Bulk synchronous parallel - Revision history

en>Revent: Cleaning Category:Pages containing citation needed template with deprecated parameters

2014-07-10T19:57:06Z

Cleaning Category:Pages containing citation needed template with deprecated parameters

← Older revision		Revision as of 21:57, 10 July 2014
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	~~I am Oscar and I totally dig that name. It's not~~ ~~home~~ [http://www.~~gaysphere~~.~~net~~/~~blog~~/~~167593~~ std ~~testing at home~~] ~~test a common thing but what she likes performing~~ is ~~[http://Stidiagnostics~~.~~org/hotlines/namerica.html base jumping]~~ and ~~now she is trying to make money with~~ at home [http://~~www~~.~~alemcheap.fi~~/~~show~~/~~the-way-to-cure-an-unpleasant-candidiasis over the counter~~ std test] ~~testing it. South Dakota is where I've always been residing. In her professional lifestyle she is a~~ [http://www.~~sciencedaily~~.~~com/releases/2013/10~~/~~131030185930~~.htm ~~payroll clerk~~] ~~but~~ she~~'s usually wanted~~ her ~~own company~~.<br><br>~~Stop by~~ my blog at home std ~~testing -~~ [http://~~www~~.~~zavodpm.ru~~/~~blogs~~/~~ashelykeplerlkws/9653-valuable-guidance-successfully-treating-candida-albicans home std test kit~~]		Hi [http://www.hard-ass-porn.com/user/MPolglaze home std test] there. Allow me begin by introducing the author, her title is Myrtle Cleary. For many years he's been living in North Dakota and his home [http://ghaziabadmart.com/oxwall/blogs/post/5933 std home test] [http://www.std-Gov.org/faq_1.htm test kit] family members loves it. One of the issues she loves most is to do aerobics and now she is attempting to earn money with it. Hiring is her working day job now and she will not alter it whenever quickly.<br><br>Have a look at my blog: at home std test, [http://youulike.com/profile/116412 click the next web site],

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9949)

2014-02-18T08:02:33Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9949)

← Older revision		Revision as of 10:02, 18 February 2014
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	~~In [[logic]], especially [[mathematical logic]], a '''signature''' lists~~ and ~~describes the [[non-logical symbol]]s of a [[formal language]]. In [[universal algebra]], a signature lists the operations~~ that ~~characterize an [[algebraic structure]]~~. ~~In [[model theory]], signatures are used for both purposes.~~		I am Oscar and I totally dig that name. It's not home [http://www.gaysphere.net/blog/167593 std testing at home] test a common thing but what she likes performing is [http://Stidiagnostics.org/hotlines/namerica.html base jumping] and now she is trying to make money with at home [http://www.alemcheap.fi/show/the-way-to-cure-an-unpleasant-candidiasis over the counter std test] testing it. South Dakota is where I've always been residing. In her professional lifestyle she is a [http://www.sciencedaily.com/releases/2013/10/131030185930.htm payroll clerk] but she's usually wanted her own company.<br><br>Stop by my blog at home std testing - [http://www.zavodpm.ru/blogs/ashelykeplerlkws/9653-valuable-guidance-successfully-treating-candida-albicans home std test kit]

	~~Signatures play the same role in [[mathematics]] as [[type signature]]~~s ~~in [[computer programming]]. They are rarely made explicit in more philosophical treatments of logic.~~

	~~== Definition ==~~

	~~Formally, a (single-sorted) '''signature''' can be defined as a triple σ = (''S''<sub>func</sub>, ''S''<sub>rel</sub>, ar), where ''S''<sub>func</sub> and ''S''<sub>rel</sub> are disjoint sets~~ not ~~containing any other basic logical symbols, called respectively~~
	* ''function symbols'' (examples: +, ×, 0, 1) and
	* ''relation symbols'' ~~or ''predicates'' (examples~~: ~~≤, ∈),~~
	~~and a function ar: ''S''<sub>func<~~/~~sub> <math>\cup<~~/~~math> ''S''<sub>rel</sub> → <math>\mathbb N_0</math> which assigns a non-negative integer called ''[[arity]]'' to every function or relation symbol~~. ~~A function or relation symbol is called ''n''-ary if its arity is ''n''. A nullary (''0''-ary) function symbol is called a ''constant symbol''~~.

