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| In the [[lambda calculus]], a term is in '''beta normal form''' if no ''[[lambda calculus#β-reduction|beta reduction]]'' is possible.<ref>{{cite web| url=http://encyclopedia2.thefreedictionary.com/Beta+normal+form | title=Beta normal form | work=[http://encyclopedia2.thefreedictionary.com/ Encyclopedia] | publisher=[[The Free Dictionary]] |accessdate=18 November 2013 }}</ref> A term is in '''beta-eta normal form''' if neither a beta reduction nor an ''[[lambda calculus#η-conversion|eta reduction]]'' is possible. A term is in '''head normal form''' if there is no ''beta-redex in head position''.
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| ==Beta reduction==
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| In the lambda calculus, a '''beta redex''' is a term of the form
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| :<math> ((\mathbf{\lambda} x . A(x)) t) </math>
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| where <math>A(x)</math> is a term (possibly) involving variable <math>x</math>.
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| A ''beta reduction'' is an application of the following rewrite rule to a beta redex
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| :<math> ((\mathbf{\lambda} x . A(x)) t) \rightarrow A(t)</math> | |
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| where <math>A(t)</math> is the result of substituting the term <math>t</math> for the variable <math>x</math> in the term <math>A(x)</math>.
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| A beta reduction is in head position if it is of the following form:
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| * <math> \lambda x_0 \ldots \lambda x_{i-1} . (\lambda x_i . A(x_i)) M_1 M_2 \ldots M_n \rightarrow
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| \lambda x_0 \ldots \lambda x_{i-1} . A(M_1) M_2 \ldots M_n </math>, where <math> i \geq 0, n \geq 1 </math>.
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| Any reduction not in this form is an internal beta reduction.
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| ==Reduction strategies==
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| In general, there can be several different beta reductions possible for a given term. '''Normal-order reduction''' is the [[evaluation strategy]] in which one continually applies the rule for ''beta reduction in head position'' until no more such reductions are possible. At that point, the resulting term is in ''head normal form''.
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| In contrast, in '''applicative order reduction''', one applies the internal reductions first, and then only applies the head reduction when no more internal reductions are possible.
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| Normal-order reduction is complete, in the sense that if a term has a head normal form, then normal order reduction will eventually reach it. In contrast, applicative order reduction may not terminate, even when the term has a normal form. For example, using applicative order reduction, the following sequence of reductions is possible:
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| :<math> (\mathbf{\lambda} x . z) ((\lambda w. w w w) (\lambda w. w w w)) </math>
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| :<math> \rightarrow (\lambda x . z) ((\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w))</math>
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| :<math> \rightarrow (\lambda x . z) ((\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w))</math>
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| :<math> \rightarrow (\lambda x . z) ((\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w) (\lambda w. w w w))</math>
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| :<math>\ldots</math>
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| But using normal-order reduction, the same starting point reduces quickly to normal form:
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| :<math> (\mathbf{\lambda} x . z) ((\lambda w. w w w) (\lambda w. w w w)) </math>
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| :<math> \rightarrow z </math>
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| Sinot's [[director string]]s is one method by which the computational complexity of beta reduction can be optimized.
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| ==See also==
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| * [[Lambda calculus]]
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| * [[Normal form (disambiguation)]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Beta Normal Form}}
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| [[Category:Lambda calculus]]
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| [[Category:Normal forms (logic)]]
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