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
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[edit] 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.
- 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.
[edit] 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
of L - the Kleene star L * of L
- the image φ(L) of L under a homomorphism
- the concatenation
of L and P - the union
of L and P - the intersection
of L and P - the difference L − P of L and P
- the reverse LR of L
- HALF(L), the set of strings that are the first half of a string in L
[edit] 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.
[edit] 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 regular.
[edit] See also
[edit] References
- Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Chapter 1: Regular Languages, pp.31–90. Subsection "Decidable Problems Concerning Regular Languages" of section 4.1: Decidable Languages, pp.152–155.
[edit] External links
- 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.
- Chalchalero! http://www.ucse.edu.ar/fma/sepa/chalchalero.htm. A free visual software to manipulate regular expressions, regular grammars, finite-state machines and finite languages developed by the SEPa! Project Team (Universidad Católica de Santiago del Estero)
| Automata theory: formal languages and formal grammars | |||
|---|---|---|---|
| Chomsky hierarchy |
Grammars | Languages | Minimal automaton |
| Type-0 | Unrestricted | Recursively enumerable | Turing machine |
| n/a | (no common name) | Recursive | Decider |
| Type-1 | Context-sensitive | Context-sensitive | Linear-bounded |
| Type-2 | Context-free | Context-free | Pushdown |
| Type-3 | Regular | Regular | Finite |
| Each category of languages or grammars is a proper subset of the category directly above it. | |||
