Normal form (natural deduction)

Template:Use dmy dates In mathematical logic and proof theory, a derivation in normal form in the context of natural deduction refers to a proof which contains no detours — steps in which a formula is first introduced and then immediately eliminated.

The concept of normalization in natural deduction was introduced by Dag Prawitz in the 1960s as part of a general effort to analyze the structure of proofs and eliminate unnecessary reasoning steps.Template:Sfn The associated normalization theorem establishes that every derivation in natural deduction can be transformed into normal form.

Definition

Natural deduction is a system of formal logic that uses introduction and elimination rules for each logical connective. Introduction rules describe how to construct a formula of a particular form, while elimination rules describe how to infer information from such formulas. A derivation is in normal form if it contains no formula which is both:

• the conclusion of an introduction rule, and
• the major premise of an elimination rule.

A derivation containing such a structure is said to include a detour. Normalization involves transforming a derivation to remove all such detours, thereby producing a proof that directly reflects the logical dependencies of the conclusion on the assumptions.

Another definition of normal derivation in classical logic is:Template:Sfn

A derivation in NK is normal if all major premisses of E-rules are assumptions.

Normalization theorem

The normalization theorem for natural deduction states that:

Every derivation in natural deduction can be converted into a derivation in normal form.

This result was first proved by Dag Prawitz in 1965.Template:Sfn The normalization process typically involves identifying and eliminating maximal formulas — formulas introduced and immediately eliminated—through a sequence of local reduction steps.

Normalization has several important consequences:

• It implies the subformula property: any formula occurring in the proof is a subformula of the assumptions or conclusion.
• It guarantees consistency of the system: there is no derivation of a contradiction from no assumptions.
• It supports constructive content in logic: proofs correspond to explicit constructions or computations.

Examples

Implication

A derivation of <math>A \rightarrow A</math> that includes a detour:

1. [A]    (assumption)
2. A → A  (→ introduction, discharging 1)
3. [A]    (assumption)
4. A      (→ elimination on 2 and 3)

This introduces and then immediately eliminates an implication. A normal derivation is:

1. [A]
2. A → A  (→ introduction)

Conjunction

A derivation of <math>A, B \vdash A</math> that includes a detour:

<math>

\frac{

 \frac{
   A \quad B
 }{
   A \land B
 }[\land\text{I}]

}{

 A

}[\land\text{E}] \quad \Rightarrow \quad A </math>

The elimination is unnecessary if <math>A</math> is already available.

Applications

Normalization is central to several areas of logic and computer science:

• In proof theory, it ensures that logical systems have desirable meta-properties such as consistency and the subformula property.
• In type theory, it underlies the soundness and completeness of type-checking algorithms.
• In proof assistants (e.g. Coq, Agda), normalization is used to verify that formal proofs are constructive and terminating.
• In functional programming, the normalization process corresponds to evaluation strategies for typed lambda calculi.

See also

• Natural deduction
• Curry–Howard correspondence
• Cut-elimination theorem
• Sequent calculus

References

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