Home > Recursively enumerable language

Alternatively we can define them as those languages for which an algorithm exists, that when given string w as input, halts and outputs YES if and only if w belongs to the language L. If w does not belong to the language L, the algorithm either runs forever, or halts and outputs NO. If the algorithm always halts the language is called recursive.

Note that, if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is "already" produced for a number which is less than n. If it already is produced, use the output for input n+1 instead (recursively), but again, test whether it is "new".

All regular, context-free, context-sensitive and recursive languages are recursively enumerable.

1 Definition

A formal language is called recursively enumerable if and only if it is a recursively enumerable subset in the setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now of all possible words over the alphabetAn alphabet is a complete standardized set of letters—basic written symbols—each of which roughly represents or represented historically a phoneme of a spoken language. This as distinguished from other writing systems such as ideograms, in which symbols r of the language.

2 Properties

If L and P are two recursively enumerable languages, then the following languages are recursively enumerable as well:

• the Kleene starIn mathematical logic and computer science, the Kleene star (or Kleene closure is a unary operation, either on sets of strings or on sets of symbols or characters. The application of the Kleene star to a set V is written as V . It is widely used for regul L^* of L
• the concatentation LP of L and P
• the union L∪P
• the intersection L∩P

Note that the set difference L\P and in particular the complement of L need not be recursively enumerable.