Regular language

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A regular language is a formal language (i.e., 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
it can be accepted by a nondeterministic finite state machine
it can be accepted by an alternating finite automaton
it can be described by a regular expression
it can be generated by a regular grammar
it can be generated by a prefix grammar
it can be accepted by a read-only Turing machine
it can be defined in monadic second-order logic

1 Regular languages over an alphabet
2 Closure Properties
3 Deciding whether a language is regular
4 Finite languages
5 See also
6 References
7 External resources

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Regular languages over an alphabet

The collection of regular languages over an alphabet Σ is defined recursively as follows:

the empty language Ø is a regular language.
the empty string language { ε } is a regular language.
For each a ∈ Σ, the singleton language { a } is a regular language.
If A and B are regular languages, then A U B (union), A • B (concatenation), and A* (Kleene star) are regular languages.
If A is a regular language, then (A) denotes the same regular language
No other languages over Σ are regular.

All finite languages are regular. Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of a's, or the language consisting of all strings of the form: several a's followed by several b's.

If a language is not regular, it requires a machine with at least Ω(log log n) space to recognize (where n is the input size). In other words, DSPACE(o(log log n)) equals the class of regular languages. In practice, most nonregular problems are solved by machines taking at least logarithmic space.

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Closure Properties

The Regular Languages are closed under the following operations: That is, if "L" and "P" are Regular Languages, the following languages are Regular as well:

the complement <math>\bar{L}</math> of L
the Kleene star L^* of L
the homomorphism φ(L) of L
the concatenation LP of L and P
the union L∪P of L and P
the intersection L∩P of L and P
the difference <math>L</math>\<math>P</math> of L and P
the half(<math>L</math>), log(<math>L</math>) and sqrt(<math>L</math>) of L

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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 not regular, one uses the Myhill-Nerode theorem or the pumping lemma.

There are two purely algebraic approaches to defining 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 ~ 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; if this is the case, this number is equal to the number of states of the minimal deterministic finite automaton accepting L.

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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 is the union of every word in the language, and thus are provably regular.

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References

Template:Cite book Chapter 1: Regular Languages, pp.31–90. Subsection "Decidable Problems Concerning Regular Languages" of section 4.1: Decidable Languages, pp.152–155.

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External resources

Department of Computer Science at the University of Western Ontario: Grail+, http://www.csd.uwo.ca/Research/grail/. A software package to manipulate regular expressions, finite-state machines and finite languages. Free for non-commercial use.