PlanetMath: regular language

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regular language

(Definition)

A regular grammar is a context-free grammar where all productions must take one of the following forms (specified here in BNF, $\lambda$ is the empty string):

$\displaystyle \verb.<.\textit{non-terminal}\verb.>.$	$\displaystyle ::=$	$\displaystyle terminal$
$\displaystyle \verb.<.\textit{non-terminal}\verb.>.$	$\displaystyle ::=$	$\displaystyle terminal\,\verb.<.\textit{non-terminal}\verb.>.$
$\displaystyle \verb.<.\textit{non-terminal}\verb.>.$	$\displaystyle ::=$	$\displaystyle \lambda$

A regular language (also known as a regular set or a regular event) is the set of strings generated by a regular grammar. Regular grammars are also known as Type-3 grammars in the Chomsky hierarchy.

A regular grammar can be represented by a deterministic or non-deterministic finite automaton. Such automata can serve to either generate or accept sentences in a particular regular language. Note that since the set of regular languages is a subset of context-free languages, any deterministic or non-deterministic finite automaton can be simulated by a pushdown automaton.

There is also a close relationship between regular languages and regular expressions. With every regular expression we can associate a regular language. Conversely, every regular language can be obtained from a regular expression. For example, over the alphabet $\lbrace a,b,c\rbrace$ , the regular language associated with the regular expression $a(b\cup c)^*a$ is the set

$\displaystyle \lbrace a\rbrace \circ \lbrace b,c\rbrace^* \circ \lbrace a\rbrace=\lbrace awa\mid w$ is a word in two letters $\displaystyle b$ and $\displaystyle c\rbrace,$

where $\circ$ is the concatenation operation, and

is the Kleene star operation. Note that

could be the empty word $\lambda$ .

Yet another way of describing a regular language is as follows: take any alphabet $\Sigma$ . Let $\mathcal{R}(\Sigma)$ be the smallest subset of $P(\Sigma^*)$ (the power set of the set of words over $\Sigma$ , in other words, the set of languages over $\Sigma$ ), among all subsets of $P(\Sigma^*)$ with the following properties:

$\mathcal{R}(\Sigma)$ contains all sets of cardinality no more than 1 (i.e., $\varnothing$ and singletons);
$\mathcal{R}(\Sigma)$ is closed under set-theoretic union, concatenation, and Kleene star operations.

Then

is a regular language over $\Sigma$ iff $L\in \mathcal{R}(\Sigma)$ .

"regular language" is owned by mps. [ full author list (3) | owner history (1) ]

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Other names:

regular grammar, Type-3 grammar, regular set, regular event

Keywords:

regular expression, Chomsky

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Cross-references: iff, union, closed under, singletons, cardinality, contains, properties, languages, power set, empty word, Kleene star, operation, concatenation, alphabet, associate, regular expressions, pushdown automaton, subset, sentences, generate, automata, non-deterministic finite automaton, Chomsky hierarchy, generated by, strings, empty string, BNF, productions, context-free grammar
There are 15 references to this entry.

This is version 13 of regular language, born on 2002-02-23, modified 2007-11-09.
Object id is 2548, canonical name is RegularLanguage.
Accessed 12165 times total.

Classification:

AMS MSC:	68Q45 (Computer science :: Theory of computing :: Formal languages and automata)
	68Q42 (Computer science :: Theory of computing :: Grammars and rewriting systems)

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