One is interested here in the means of formalizing information in order to be able to handle it (mainly to transmit it). One will thus not be interested in the contents but only in the form.

Alphabet, word, languages

Definitions

One defines a alphabet as a nonempty whole of symbols, for example:

• has = {has, B, C,…, Z}, the Latin alphabet;
• has = {0,1,2,…, 9}, the figures known as Arab
• has = {0,1,…, 9, has, B,…, F}, the hexadecimal figures.
• has = {0,1}, the alphabet of the Boolean Logique.
• has = {has, T, G, C}, the bases of DNA which code our Génome (this alphabet is the principal subject of the Bio-data processing).

One names lettre an element of a alphabet.
One names mot a finished continuation of lettres.
The continuation of 0 letter is named the mot vide, noted ε.
One names langage a whole of mots associated with some rules of interpretation (without this last restriction, any table of random values could be named language ). In the case of the DNA, these rules are contained in the Ribosome, in the natural languages, they are contained in their Lexique, on a computer, they are present in the circuits of the Central processing unit.

Operations

That is to say an alphabet $A$ and a natural entirety $n$.
One notes $A^n$ the whole of all the words length $n$ on $A$ and $A^*$ the whole of all the words of $A$.
One has: $A^*\; =\; \backslash \; bigcup\_\; \{N\; \backslash \; geq\; 0\}\; ^\; \{\backslash \; infty\}\; A^n$ (Closing of Kleene).
The operation of concatenation is defined $\backslash \; cdot:\; A^*\; \backslash \; times\; A^*\; \backslash \; rightarrow\; A^*$ which with $(U,\; v)$ associates a word $w$ which is consisted of the continuation of letters of $u$ then that of $v$.
Example: “marc” $\backslash \; cdot$ “and Sophie” = “marc and Sophie” (the quotation marks are used to delimit the symbols, they are not elements of $A$).

• Properties :
• $\backslash \; cdot$ is associative: $\backslash \; forall\; U,\; v,\; W\; \backslash \; in\; A^*,\; (U\; \backslash \; cdot\; v)\; \backslash \; cdot\; W\; =\; U\; \backslash \; cdot\; (v\; \backslash \; cdot\; W)$
• $\backslash \; cdot$ admits ε like neutral element: $\backslash \; forall\; U\; \backslash \; in\; A^*,\; U\; \backslash \; cdot\; \backslash \; epsilon\; =\; \backslash \; epsilon\; \backslash \; cdot\; U\; =\; u$
• $\backslash \; cdot$ is not not commutative.

Codings and codes

Coding

That is to say L and M two langages.
A codage C of L in M is a morphism (for the operation $\backslash \; cdot$) injective. In other words, it is a correspondence between the words of L and those of M, where with any word of L a single word of M is associated and such as coding with concaténée either equal to concaténée with codings. ($\backslash \; forall\; U,\; v\; \backslash \; in\; L,\; C\; (u.v)\; =\; C\; (U)\; .c\; (v)$).

Code

A language L on an alphabet has is a code if and only if there do not exist two factorizations different from the words $A^*$ with words of L.

Applications, examples

• Coding Manchester
• Coding NRZ
• Coding Miller
• Code Baudot

Random links:

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