	A signature with no function symbols is called a '''relational signature''', and a signature with no relation symbols is called an '''algebraic signature'''. A '''finite signature''' is a signature such that ''S''<sub>func</sub> and ''S''<sub>rel</sub> are finite. More generally, the '''cardinality''' of a signature σ = (''S''<sub>func</sub>, ''S''<sub>rel</sub>, ar) is defined as \|σ\| = \|''S''<sub>func</sub>\| + \|''S''<sub>rel</~~sub>\|.~~

	~~The '''language of a signature''' is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.~~

	~~== Other conventions ==~~

	In universal algebra the word '''type''' or '''similarity type''' is often used as a synonym for "signature". In model theory, a signature σ is often called '''vocabulary''', or identified with the [[first-order language\|(first-order) language]] ''L'' to which it provides the [[non-logical symbols]]. However, the [[cardinality]] of the language ''L'' will always be infinite; if σ is finite then \|L\| will be [[Aleph-naught\|&alefsym;<sub>0</~~sub>~~]].

	~~As the formal definition is inconvenient for everyday use, the definition of~~ a ~~specific signature~~ is ~~often abbreviated in an informal way, as in:~~

	~~:"The standard signature for~~ [~~[abelian group]]s is σ = (+,–,0), where – is a unary operator."~~

	~~Sometimes an algebraic signature is regarded as just a list of arities, as in~~:

	~~:"The similarity type for abelian groups is σ = (2,1,0)."~~

	~~Formally this would define the function symbols of the signature as something like ''f''<sub>0<~~/~~sub>  (nullary), ''f''<sub>1<~~/~~sub> (unary) and ''f''<sub>2</sub> (binary), but in reality the usual names are used even in connection with this convention~~.

	In [[mathematical logic]], very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. They form a set ''S''<sub>const</~~sub> disjoint from ''S''<sub>func<~~/~~sub>, on which the arity function ''ar'' is not defined~~. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula of [[propositional logic]] is ~~also a formula of [[first-order logic]].~~

	~~== Use of signatures in logic and algebra ==~~

	~~In the context of [[first-order logic]], the symbols in a signature are also known as the [[non-logical symbols]], because together~~ with ~~the logical symbols they form the underlying alphabet over which two~~ [~~[formal language]]s are inductively defined: The set of ''terms'' over the signature and the set of (well-formed) ''formulas'' over the signature.~~

	In a [[structure (mathematical logic)\|structure]], an ''interpretation'' ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an ''n''-ary function symbol ''f'' in a structure '''''A''''' with ''domain'' ''A'' is a function ''f<sup>'''A'''</sup>'':~~ ''A<sup>n<~~/~~sup>'' → ''A'', and the interpretation of an ''n''-ary relation symbol is a relation ''R<sup>'''A'''<~~/~~sup>'' ⊆ ''A<sup>n<~~/~~sup>''. Here ''A''<sup>''n''<~~/~~sup> = ''A'' × ''A'' × ... × ''A'' denotes~~ the ~~''n''~~-~~fold [[cartesian product]] of the domain ''A'' with itself, and so ''f'' is in fact an ''n''~~-~~ary function, and ''R'' an ''n''~~-~~ary relation.~~

	~~== Many~~-~~sorted signatures ==~~

	~~For many~~-~~sorted logic and for [[structure (mathematical logic)#Many~~-~~sorted structures\|many-sorted structures~~]~~] signatures must encode information about the sorts~~. ~~The most straightforward way of doing this~~ is ~~via~~ '~~''symbol types''' that play the role of generalized arities~~.~~<ref>~~[http://~~replay~~.~~waybackmachine~~.~~org~~/~~20070929131504~~/~~http:~~//~~react~~.~~cs.uni-sb.de/%7Ezarba/snow/ch01.pdf Many-Sorted Logic~~], the first chapter in [http://replay.waybackmachine.org/20081006212815/http://react.cs.uni-sb.de/~zarba/notes.html Lecture notes on Decision Procedures], written by [http://theory.stanford.edu/~zarba/ Calogero G. Zarba].</ref>

