Regular Language - Deciding Whether A Language Is Regular

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

uwL if and only if vwL 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:

    Once again, we’ve wasted time and money by dealing with the homeless backward. Too much energy has gone into deciding where we do not want them to be, and making sure that they would not be there.
    Anna Quindlen (b. 1952)

    The necessity of poetry has to be stated over and over, but only to those who have reason to fear its power, or those who still believe that language is “only words” and that an old language is good enough for our descriptions of the world we are trying to transform.
    Adrienne Rich (b. 1929)

    A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.
    Henry David Thoreau (1817–1862)