

I

1

×

I

2

,

F

1

×

F

2

)

,

where

δ

is a

function

from

S

×

Σ

to

P

⁢

(

S

1

)

×

P

⁢

(

S

2

)

⊆

P

⁢

(

S

)

, given by

δ

⁢

(

(

s

1

,

s

2

)

,

(

α

1

,

α

2

)

)

:=

δ

1

⁢

(

s

1

,

α

1

)

×

δ

2

⁢

(

s

2

,

α

2

)

.

Since

S

,

Σ

,

I

,

F

are non-empty,

A

is an automaton. The automaton

A

can be thought of as a

machine

that runs automata

A

1

and

A

2

simultaneously. A pair

(

α

1

,

α

2

)

of symbols being fed into

A

at start state

(

q

1

,

q

2

)

∈

I

is the same as

A

1

reading

α

1

at state

q

1

and

A

2

reading

α

2

at state

q

2

. The set of all possible next states for the

configuration

(

(

s

1

,

s

2

)

,

(

α

1

,

α

2

)

)

in

A

is the same as the set of all possible

combinations

(

t

1

,

t

2

)

, where

t

1

is a next state for the configuration

(

s

1

,

α

1

)

in

A

1

and

t

2

is a next state for the configuration

(

s

2

,

α

2

)

in

A

2

.

If

A

1

and

A

2

are FSA, so is

A

. In

addition

, if both

A

1

and

A

2

are

deterministic

, so is

A

, because

δ

⁢

(

(

s

1

,

s

2

)

,

(

α

1

,

α

2

)

)

=

(

δ

1

⁢

(

s

1

,

α

1

)

,

δ

2

⁢

(

s

2

,

α

2

)

)

,

and

I

is a

singleton

.

As usual,

δ

can be extended to read words over

Σ

, and it is easy to see that

δ

⁢

(

(

s

1

,

s

2

)

,

(

a

1

,

a

2

)

)

=

δ

1

⁢

(

s

1

,

a

1

)

×

δ

2

⁢

(

s

2

,

a

2

)

,

where

a

1

and

a

2

are words over

Σ

1

and

Σ

2

respectively. A word

(

a

1

,

a

2

)

is accepted by

A

iff

a

1

is accepted by

A

1

and

a

2

is accepted by

A

2

.

Intersection of Two Automata

Again, we assume

A

1

and

A

2

are automata specified above. Now, suppose

Σ

1

=

Σ

2

=

Δ

. Then

Δ

can be identified as the diagonal in

Σ

=

Σ

1

×

Σ

2

=

Δ

2

. We are then led to an automaton

A

1

∩

A

2

:=

(

S

,

Δ

,

δ

,

I

,

F

)

,

where

S

,

I

, and

F

are defined previously, and

δ

is given by

δ

⁢

(

(

s

1

,

s

2

)

,

α

)

=

δ

1

⁢

(

s

1

,

α

)

×

δ

2

⁢

(

s

2

,

α

)

.

Suppose in addition that

Δ

is finite. From the discussion in the previous

section

, it is evident that the

language

accepted by

A

1

∩

A

2

is the same as the

intersection

of the language accepted by

A

1

and the language accepted by

A

2

:

L

⁢

(

A

1

∩

A

2

)

=

L

⁢

(

A

1

)

∩

L

⁢

(

A

2

)

.

Title

product of automata

Canonical

name

ProductOfAutomata

Date of creation

2013-03-22 18:03:06

Last modified on

2013-03-22 18:03:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D05

Classification

msc 68Q45

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