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Formal language

A formal language is a set of finite-length words (or "strings") over some finite alphabet. A typical alphabet would be {a, b}, a typical string over that alphabet would be "ababba", and a typical language over that alphabet containing that string would be the set of all strings which contain the same number of a's as b's. The empty word is allowed and is usually denoted by e, ε or λ. Note that while the alphabet is a finite set and every string has finite length, a language may very well have infinitely many member strings.

Some examples of formal languages:

  • the set of all words over {a, b},
  • the set { an | n is a prime number },
  • the set of syntactically correct programs in some programming language, or
  • the set of inputs upon which a certain Turing machine halts.

A formal language can be specified in a great variety of ways, such as:

Several operations can be used to produce new languages from given ones. Suppose L1 and L2 are languages over some common alphabet.

  • The concatenation L1L2 consists of all strings of the form vw where v is a string from L1 and w is a string from L2.
  • The intersection of L1 and L2 consists of all strings which are contained in L1 and also in L2.
  • The union of L1 and L2 consists of all strings which are contained in L1 or in L2.
  • The complement of the language L1 consists of all strings over the alphabet which are not contained in L1.
  • The right quotient L1/L2 of L1 by L2 consists of all strings v for which there exists a string w in L2 such that vw is in L1.
  • The Kleene star L1* consists of all strings which can be written in the form w1w2...wn with strings wi in L1 and n ≥ 0. Note that this includes the empty string ε because n = 0 is allowed.
  • The reverse L1R contains the reversed versions of all the strings in L1.
  • The shuffle of L1 and L2 consists of all strings which can be written in the form v1w1v2w2...vnwn where n ≥ 1 and v1,...,vn are strings such that the concatenation v1...vn is in L1 and w1,...,wn are strings such that w1...wn is in L2.

A typical questions asked about a formal language is how difficult it is to decide whether a given word belongs to the language. This is the domain of computability theory and complexity theory.

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