closure properties on languages

closure properties on languages

This entry lists some common

closure properties

on the families of

languages

corresponding to the

Chomsky hierarchy

, as well as other related families.

operation

REG

DCFL

CFL

CSL

RC

RE

union

Y

N

Y

Y

Y

Y

intersection

Y

N

N

Y

Y

Y

set difference

Y

N

N

Y

Y

N

complementation

Y

Y

N

Y

Y

N

intersection with a

regular language

Y

Y

Y

Y

Y

Y

concatenation

Y

N

Y

Y

Y

Y

Kleene star

Y

N

Y

Y

Y

Y

Kleene plus

Y

N

Y

Y

Y

Y

reversal

Y

Y

Y

Y

Y

Y

λ

-free

homomorphism

Y

N

Y

Y

Y

Y

homomorphism

Y

N

Y

N

N

Y

inverse

homomorphism

Y

Y

Y

Y

Y

Y

λ

-free

substitution

Y

N

Y

Y

Y

Y

substitution

Y

N

Y

N

N

Y

λ

-free

GSM mapping

Y

N

Y

Y

Y

Y

GSM mapping

Y

N

Y

N

N

Y

inverse GSM mapping

Y

Y

Y

Y

Y

Y

k

limited erasing

Y

Y

Y

Y

rational transduction

Y

N

Y

N

N

Y

right quotient

with a regular language

Y

Y

Y

N

Y

left quotient with a regular language

Y

Y

Y

N

Y

w

h

e

r

e

t

h

e

d

e

f

i

n

i

t

i

o

n

s

o

f

t

h

e

c

e

l

l

s

i

n

t

h

e

t

o

p

r

o

w

a

r

e

t

h

e

f

o

l

l

o

w

i

n

g

l

a

n

g

u

a

g

e

f

a

m

i

l

i

e

s

:

abbreviation

name

REG

regular

DCFL

deterministic context-free

CFL

context-free

CSL

context-sensitive

RC

recursive

RE

recursively enumerable

𝐑𝐞𝐦𝐚𝐫𝐤𝐬

.

B

a

s

e

d

o

n

t

h

e

t

a

b

l

e

a

b

o

v

e

,

s

t

u

d

i

e

s

i

n

c

l

o

s

u

r

e

p

r

o

p

e

r

t

i

e

s

h

a

v

e

b

e

e

n

e

x

p

a

n

d

e

d

i

n

o

t

h

e

r

d

i

r

e

c

t

i

o

n

s

,

s

u

c

h

a

s

•

closure properties of the above operations on more families of languages, such as linear languages, indexed languages, and mildly context-sensitive languages

•

given that an arbitrary family of languages is closed under a subset of operations above, what more can one say about the family? what other closure properties can be deduced? This is known as AFL (abstract family of languages) theory.

•

closure properties of yet more operations on languages, such as commutative closures, shuffle closures, insertion closures, deletion closures, subword extensions, prefix extensions, and suffix extensions.

•

knowing that a closure property of an operation is not satisfied for a given language family, study the closure of the property on the family. For example, what is the Boolean closure of context-free languages?

•

knowning the a closure property of an operation is not satisfied for a given language family, study whether it is decidable that taking the operation on language(s) leads to a language in the family. For example, is it decidable that the intersection of two context-free languages is context-free? Furthermore, if the problem is decidable, what is its complexity?

References

1

A

.

P

.

P

a

r

k

e

s

,

I

n

t

r

o

d

u

c

t

i

o

n

t

o

L

a

n

g

u

a

g

e

s

,

M

a

c

h

i

n

e

s

,

a

n

d

L

o

g

i

c

,

S

p

r

i

n

g

e

r

(

2002

)

.

2

A

.

S

a

l

o

m

a

a

,

F

o

r

m

a

l

L

a

n

g

u

a

g

e

s

,

A

c

a

d

e

m

i

c

P

r

e

s

s

,

N

e

w

Y

o

r

k

(

1973

)

.

3

J

.

E

.

H

o

p

c

r

o

f

t

,

J

.

D

.

U

l

l

m

a

n

,

F

o

r

m

a

l

L

a

n

g

u

a

g

e

s

a

n

d

T

h

e

i

r

R

e

l

a

t

i

o

n

t

o

A

u

t

o

m

a

t

a

,

A

d

d

i

s

o

n

-

W

e

s

l

e

y

,

(

1969

)

.

Title

closure properties on languages

Canonical name

ClosurePropertiesOnLanguages

Date of creation

2013-03-22 18:59:36

Last modified on

2013-03-22 18:59:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Feature

Classification

msc 68Q15

Classification

msc 03D55

Related topic

AbstractFamilyOfLanguages

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