Theory of the languages

The theory of the languages aims to include/understand the operation of the Langage S, seen like means of communication, from a mathematical point of view.

A language is a whole of Mot S. a word (or Lexème) is a combination of elementary signs. The whole of these elementary signs is called Alphabet. The function associating the alphabet with the language is called Grammaire. One can associate with a grammar a automat allowing to determine if a word belongs to a language.

Among the practical applications of the theory of the languages, one finds in particular the Compilateur S, in Informatique

Words

One gives oneself a unit X , called alphabet whose elements are called letters .

• a word length K is a k-uplet of letters $u=\; (a\_1,\; a\_2,\dots ,\; a\_k)$.
• $\backslash \; displaystyle\; X^k=\; \backslash \; underbrace\; \{X.X\dots \; X\}\; \_\; \{K\; \backslash ,\; time\}$ is the whole of the words length K
• $X^*=\; \backslash \; bigcup\_\; \{K\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}\; X^k\; \backslash ,$ is the whole of the words.
• $\backslash \; epsilon$ or () is the blank word length 0.
• One defines on $X^*\; \backslash ,$, a Law of composition interns called Concaténation.

$(a\_1,\dots ,\; a\_n).\; (b\_1,\dots ,\; b\_m)\; =\; (a\_1,\dots ,\; a\_n,\; b\_1,\dots ,\; b\_m)\; \backslash ,$ (length $n+m$).

This internal law of composition is associative and admits the blank word for neutral element, consequently $\backslash \; langle\; X^*,\; \backslash \; cdot,\; \backslash \; epsilon\; \backslash \; rangle\; \backslash ,$ is a monoid, called free monoid on $X\; \backslash ,$.

Remark : Any word $(a\_1,\dots ,\; a\_n)$ equal to is concaténé $(a\_1).\; (a\_2)\dots \; (a\_n)$. By identifying the words length 1 with the letters, one thus writes the word in the form: $u=a\_1.a\_2\dots \; a\_n\; \backslash ,$

Languages

A whole of words on X is called a language . The languages can be characterized by the means which make it possible to describe them, for example:

• the rational languages can be described by finite-state machines or rational expressions;
• the algebraic languages can be described by grammars out-context or automats with piles;
• …

See too

• Finite-state machine
• Closing of Kleene
• formal Grammar
• Hierarchy of Chomsky
• formal Language
• regular Language
• Linguistic

• Noam Chomsky

• Marcel-Paul Schützenberger

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