en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:02:14Z

WP:CHECKWIKI error fixes using AWB (10093)

en>Andersenman: /* Units */ Wikilinked a unit used in the article

2013-12-31T13:28:30Z

Units: Wikilinked a unit used in the article

← Older revision		Revision as of 15:28, 31 December 2013
Line 1:		Line 1:
	~~Alyson~~ is the title ~~individuals use to call me~~ and ~~I believe it seems fairly good when you say it~~. I'~~ve always cherished living~~ in ~~Alaska~~. ~~To perform lacross is~~ the ~~thing I adore most~~ of ~~all~~. ~~Invoicing is what~~ I do for ~~a residing but~~ I~~'ve usually needed my personal company~~.<br><br>~~My web page~~ ... ~~online psychic readings~~ ([~~http~~://si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl~~=~~48014&mid~~=~~c0102 why~~ not ~~try here~~])		{{for\|the representation of variables with natural numbers\|De Bruijn index}}
			In [[mathematical logic]], the '''De Bruijn notation''' is a [[syntax (logic)\|syntax]] for terms in the [[λ calculus]] invented by the [[Netherlands\|Dutch]] [[mathematician]] [[Nicolaas Govert de Bruijn]].<ref name="de bruijn 80">{{cite book\|author=De Bruijn, Nicolaas Govert\|chapter=A survey of the project AUTOMATH\|title=To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism\|editor=Hindley J. R. and Seldin J. P.\|pages=29–61\|publisher=[[Academic Press]]\|year=1980\|isbn=0-12-349050-2\|oclc=6305265}}</ref> It can be seen as a reversal of the usual syntax for the λ calculus where the [[argument (computer science)\|argument]] in an application is placed next to its corresponding binder in the [[function (mathematics)\|function]] instead of after the latter's body.

			== Formal definition ==

			Terms (<math>M, N, \ldots</math>) in the De Bruijn notation are either variables (<math>v</math>), or have one of two ''wagon'' prefixes. The ''abstractor wagon'', written <math>[v]</math>, corresponds to the usual λ-binder of the [[λ calculus]], and the ''applicator wagon'', written <math>(M)</math>, corresponds to the argument in an application in the λ calculus.

			:<math>M,N,... ::=\ v\ \|\ [v]\;M\ \|\ (M)\;N</math>

			Terms in the traditional syntax can be converted to the De Bruijn notation by defining an inductive function <math>\mathcal{I}</math> for which:
			<blockquote><math>
			\begin{align}
			\mathcal{I}(v) &= v \\
			\mathcal{I}(\lambda v.\ M) &= [v]\;\mathcal{I}(M) \\
			\mathcal{I}(M\;N) &= (\mathcal{I}(N))\mathcal{I}(M).
			\end{align}
			</math></blockquote>

			All operations on λ-terms commute with respect to the <math>\mathcal{I}</math> translation. For example, the usual β-reduction,
			:<math>(\lambda v.\ M)\;N\ \ \longrightarrow_\beta\ \ M[v := N]</math>
			in the De Bruijn notation is, predictably,
			:<math>(N)\;[v]\;M\ \ \longrightarrow_\beta\ \ M[v := N].</math>

			A feature of this notation is that abstractor and applicator wagons of β-redexes are paired like parentheses. For example, consider the stages in the β-reduction of the term <math>(M)\;(N)\;[u]\;(P)\;[v]\;[w]\;(Q)\;z</math>, where the redexes are underlined:<ref name="kamaredding 2003">{{cite journal\|author=Kamareddine, Fairouz\|title=Reviewing the classical and the De Bruijn notation for λ-calculus and pure type systems\|journal=[[Logic and Computation]]\|volume=11\|issue=3\|pages=363–394\|year=2001\|issn=0955-792X\|doi=10.1093/logcom/11.3.363}} The example is from page 384.</ref>

			<blockquote><math>
			\begin{align}
			(M)\;\underline{(N)\;[u]}\;(P)\;[v]\;[w]\;(Q)\;z
			&{\ \longrightarrow_\beta\ }
			(M)\;\underline{(P[u:=N])\;[v]}\;[w]\;(Q[u:=N])\;z \\
			&{\ \longrightarrow_\beta\ }
			\underline{(M)\;[w]}\;(Q[u:=N,v:=P[u:=N]])\;z \\
			&{\ \longrightarrow_\beta\ }
			(Q[u:=N,v:=P[u:=N],w:=M])\;z.
			\end{align}
			</math></blockquote>

			Thus, if one views the applicator as an open paren ('<code>(</code>') and the abstractor as a close bracket ('<code>]</code>'), then the pattern in the above term is '<code>((](]]</code>'. De Bruijn called an applicator and its corresponding abstractor in this interpretation ''partners'', and wagons without partners ''bachelors''. A sequence of wagons, which he called a ''segment'', is ''well balanced'' if all its wagons are partnered.

			== Advantages of the De Bruijn notation ==

			In a well balanced segment, the partnered wagons may be moved around arbitrarily and, as long as parity is not destroyed, the meaning of the term stays the same. For example, in the above example, the applicator <math>(M)</math> can be brought to its abstractor <math>[w]</math>, or the abstractor to the applicator. In fact, ''all'' [[commutative conversion\|commutative]]s and [[permutative conversion]]s on lambda terms may be described simply in terms of parity-preserving reorderings of partnered wagons. One thus obtains a ''generalised conversion'' primitive for λ-terms in the De Bruijn notation.

			Several properties of λ-terms that are difficult to state and prove using the traditional notation are easily expressed in the De Bruijn notation. For example, in a [[type theory\|type-theoretic]] setting, one can easily compute the canonical class of types for a term in a typing context, and restate the [[type checking]] problem to one of verifying that the checked type is a member of this class.<ref name="kamareddine nederpelt 96">{{cite journal\|author=Kamareddine, Fairouz\|coauthors=Nederpelt, Rob\|title=A useful λ-notation\|journal=[[Theoretical Computer Science (journal)\|Theoretical Computer Science]]\|volume=155\|year=1996\|pages=85–109\|issn=0304-3975\|doi=10.1016/0304-3975(95)00101-8}}</ref> De Bruijn notation has also been shown to be useful in calculi for [[explicit substitution]] in [[pure type system]]s.<ref name="leuw 95">{{Cite journal\|last=De Leuw\|first=B.-J.\|title=Generalisations in the λ-calculus and its type theory\|year=1995\|place=Masters Thesis, [[University of Glasgow]]\|postscript=<!--None-->}}.</ref>

			== See also ==
			* [[Mathematical notation]]

			== References ==

			{{reflist}}

			[[Category:Lambda calculus]]

76.30.80.101 at 18:10, 23 August 2012

2012-08-23T18:10:33Z

New page

Alyson is the title individuals use to call me and I believe it seems fairly good when you say it. I've always cherished living in Alaska. To perform lacross is the thing I adore most of all. Invoicing is what I do for a residing but I've usually needed my personal company.<br><br>My web page ... online psychic readings ([http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 why not try here])

Gravity gradiometry - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

en>Andersenman: /* Units */ Wikilinked a unit used in the article

76.30.80.101 at 18:10, 23 August 2012