Difference between revisions of "Beta reduction"

Revision as of 18:22, 3 February 2007

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 General: Mathematics - Category theory Research - Curry/Howard/Lambek Lambda calculus: Alpha conversion - Beta reduction Eta conversion - Lambda abstraction Other: 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 free 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 (*).)

@@ Line 1: / Line 1: @@
-A ''beta reduction'' (also written ''&beta; reduction'') is where you actually apply a lambda function to an expression to generate a result.
+A ''beta reduction'' (also written ''&beta; reduction'') is the process of calculating a result from the application of a function to an expression.
+{{Foundations infobox}}
-For example, suppose we have
+For example, suppose we apply the function
 <haskell>
-*x*x + y
+(\x -> 2*x*x + y)
 </haskell>
+to the value <hask>7</hask>.  To calculate the result, we substitute <hask>7</hask> for every [[Free variable|free occurrence]] of <hask>x</hask>, and so the application of the function
-If we now replace every occurance of <hask>x</hask> with 7, we arrive at
+<haskell>
+(\x -> 2*x*x + y)(7)
+</haskell>
+is ''reduced'' to the result
 <haskell>
 *7*7 + y
 </haskell>
-We have thus performed a ''beta reduction''.
+This is a ''beta reduction''.
+(Further reductions could be applied to reduce <hask>2*7*7</hask> to <hask>98</hask>.  Although the lambdas are not explicit, they exist hidden in the definition of <hask>(*)</hask>.)
 Also see [[Lambda calculus]] and the [http://en.wikipedia.org/wiki/Lambda_calculus wikipedia lambda calculus article].