	~~;Symbol types~~

	~~Let ''S'' be a set (of sorts) not containing the symbols × or →.~~

	~~The symbol types over ''S'' are certain words over the alphabet ''S'' <math>\cup</math> {×, →}: the relational symbol types '~~'s~~''<sub>1</sub> × .~~.~~. × ''s''~~<~~sub~~>~~''n''~~<~~/sub~~>, and the functional symbol types ''s''<sub>1</sub> × ... × ''s''<sub>''n''</sub>→''s<nowiki>'</nowiki>'', for non-negative integers ''n'' and ''s''<sub>1</sub>,''s''<sub>2</sub>,...,''s''<sub>''n''</sub>,''s<nowiki>'</nowiki>'' <math>\in</math> ''S''. (For ''n'' = 0, the expression ''s''<sub>1</sub> × ... × ''s''<sub>''n''</sub> denotes the empty word.)

	~~;Signature~~

	~~A (many~~-~~sorted) signature is a triple (''S'', ''P'', type) consisting of~~
	* a set ''S'' of sorts,
	* a set ''P'' of symbols, and
	* a map type which associates to every symbol in ''P'' a symbol type over ''S''.

	~~== Notes ==~~
	~~<references/>~~

	~~== References ==~~
	* {{ cite book \| last=Burris \| first =Stanley N. \| coauthors=Sankappanavar, H.P. \| publisher=[[Springer Science+Business Media\|Springer]] \| title=A Course in Universal Algebra \| year=1981 \| isbn=3-540-90578-2 }} [http://www.~~thoralf~~.~~uwaterloo.ca~~/~~htdocs~~/~~ualg.html Free online edition].~~
	* {{ cite book \| last=Hodges \| first=Wilfrid \| publisher=[[Cambridge University Press]] \| title=A shorter model theory \| year=1997 \| isbn=0-~~521~~-~~58713~~-~~1 }}~~

	~~== External links ==~~
	* [http://www.seop.leeds.ac.uk/ Stanford Encyclopedia of Philosophy]: "[http://www.seop.leeds.ac.uk/entries/model-~~theory/ Model theory]"~~ -- ~~by [[Wilfred Hodges]].~~
	* [http://planetmath.org/ PlanetMath:] Entry "[http://planetmath.org/encyclopedia/Signature.html Signature]" describes the concept for the case when no sorts are introduced.
	* [http://homepages.feis.herts.ac.uk/~comqejb/algspec/pr.html Baillie, Jean], "[http://homepages.feis.herts.ac.uk/~comqejb/algspec/pr.html An Introduction to the Algebraic Specification of Abstract Data Types.]"

	~~[[Category:Model theory]]~~
	~~[[Category:Universal algebra]~~]

