Editing Computable function (section)

==  Formal languages ==
{{main|Formal language}}

In [[Computability theory (computer science)|computability theory in computer science]], it is common to consider [[formal language]]s. An '''alphabet''' is an arbitrary set. A '''word''' on an alphabet is a finite sequence of symbols from the alphabet; the same symbol may be used more than once. For example, binary strings are exactly the words on the alphabet {{math|{0, 1}}}. A '''language''' is a subset of the collection of all words on a fixed alphabet. For example, the collection of all binary strings that contain exactly 3 ones is a language over the binary alphabet.

A key property of a formal language is the level of difficulty required to decide whether a given word is in the language. Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine. A language is called '''computable''' (synonyms: '''recursive''', '''decidable''') if there is a computable function {{math|''f''}} such that for each word {{math|<var>w</var>}} over the alphabet, {{math|''f''(<var>w</var>) {{=}} 1}} if the word is in the language and {{math|''f''(<var>w</var>) {{=}} 0}} if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language.

A language is '''computably enumerable''' (synonyms: '''recursively enumerable''', '''semidecidable''') if there is a computable function {{math|''f''}} such that {{math|''f''(<var>w</var>)}} is defined if and only if the word {{math|<var>w</var>}} is in the language. The term ''enumerable'' has the same etymology as in computably enumerable sets of natural numbers.