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.

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 relational database theory

• first normal form
• second normal form
• third normal form
• fourth normal form
• fifth normal form

see database normalization for all four.

In linear algebra:

• Jordan normal form
• Frobenius normal form

In proof theory

• Normal form for proofs in natural deduction

In the lambda calculus

• Beta normal form and Beta-eta normal form

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.

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

See also

• canonical form
• Smith normal form