24.113.76.81: fixed the reference

2014-01-24T13:50:48Z

fixed the reference

← Older revision		Revision as of 15:50, 24 January 2014
Line 1:		Line 1:
	~~Let me~~ first ~~begin by introducing myself~~. ~~My title~~ is ~~std testing at home Boyd Butts although it~~ is ~~not~~ [~~http~~://~~welkinn~~.~~com~~/~~profile~~-~~17521~~/~~info~~/ ~~over~~ the ~~counter std test~~] ~~title on my birth certificate~~. ~~Hiring~~ is ~~her day job now and she will not change it~~ [http://~~www~~.~~animecontent~~.~~com~~/~~blog~~/~~348813 at home std testing~~] ~~anytime soon. What I adore~~ [http://~~Dermatology~~.~~About~~.~~com~~/od/~~dermphotos~~/ig/~~Herpes~~-~~Gallery~~/ ~~performing~~] ~~is doing~~ [http://~~articles~~.~~Baltimoresun~~.~~com~~/~~2011-02-21~~/~~specialsection~~/~~bs-hs-std-testing-20110221_1_std-testing-gonorrhea-~~and~~-chlamydia-chlamydia-cases ceramics] but I home std test haven~~'~~t made a dime with it~~. ~~Years ago we moved to North Dakota~~.<br><br>~~Also visit my website~~ - ~~[https:~~//~~alphacomms~~.~~zendesk~~.~~com~~/~~entries~~/~~53455304~~-~~Curing~~-~~Your~~-~~Candida~~-~~Albicans~~-~~How~~-To-~~Make~~-It-~~Happen~~-~~Easily alphacomms~~.~~zendesk~~.~~com~~]		In [[logic]], especially [[mathematical logic]], a '''signature''' lists and describes the [[non-logical symbol]]s of a [[formal language]]. In [[universal algebra]], a signature lists the operations that characterize an [[algebraic structure]]. In [[model theory]], signatures are used for both purposes.

			Signatures play the same role in [[mathematics]] as [[type signature]]s in [[computer programming]]. They are rarely made explicit in more philosophical treatments of logic.

			== Definition ==

			Formally, a (single-sorted) '''signature''' can be defined as a triple σ = (''S''<sub>func</sub>, ''S''<sub>rel</sub>, ar), where ''S''<sub>func</sub> and ''S''<sub>rel</sub> are disjoint sets not containing any other basic logical symbols, called respectively
			* ''function symbols'' (examples: +, ×, 0, 1) and
			* ''relation symbols'' or ''predicates'' (examples: ≤, ∈),
			and a function ar: ''S''<sub>func</sub> <math>\cup</math> ''S''<sub>rel</sub> → <math>\mathbb N_0</math> which assigns a non-negative integer called ''[[arity]]'' to every function or relation symbol. A function or relation symbol is called ''n''-ary if its arity is ''n''. A nullary (''0''-ary) function symbol is called a ''constant symbol''.

			A signature with no function symbols is called a '''relational signature''', and a signature with no relation symbols is called an '''algebraic signature'''. A '''finite signature''' is a signature such that ''S''<sub>func</sub> and ''S''<sub>rel</sub> are finite. More generally, the '''cardinality''' of a signature σ = (''S''<sub>func</sub>, ''S''<sub>rel</sub>, ar) is defined as \|σ\| = \|''S''<sub>func</sub>\| + \|''S''<sub>rel</sub>\|.

			The '''language of a signature''' is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.

			== Other conventions ==

			In universal algebra the word '''type''' or '''similarity type''' is often used as a synonym for "signature". In model theory, a signature σ is often called '''vocabulary''', or identified with the [[first-order language\|(first-order) language]] ''L'' to which it provides the [[non-logical symbols]]. However, the [[cardinality]] of the language ''L'' will always be infinite; if σ is finite then \|L\| will be [[Aleph-naught\|&alefsym;<sub>0</sub>]].

			As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:

			:"The standard signature for [[abelian group]]s is σ = (+,–,0), where – is a unary operator."

			Sometimes an algebraic signature is regarded as just a list of arities, as in:

			:"The similarity type for abelian groups is σ = (2,1,0)."

			Formally this would define the function symbols of the signature as something like ''f''<sub>0</sub>  (nullary), ''f''<sub>1</sub> (unary) and ''f''<sub>2</sub> (binary), but in reality the usual names are used even in connection with this convention.

			In [[mathematical logic]], very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. They form a set ''S''<sub>const</sub> disjoint from ''S''<sub>func</sub>, on which the arity function ''ar'' is not defined. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula of [[propositional logic]] is also a formula of [[first-order logic]].

			== Use of signatures in logic and algebra ==

			In the context of [[first-order logic]], the symbols in a signature are also known as the [[non-logical symbols]], because together with the logical symbols they form the underlying alphabet over which two [[formal language]]s are inductively defined: The set of ''terms'' over the signature and the set of (well-formed) ''formulas'' over the signature.

