Regular Languages Article Index for
Regular 		Articles about
Regular Language 		Website Links For
Regular 		 

Information About

Regular Languages
  CATEGORIES ABOUT REGULAR LANGUAGE
   
formal languages
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




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 Language s 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 .


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:



DECIDING WHETHER A LANGUAGE IS REGULAR


To locate the regular languages in the or the Pumping Lemma .

  • denotes the 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.


  • , one defines an Equivalence Relation ~ on Σ--- as follows: ''u'' ~ ''v'' is defined to mean


    • 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''.



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.


SEE ALSO



REFERENCES

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



EXTERNAL LINKS