Beta reduction

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Revision as of 12:55, 30 January 2007 (edit) MathematicalOrchid (Talk | contribs) (I think my example actually included an eta-reduction as well as a beta conversion. Edited example.) ← Previous diff		Current revision (18:22, 3 February 2007) (edit) (undo) Nedervold (Talk | contribs)
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-	A ''beta reduction'' (also written ''β reduction'') is where you actually apply a lambda function to an expression to generate a result.	+	A ''beta reduction'' (also written ''β reduction'') is the process of calculating a result from the application of a function to an expression.
-	For example, suppose we have	+	{{Foundations infobox}}
		+	For example, suppose we apply the function
	<haskell>		<haskell>
-	2*x*x + y	+	(\x -> 2*x*x + y)
	</haskell>		</haskell>
-	If we now replace every occurance of <hask>x</hask> with 7, we arrive at	+	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
		+	<haskell>
		+	(\x -> 2*x*x + y)(7)
		+	</haskell>
		+	is ''reduced'' to the result
	<haskell>		<haskell>
	2*7*7 + y		2*7*7 + y
	</haskell>		</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].		Also see [[Lambda calculus]] and the [http://en.wikipedia.org/wiki/Lambda_calculus wikipedia lambda calculus article].
	[[Category:Glossary]]		[[Category:Glossary]]

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Current revision

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

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

is reduced to the result

This is a beta reduction.

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

Also see Lambda calculus and the wikipedia lambda calculus article.

Category: Glossary