			In a [[structure (mathematical logic)\|structure]], an ''interpretation'' ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an ''n''-ary function symbol ''f'' in a structure '''''A''''' with ''domain'' ''A'' is a function ''f<sup>'''A'''</sup>'': ''A<sup>n</sup>'' → ''A'', and the interpretation of an ''n''-ary relation symbol is a relation ''R<sup>'''A'''</sup>'' ⊆ ''A<sup>n</sup>''. Here ''A''<sup>''n''</sup> = ''A'' × ''A'' × ... × ''A'' denotes the ''n''-fold [[cartesian product]] of the domain ''A'' with itself, and so ''f'' is in fact an ''n''-ary function, and ''R'' an ''n''-ary relation.

			== Many-sorted signatures ==

			For many-sorted logic and for [[structure (mathematical logic)#Many-sorted structures\|many-sorted structures]] signatures must encode information about the sorts. The most straightforward way of doing this is via '''symbol types''' that play the role of generalized arities.<ref>[http://replay.waybackmachine.org/20070929131504/http://react.cs.uni-sb.de/%7Ezarba/snow/ch01.pdf Many-Sorted Logic], the first chapter in [http://replay.waybackmachine.org/20081006212815/http://react.cs.uni-sb.de/~zarba/notes.html Lecture notes on Decision Procedures], written by [http://theory.stanford.edu/~zarba/ Calogero G. Zarba].</ref>

			;Symbol types

			Let ''S'' be a set (of sorts) not containing the symbols × or →.

			The symbol types over ''S'' are certain words over the alphabet ''S'' <math>\cup</math> {×, →}: the relational symbol types ''s''<sub>1</sub> × ... × ''s''<sub>''n''</sub>, and the functional symbol types ''s''<sub>1</sub> × ... × ''s''<sub>''n''</sub>→''s<nowiki>'</nowiki>'', for non-negative integers ''n'' and ''s''<sub>1</sub>,''s''<sub>2</sub>,...,''s''<sub>''n''</sub>,''s<nowiki>'</nowiki>'' <math>\in</math> ''S''. (For ''n'' = 0, the expression ''s''<sub>1</sub> × ... × ''s''<sub>''n''</sub> denotes the empty word.)

			;Signature

			A (many-sorted) signature is a triple (''S'', ''P'', type) consisting of
			* a set ''S'' of sorts,
			* a set ''P'' of symbols, and
			* a map type which associates to every symbol in ''P'' a symbol type over ''S''.

			== Notes ==
			<references/>

			== References ==
			* {{ cite book \| last=Burris \| first =Stanley N. \| coauthors=Sankappanavar, H.P. \| publisher=[[Springer Science+Business Media\|Springer]] \| title=A Course in Universal Algebra \| year=1981 \| isbn=3-540-90578-2 }} [http://www.thoralf.uwaterloo.ca/htdocs/ualg.html Free online edition].
			* {{ cite book \| last=Hodges \| first=Wilfrid \| publisher=[[Cambridge University Press]] \| title=A shorter model theory \| year=1997 \| isbn=0-521-58713-1 }}

			== External links ==
			* [http://www.seop.leeds.ac.uk/ Stanford Encyclopedia of Philosophy]: "[http://www.seop.leeds.ac.uk/entries/model-theory/ Model theory]" -- by [[Wilfred Hodges]].
			* [http://planetmath.org/ PlanetMath:] Entry "[http://planetmath.org/encyclopedia/Signature.html Signature]" describes the concept for the case when no sorts are introduced.
			* [http://homepages.feis.herts.ac.uk/~comqejb/algspec/pr.html Baillie, Jean], "[http://homepages.feis.herts.ac.uk/~comqejb/algspec/pr.html An Introduction to the Algebraic Specification of Abstract Data Types.]"

			[[Category:Model theory]]
			[[Category:Universal algebra]]

24.61.47.87: /* Extensions and uses */

2012-08-30T19:08:23Z

Extensions and uses

New page

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