Math 2.0

Computer science Definition

Evaluation strategy

Builds on Lambda calculusBeta reductionBeta normal form

Definition

Let Λ\Lambda be the set of terms of the lambda calculus. A reduction strategy for β\beta-reduction is a partial map S ⁣:Λ→ΛS \colon \Lambda \to \Lambda with these properties.

  1. S(M)S(M) is defined exactly when MM is not in beta normal form.
  2. If S(M)S(M) is defined, then M→βS(M)M \to_\beta S(M) by one step of beta reduction.

The strategy computes from MM the sequence M,S(M),S(S(M)),…M, S(M), S(S(M)), \ldots, which stops when a normal form is reached.

Two standard strategies are these. The normal order strategy contracts the leftmost outermost redex, the one whose λ\lambda stands furthest to the left in the written term. The applicative order strategy contracts the leftmost innermost redex, a redex that contains no other redex.

Remarks

Normalization theorem: if MM has a β\beta-normal form, the normal order strategy reaches it after finitely many steps.

Example: let Ω=(λx. x x)(λx. x x)\Omega = (\lambda x.\, x\, x)(\lambda x.\, x\, x) and M=(λx. y) ΩM = (\lambda x.\, y)\, \Omega. Normal order contracts the outer redex and gives yy in one step. Applicative order contracts the redex Ω\Omega inside the argument and gets MM again, so it never terminates.

Non-example: a map with S((λx. x) y)=(λx. x) yS((\lambda x.\, x)\, y) = (\lambda x.\, x)\, y is not a reduction strategy. It fails clause 2, since the only one-step reduct of (λx. x) y(\lambda x.\, x)\, y is yy.

In programming languages, call-by-name and call-by-value are weak versions of these two strategies: they never reduce inside a λ\lambda-abstraction.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Beta normal form, concept, no evidence levelBeta normal formconceptBeta reduction, concept, no evidence levelBeta reductionconceptLambda calculus, concept, no evidence levelLambda calculusconceptEvaluation strategy, definition, no evidence levelEvaluation strategydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Beta normal form concept
  2. Beta reduction concept
  3. Lambda calculus concept
  4. Evaluation strategy definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Evaluation_strategy
About this page
TagFNWZCVBQFQ
URIm2://FNWZCVBQFQ
Kinddefinition
WritingW0 stub
MSC68N18, 03B40, 68Q55
DefinesEvaluation strategy
UsesLambda calculus, Beta reduction, Beta normal form
Changed3 November 2026

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