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Mathematical logic and foundations Definition

Beta reduction

Builds on Lambda calculusTransitive closure

Definition

Let Λ\Lambda be the set of terms of the lambda calculus, taken up to renaming of bound variables. For terms M,NM, N and a variable xx, let M[x:=N]M[x := N] denote the capture-avoiding substitution of NN for the free occurrences of xx in MM. A β\beta-redex is a term of the form (λx. M) N(\lambda x.\, M)\, N, and its contractum is M[x:=N]M[x := N].

One-step β\beta-reduction →β\to_\beta is the least binary relation on Λ\Lambda with these properties.

  1. (λx. M) N→βM[x:=N](\lambda x.\, M)\, N \to_\beta M[x := N] for all xx, MM and NN.
  2. If M→βM′M \to_\beta M', then M N→βM′ NM\, N \to_\beta M'\, N, N M→βN M′N\, M \to_\beta N\, M' and λx. M→βλx. M′\lambda x.\, M \to_\beta \lambda x.\, M'.

Multi-step β\beta-reduction ↠β\twoheadrightarrow_\beta is the reflexive transitive closure of →β\to_\beta. β\beta-conversion =β=_\beta is the least equivalence relation that contains →β\to_\beta.

Remarks

Example: (λx. x y)(λz. z)→β(λz. z) y→βy(\lambda x.\, x\, y)(\lambda z.\, z) \to_\beta (\lambda z.\, z)\, y \to_\beta y. Each step contracts a redex by clause 1, so (λx. x y)(λz. z)↠βy(\lambda x.\, x\, y)(\lambda z.\, z) \twoheadrightarrow_\beta y.

Non-example: the term x (λy. y)x\, (\lambda y.\, y) has no reduction step. It is an application, but its function part is the variable xx and not an abstraction, so it is not a redex, and its subterms contain no redex either.

Capture avoidance matters: (λx. λy. x) y(\lambda x.\, \lambda y.\, x)\, y reduces to λz. y\lambda z.\, y, not to λy. y\lambda y.\, y.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Lambda calculus, concept, no evidence levelLambda calculusconceptTransitive closure, concept, no evidence levelTransitive closureconceptBeta reduction, definition, no evidence levelBeta reductiondefinition
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  1. Lambda calculus concept
  2. Transitive closure concept
  3. Beta reduction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lambda_calculus, β-reduction
About this page
TagZ487CD38WT
URIm2://Z487CD38WT
Kinddefinition
WritingW0 stub
MSC03B40, 68N18
DefinesBeta reduction
UsesLambda calculus, Transitive closure
Changed3 November 2026

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