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 is not 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 is regular, one often uses the Myhill–Nerode theorem or the pumping lemma among other methods.
There are two purely algebraic approaches to define regular languages. If:
- Σ is a finite alphabet,
- 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,
- S is a subset of M
then the set 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.
Read more about this topic: Regular Language
Famous quotes containing the words deciding, language and/or regular:
“Our passional nature not only lawfully may, but must, decide an option between propositions, whenever it is a genuine option that cannot by its nature be decided on intellectual grounds; for to say, under such circumstances, “Do not decide, but leave the question open,” is itself a passional decision—just like deciding yes or no—and is attended with the same risk of losing the truth.”
—William James (1842–1910)
“Translate a book a dozen times from one language to another, and what becomes of its style? Most books would be worn out and disappear in this ordeal. The pen which wrote it is soon destroyed, but the poem survives.”
—Henry David Thoreau (1817–1862)
“While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.”
—Anton Pavlovich Chekhov (1860–1904)