Formal language

In mathematics, logic, and computer science, a formal language L is a set of finite-length sequences of elements drawn from a specified finite set A of symbols. Among the more common options that are found in applications, a formal language may be viewed as being analogous to (1) a collection of words or (2) a collection of sentences. In Case 1, the set A is called the alphabet of L, whose elements are called words. In Case 2, the set A is called the lexicon or the vocabulary of L, whose elements are then called sentences. In any case, the mathematical theory that treats formal languages in general is known as formal language theory.

Although it is common to hear the term formal language used in other contexts to refer to a mode of expression that is more disciplined or more precise than everyday speech, the sense of formal language discussed in this article is restricted to its meaning in formal language theory.

An alphabet might be $$\left \{ a, b \right \}$$, and a string over that alphabet might be $$ababba$$.

A typical language over that alphabet, containing that string, would be the set of all strings which contain the same number of symbols $$a$$ and $$b$$.

The empty word (that is, length-zero string) is allowed and is often denoted by $$e$$, $$\epsilon$$ or $$\Lambda$$. While the alphabet is a finite set and every string has finite length, a language may very well have infinitely many member strings (because the length of words in it may be unbounded).

A question often asked about formal languages is "how difficult is it to decide whether a given word belongs to a particular language?" This is the domain of computability theory and complexity theory.

Examples
Some examples of formal languages:


 * the set of all words over $${a, b}$$
 * the set $$\left \{ a^{n}\right\}$$, n is a natural number and $$a^{n}$$ means $$a$$ repeated $$n$$ times
 * Finite languages, such as $${a, aa, bba}$$ -
 * the set of syntactically correct programs in a given programming language; or
 * the set of inputs upon which a certain Turing machine halts.

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


 * Strings produced by some formal grammar (see Chomsky hierarchy);
 * Strings produced by a regular expression;
 * Strings accepted by some automaton, such as a Turing machine or finite state automaton;
 * From a set of related YES/NO questions, those questions for which the answer is YES &mdash; see decision problem.

Operations
Several operations can be used to produce new languages from given ones. Suppose $$L_{1}$$ and $$L_{2}$$ are languages over some common alphabet.
 * The concatenation $$L_{1}L_{2}$$ consists of all strings of the form $$vw$$ where $$v$$ is a string from $$L_{1}$$ and $$w$$ is a string from $$L_{2}$$.
 * The intersection $$L_1 \cap L_2$$ of $$L_{1}$$ and $$L_{2}$$ consists of all strings which are contained in $$L_1$$ and also in $$L_{2}$$.
 * The union $$L_1 \cup L_2$$ of $$L_{1}$$ and $$L_{2}$$ consists of all strings which are contained in $$L_{1}$$ or in $$L_{2}$$.
 * The complement of the language $$L_{1}$$ consists of all strings over the alphabet which are not contained in $$L_{1}$$.
 * The right quotient $$L_{1}/L_{2}$$ of $$L_{1}$$ by $$L_{2}$$ consists of all strings $$v$$ for which there exists a string $$w$$ in $$L_{2}$$ such that $$vw$$ is in $$L_{1}$$.
 * The Kleene star $$L_{1}^{*}$$ consists of all strings which can be written in the form $$w_{1}w_{2}...w_{n}$$ with strings $$w_{i}$$ in $$L_{1}$$ and $$n \ge 0$$. Note that this includes the empty string $$\epsilon$$ because $$n = 0$$ is allowed.
 * The reverse $$L_{1}^{R}$$ contains the reversed versions of all the strings in $$L_{1}$$.
 * The shuffle of $$L_{1}$$ and $$L_{2}$$ consists of all strings which can be written in the form $$v_{1}w_{1}v_{2}w_{2}...v_{n}w_{n}$$ where $$n \ge 1$$ and $$v_{1},...,v_{n}$$ are strings such that the concatenation $$v_{1}...v_{n}$$ is in $$L_{1}$$ and $$w_{1},...,w_{n}$$ are strings such that $$w_{1}...w_{n}$$ is in $$L_{2}$$.