Everything about Regular Language totally explained
In
theoretical computer science, a
regular language is a
formal language (for example, a possibly infinite set of finite sequences of symbols from a finite alphabet) that satisfies the following equivalent properties:* it can be accepted by a
deterministic finite state machine
Deciding whether a language is regular
To locate the regular languages in the
Chomsky hierarchy, one notices that every regular language is
context-free. The converse isn't true: for example the language consisting of all strings having the same number of
a's as
b's is context-free but not regular. To prove that a language such as this isn't regular, one uses the
Myhill-Nerode theorem or the
pumping lemma.
There are two purely algebraic approaches to define regular languages. If Σ is a finite alphabet and Σ* denotes the
free monoid over Σ consisting of all strings over Σ,
f : Σ* →
M is a
monoid homomorphism where
M is a
finite monoid, and
S is a subset of
M, then the set
f −1(
S) is regular. Every regular language arises in this fashion.
If
L is any subset of Σ*, one defines an
equivalence relation ~ (called the
syntactic relation) on Σ* as follows:
u ~
v is defined to mean
» uw ∈
L if and only if
vw ∈
L for all
w ∈ Σ*
The language
L is regular if and only if the number of equivalence classes of ~ is finite (A proof of this is provided in the article on the
syntactic monoid). When a language is regular, then the number of equivalence classes is equal to the number of states of the
minimal deterministic finite automaton accepting
L.
A similar set of statements can be formulated for a monoid
. In this case, equivalence over
M leads to the concept of a
recognizable language.
Finite languages
A specific subset within the class of regular languages is the finite languages – those containing only a finite number of words. These are obviously regular as one can create a
regular expression that's the
union of every word in the language, and thus are regular.
Further Information
Get more info on 'Regular Language'.
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