##  [Normalization by Evaluation](/normalization-evaluation-0) 

 Definition

A technique for computing the normal form of a term by interpreting the term in a suitable semantic model (evaluation) and then reifying or reading back the semantic value into a syntactic normal form.

 

 

 

 

 

 





## Principle

Principle

Evaluate syntactic terms into a semantic domain where reductions are performed implicitly by the model, then reify semantic values back to syntax to obtain a normal or canonical representative.

 

 

 

 

 





## Demonstration

Demonstration

For simply typed λ-calculus, interpret terms in a model of functions and neutral terms; evaluating (λx. M) N yields the semantic result and reification produces the β-normal, η-long form without performing explicit syntactic β-steps.

 

 

 

 

## Misapplication

Misapplication

Using a semantic domain that does not respect the operational congruences (for example ignoring neutral terms) or failing to implement correct reification leads to incorrect or incomplete normal forms.

 

 

 

 

 





## Consequence

Consequence

NbE often yields efficient normalization algorithms, handles extensional equalities cleanly (η-laws), and separates computational content (evaluation) from syntactic reconstruction (reification), facilitating proofs of normalization and decidability of equality.

 

 

 

 

## Reversal

Reversal

The reversal is pure syntactic normalization by repeated local rewrite steps (β-reduction, η-expansion) which may be less modular and harder to relate to models of computation.

 

 

 

 

 





## Boundary

Boundary

Applies when a faithful semantic interpretation and effective reification procedure exist; it is not applicable when the semantics is not computable or when reification cannot produce finite syntactic representatives.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between building rich semantic domains that make evaluation simple and keeping reification feasible; more expressive models simplify evaluation but complicate reification back to syntax.

 

 

 

 

 





## Synthesis

Synthesis

Normalization by evaluation computes canonical syntactic forms by delegating reduction to a semantic model and then systematically reconstructing syntax, yielding modular, often efficient normalization and clear semantic justification of normal forms.