rational transducer

rational transducer

Definition

A

rational transducer

is a

generalization

of a

generalized sequential machine

(gsm). Recall that a gsm

M

is a quadruple

(

S

,

Σ

,

Δ

,

τ

)

where

S

is a

finite set

of states,

Σ

and

Δ

are the input and output

alphabets

respectively, and

τ

is the

transition function

taking an input symbol from one state to an output word in another state. A rational transducer has all of the

components

above, except that the transition function

τ

is more general: its domain consists of a pair of a state and an input word, rather than an input symbol.

Formally, a

rational transducer

M

is a quadruple

(

S

,

Σ

,

Δ

,

τ

)

where

S

,

Σ

,

Δ

are defined just as those in a gsm, except that the transition function

τ

has domain a finite subset of

S

×

Σ

*

such that

τ

⁢

(

s

,

u

)

is finite for each

(

s

,

u

)

∈

dom

⁡

(

τ

)

. One can think of

τ

as a finite subset of

S

×

Σ

*

×

S

×

Δ

*

, or equivalently a finite

relation

between

S

×

Σ

*

and

S

×

Δ

*

.

Like a gsm, a rational transducer can be turned into a

language

acceptor by fixing an initial state

s

0

∈

S

and a non-empty set

F

of finite states

F

⊆

S

. In this case, a rational transducer turns into a 6-tuple

(

S

,

Σ

,

Δ

,

τ

,

s

0

,

F

)

. An input

configuration

(

s

0

,

u

)

is said to be

initial

, and an output configuration

(

t

,

v

)

is said to be

final

if

t

∈

F

. The language accepted by a rational transducer

M

is defined as the set

L

⁢

(

M

)

:=

{

u

∈

Σ

*

∣

τ

⁢

(

s

0

,

u

)

⁢

contains an final output configuration.

}

.

Rational Transductions

Additionally, like a gsm, a rational transducer can be made into a language translator. The initial state

s

0

and the set

F

of final states are needed. Given a rational transducer

M

, for every input word

u

, let

RT

M

⁡

(

u

)

:=

{

v

∈

Δ

*

∣

(

t

,

v

)

∈

τ

⁢

(

s

0

,

u

)

⁢

is a final output configuration.

}

.

Thus,

RT

M

is a

function

from

Σ

*

to

P

⁢

(

Δ

*

)

, and is called the

rational transduction

(defined) for the rational transducer

M

. The rational transduction for

M

can be extended: for any language

L

over the input alphabet

Σ

,

RT

M

⁡

(

L

)

:=

⋃

{

RT

M

⁡

(

u

)

∣

u

∈

L

}

.

In this way,

RT

M

may be thought of as a language translator.

As with GSM mappings, one can define the

inverse

of a rational transduction, given a rational transduction

RT

M

:

RT

M

-

1

⁡

(

v

)

:=

{

u

∣

v

∈

RT

M

⁡

(

u

)

}

and

RT

M

-

1

⁡

(

L

)

:=

⋃

{

RT

M

-

1

⁡

(

v

)

∣

v

∈

L

}

.

Here are some examples of rational transductions

•

Every GSM mapping is clearly a rational transduction, since every gsm is a rational transducer. As a corollary, any

homomorphism

, as well as

intersection

with any

regular language

, is a rational transduction.

•

The inverse of a rational transduction is a transduction. Given any rational transducer

M

=

(

S

,

Σ

,

Δ

,

τ

,

s

0

,

F

)

, define a rational transducer

M

′

=

(

S

′

,

Δ

,

Σ

,

τ

′

,

t

0

,

F

′

)

as follws:

S

′

=

S

∪

⋅

{

t

0

}

(where

∪

⋅

denotes

disjoint union

),

F

′

=

{

s

0

}

, and

τ

′

⊆

S

×

Δ

*

×

S

×

Σ

*

is given by

τ

′

⁢

(

t

,

v

)

=

{

{

(

s

,

v

)

∣

(

s

,

v

)

⁢

is a final output configuration of

⁢

M

}

if

⁢

t

=

t

0

{

(

s

,

u

)

∣

(

t

,

v

)

∈

τ

⁢

(

s

,

u

)

}

otherwise.

As

τ

is finite, so is

τ

′

, so that

M

′

is well-defined. In

addition

,

RT

M

′

=

RT

M

-

1

. As a corollary, the inverse homomorphism is a rational transduction.

•

The

composition

of two rational transductions is a rational transduction. To see this, suppose

M

1

=

(

S

1

,

Σ

1

,

Δ

1

,

τ

1

,

s

1

,

F

1

)

and

M

2

=

(

S

2

,

Σ

2

,

Δ

2

,

τ

2

,

s

2

,

F

2

)

are two rational transducers such that

Δ

1

⊆

Σ

2

. Define

M

=

(

S

,

Σ

1

,

Δ

2

,

τ

,

s

1

,

F

2

)

as follows:

S

=

S

1

∪

⋅

S

2

, and

τ

⊆

S

×

Σ

1

*

×

S

×

Δ

2

*

is given by

τ

⁢

(

s

,

u

)

=

{

τ

1

⁢

(

s

,

u

)

if

⁢

(

s

,

u

)

∈

S

1

×

Σ

1

*

τ

2

⁢

(

s

,

u

)

if

⁢

(

s

,

u

)

∈

S

2

×

Σ

2

*

{

(

s

2

,

u

)

}

if

⁢

(

s

,

u

)

⁢

is a final output configuration of

⁢

M

1

∅

otherwise.

Again, since both

τ

1

and

τ

2

are finite, so is

τ

, and thus

M

well-defined. In addition,

RT

M

=

RT

M

2

∘

RT

M

1

.

A family

ℱ

of languages is said to be

closed under

rational transduction if for every

L

∈

ℱ

, and any rational transducer

M

, we have

RT

M

⁡

(

L

)

∈

ℱ

. The three properties above show that if

ℱ

is closed under rational transduction, it is a cone. The

converse

is also true, as it can be shown that every rational transduction can be expressed as a composition of inverse homomorphism, intersection with a regular language, and homomorphism. Thus, a family of languages being closed under rational transduction completely characterizes a cone.

References

1

A. Salomaa

Computation

and Automata, Encyclopedia of Mathematics and Its Applications, Vol. 25

. Cambridge (1985).

Title

rational transducer

Canonical name

RationalTransducer

Date of creation

2013-03-22 18:59:29

Last modified on

2013-03-22 18:59:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D05

Classification

msc 68Q45

Related topic

GeneralizedSequentialMachine

Defines

rational transduction

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