regular language

regular language

A

regular grammar

is a

context-free grammar

where a

production

has one of the following three forms:

A

→

λ

,

A

→

u

,

A

→

v

⁢

B

where

A

,

B

are non-terminal symbols,

u

,

v

are terminal words, and

λ

the

empty word

. In BNF, they are:

<non-terminal

>

:

:=

t

⁢

e

⁢

r

⁢

m

⁢

i

⁢

n

⁢

a

⁢

l

<non-terminal

>

:

:=

t

⁢

e

⁢

r

⁢

m

⁢

i

⁢

n

⁢

a

⁢

l

⁢

<

non-terminal

>

<non-terminal

>

:

:=

λ

A

regular language

(also known as a

regular set

or a

regular event

) is the set of strings generated by a regular grammar.
Regular grammars are also known as Type-3 grammars in the

Chomsky hierarchy

.

A regular grammar can be represented by a

deterministic

or

non-deterministic finite automaton

. Such

automata

can serve to either

generate

or accept

sentences

in a particular regular language. Note that since the set of regular languages is a subset of context-free languages, any deterministic or non-deterministic finite automaton can be simulated by a

pushdown automaton

.

There is also a close relationship between regular languages and

regular expressions

. With every regular expression we can

associate

a regular language.

Conversely

, every regular language can be obtained from a regular expression. For example, over the

alphabet

{

a

,

b

,

c

}

, the regular language associated with the regular expression

a

⁢

(

b

∪

c

)

*

⁢

a

is the set

{

a

}

∘

{

b

,

c

}

*

∘

{

a

}

=

{

a

⁢

w

⁢

a

∣

w

⁢

is a word in two letters

⁢

b

⁢

and

⁢

c

}

,

where

∘

is the

concatenation

operation

, and

*

is the

Kleene star

operation. Note that

w

could be the empty word

λ

.

Yet another way of describing a regular language is as follows: take any alphabet

Σ

. Let

ℛ

⁢

(

Σ

)

be the smallest subset of

P

⁢

(

Σ

*

)

(the

power set

of the set of words over

Σ

, in other words, the set of

languages

over

Σ

), among all subsets of

P

⁢

(

Σ

*

)

with the following

properties

:

•

ℛ

⁢

(

Σ

)

contains all sets of

cardinality

no more than 1 (i.e.,

∅

and

singletons

);

•

ℛ

⁢

(

Σ

)

is

closed under

set-theoretic union, concatenation, and Kleene star operations.

Then

L

is a regular language over

Σ

iff

L

∈

ℛ

⁢

(

Σ

)

.

Normal form

. Every regular language can be generated by a

grammar

whose productions are either of the form

A

→

a

⁢

B

or of the form

A

→

λ

, where

A

,

B

are non-terminal symbols, and

a

is a terminal symbol. Furthermore, for every pair

(

A

,

a

)

, there is exactly one production of the form

A

→

a

⁢

B

.

Remark

. Closure properties on the family of regular languages are: union,

intersection

,

complementation

,

set difference

, concatenation, Kleene star,

homomorphism

,

inverse

homomorphism, and

reversal

.

References

1

A. Salomaa

Computation

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

. Cambridge (1985).

Title

regular language

Canonical

name

RegularLanguage

Date of creation

2013-03-22 12:26:31

Last modified on

2013-03-22 12:26:31

Owner

mps (409)

Last modified by

mps (409)

Numerical id

21

Author

mps (409)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 68Q42

Synonym

type-3 language

Synonym

type-3 grammar

Synonym

regular set

Synonym

regular event

Related topic

Language

Related topic

DeterministicFiniteAutomaton

Related topic

NonDeterministicFiniteAutomaton

Related topic

RegularExpression

Related topic

KleeneAlgebra

Related topic

ContextFreeLanguage

Related topic

KleenesTheorem

Defines

regular grammar

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