Normal form

The term normal form is used in a variety of contexts. Many of the uses in mathematics are special cases of a single situation, looked at abstractly: within an equivalence class one specifies a representative element, which is in a simplest or most manageable or otherwise tidiest and most desirable form, in terms of structure or syntax. A little more loosely, an equivalence class might contain several examples of such special, distinguished elements. For example, the Jordan normal form under similarity of matrices (link below) may mean any suitable block matrix in similarity class, and in the general case there can be several such.

To transform something into a normal form is often called normalization.

In classical logic, propositions may be in:

• Negation normal form
• Conjunctive normal form
• Disjunctive normal form
• Algebraic normal form

In formal language theory:

• Chomsky normal form
• Greibach normal form
• Kuroda normal form

In game theory:

• Normal form game

In relational database theory

• First normal form
• Second normal form
• Third normal form
• Boyce-Codd normal form
• Fourth normal form
• Fifth normal form
• Sixth normal form
• Domain-key normal form

see also database normalization.

In linear algebra:

• Jordan normal form
• Frobenius normal form
• Smith normal form

In proof theory

• Normal form for proofs in natural deduction

In the lambda calculus

• Beta normal form if no beta reduction is possible

In musical set theory:

• the normal form of a pitch or pitch class set, which is the order that occupies the smallest possible span and is stacked leftmost.

In rewriting:

• a normal form is an element of a rewrite system which cannot be rewritten further.

In Scientific notation:

• normalized form is where the coefficient is $\geq$ 1 and $<\$ 10

See also

• canonical form
• Normalization property