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:
“Since civilizing children takes the better part of two decades—some twenty years of nonstop thinking, nurturing, teaching, coaxing, rewarding, forgiving, warning, punishing, sympathizing, apologizing, reminding, and repeating, not to mention deciding what to do when—I now understand that one wrong move is invariably followed by hundreds of opportunities to be wrong again.”
—Mary Kay Blakely (20th century)
“Surrealism is not a school of poetry but a movement of liberation.... A way of rediscovering the language of innocence, a renewal of the primordial pact, poetry is the basic text, the foundation of the human order. Surrealism is revolutionary because it is a return to the beginning of all beginnings.”
—Octavio Paz (b. 1914)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)