Normal form

This article describes the mathematical representation - for further meanings, see normal form (disambiguation) .

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A normal form (also canonical form ) is a mathematical representation with certain properties given by the type of normal form. If a normal form is defined, it can be achieved using any representation using an equivalence relation . If several representations lead to the same normal form, they are equivalent in terms of the type of normal form and can therefore be compared and classified. Many normal forms are unique ; there is only one normal form for each representation.

Formally, a normal form is the last element in a chain of a well-founded relation . The relation is defined by the permitted transformations. The well-foundedness of the relations follows from the finiteness of the number of manipulations.

Examples

• The abbreviated fraction of a rational number sets two given fractions in relation to their numerical values: and both have the normal form and thus the same numerical value. ${\displaystyle {\tfrac {2}{4}}}$ ${\displaystyle {\tfrac {3}{6}}}$ ${\displaystyle {\tfrac {1}{2}}}$
• The step form (see below) sets a matrix in relation to a matrix if it is evident from pivoting. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A}$

List of normal forms

Important, concrete normal forms are:

• In mathematics, a representation of an object that has certain predetermined properties and can be uniquely determined for all objects of this type. In particular:
  • the Hessian normal form of a plane
  • the step normal form of a linear system of equations, see Gaussian elimination method
  • the Jordanian normal form of a square matrix
  • the Frobenius normal form , also the rational normal form of a square matrix
  • the Smith normal form of a matrix with entries from a main ideal ring
  • the normal form of an orthogonal matrix, see Orthogonal Matrix #Diagonalisability
  • the normal form of a linear function, see linear function
  • the normal form of a quadratic equation, see quadratic equation
  • the normal shape of a quadric, see quadric #Normal Shapes
  • a fully abbreviated fraction for a rational number

• in game theory a form of representation of a game , see normal form of a game
• in theoretical computer science a simple form of context-free grammar, see Chomsky hierarchy . In particular
  • the Chomsky normal form
  • the Greibach normal form
  • the Gentzen normal form, see Gentzenscher's law

• in practical informatics for relational databases the data structure that results from the gradual removal of redundancies, see normalization (database)

• in logic a form of representation of a logical formula , in particular
  • the Shannon normal form
  • the negation normal form
  • Formulas in canonical normal form , especially as:
    • conjunctive normal form
    • disjunctive normal form
    • All round normal form

• in predicate logic
  • the adjusted normal form
  • the negation normal form
  • the prenex normal form
  • the scolemic form
  • the normal clause form

• in abstract reduction systems an object that cannot be further reduced

• in digital technology for digital filters in formal form the minimum number of their elements taking into account the desired filter properties, see digital filter

Web links

Wiktionary: normal form  - explanations of meanings, word origins, synonyms, translations