Beta reduction

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A beta reduction (also written β reduction) is the process of calculating a result from the application of a function to an expression.

Haskell theoretical foundations

Mathematics - Category theory
Research - Curry/Howard/Lambek

Lambda calculus:
Alpha conversion - Beta reduction
Eta conversion - Lambda abstraction

Recursion - Combinatory logic
Chaitin's construction - Turing machine
Relational algebra

For example, suppose we apply the function

(\x -> 2*x*x + y)

to the value 7. To calculate the result, we substitute 7 for every occurrence of x, and so the application of the function

(\x -> 2*x*x + y)(7)

is reduced to the result

2*7*7 + y

This is a beta reduction.

(Further reductions could be applied to reduce 2*7*7 to 98. Although the lambdas are not explicit, they exist hidden in the definition of (*).)

Also see Lambda calculus and the wikipedia lambda calculus article.