locally testable

locally testable

A

regular language

L

over an

alphabet

Σ

is

locally testable

if, loosely speaking, testing whether or not an arbitrary word

u

(over

Σ

) is in

L

is completely determined by its

subwords

of some fixed length. The name

locally testable

comes from the fact that properties of

u

, and not

L

, determine the membership of

u

in

L

.

To formalize this notion, we first define, for any word

u

over

Σ

, the set

sw

k

⁡

(

u

)

of all subwords of

#

⁢

u

⁢

#

of length

k

, where

#

is a symbol not in

Σ

.

Definition

. We say that a regular language

L

is

k

-testable

if for any

u

,

v

∈

Σ

*

such that

sw

k

⁡

(

u

)

=

sw

k

⁡

(

v

)

,

we have

u

∈

L

iff

v

∈

L

. The equation above says three things at once:

•

the set of subwords of

u

of length

k

is equal to the set of subwords of

v

of length

k

,

•

the prefix of

u

of length

k

is equal to the prefix of

v

of length

k

, and

•

the suffix of

u

of length

k

is equal to the suffix of

v

of length

k

.

We say that

L

is

locally testable

if it is

k

-testable for some

k

≥

0

.

If we denote

𝒯

⁢

(

k

)

the family of all

k

-testable

languages

, and

𝒯

⁢

(

∞

)

the family of all locally testable languages, then

𝒯

⁢

(

∞

)

=

⋃

k

=

0

∞

𝒯

⁢

(

k

)

.

Note that there are only two

0

-testable languages:

Σ

*

and

∅

.

Proposition 1

.

Let

D

be the family of

definite languages

. Then

1.

𝒯

⁢

(

k

)

⊂

𝒯

⁢

(

k

+

1

)

for any

k

≥

0

, and the

inclusion

is strict.

2.

𝒟

and

𝒯

⁢

(

k

)

are not

comparable

for any

k

>

0

. In other words, for every

k

, there is a

k

-testable language that is not definite, and a definite language that is not

k

-testable.

3.

𝒟

⊂

𝒯

⁢

(

∞

)

, and the inclusion is strict.

We only sketch the proof here. For the first assertion, note that for every

k

≥

0

,

sw

k

+

1

⁡

(

u

)

=

sw

k

+

1

⁡

(

v

)

implies

sw

k

⁡

(

u

)

=

sw

k

⁡

(

v

)

.

In

addition

, the language

{

a

⁢

b

k

}

*

is

(

k

+

1

)

-testable but not

k

-testable. For the second statement, note that

{

a

⁢

b

k

}

*

is not definite for any

k

≥

0

. On the other hand, a finite language containing a single word of length

k

+

1

is not

k

-testable. The last assertion is a direct

consequence

of the second one.

Thus, the families

𝒯

⁢

(

k

)

provide us with an example of an

infinite chain

of subfamilies of the family of regular languages.

With regard to the

closure properties

in

𝒯

⁢

(

k

)

, it is easily see that

𝒯

⁢

(

k

)

for all

k

≥

0

including

k

=

∞

, is

closed under

complementation

and

intersection

, and hence all Boolean operations. The star-closure of

𝒯

⁢

(

∞

)

is

ℛ

, the family of all regular languages.

Remark

. Every locally testable language is

star-free

, but not

conversely

.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

Title

locally testable

Canonical name

LocallyTestable

Date of creation

2013-03-22 18:59:03

Last modified on

2013-03-22 18:59:03

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 68Q42

Synonym

k-testable

Defines

k

-testable

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