|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[logic]], [[mathematics]], and [[computer science]], the '''arity''' {{IPAc-en|audio=en-us-arity.ogg|ˈ|æ|r|ɨ|t|i}} of a [[function (mathematics)|function]] or [[operation (mathematics)|operation]] is the number of [[argument of a function|arguments]] or [[operand]]s the function or operation accepts. The arity of a [[relation (mathematics)|relation]] (or [[predicate (mathematical logic)|predicate]]) is the dimension of the [[domain of a function|domain]] in the corresponding [[Cartesian product]]. (A function of arity ''n'' thus has arity ''n''+1 considered as a relation.) The term springs from words like unary, binary, ternary, etc. Unary functions or predicates may be also called "monadic"; similarly, binary functions may be called "dyadic".
| | Eartha just what you can call me and I feel comfortable making sure use complete name. Florida has for ages been his living place. Filing is my [http://Www.alexa.com/search?q=day+job&r=topsites_index&p=bigtop day job] now and the salary recently been really completing. What she really enjoys doing is badge collecting but she can't get it to her job. If you want to find out more check out my website: http://www.ezramedicalcare.com/?document_srl=2091148<br><br>Here is my page: [http://www.ezramedicalcare.com/?document_srl=2091148 video role] |
| | |
| In mathematics arity may also be named ''rank'',<ref name="Hazewinkel2001">{{cite book|author=[[Michiel Hazewinkel]]|title=Encyclopaedia of Mathematics, Supplement III|url=http://books.google.com/books?id=47YC2h295JUC&pg=PA3|year=2001|publisher=Springer|isbn=978-1-4020-0198-7|page=3}}</ref><ref name="Schechter1997">{{cite book|author=Eric Schechter|title=Handbook of Analysis and Its Foundations|url=http://books.google.com/books?id=eqUv3Bcd56EC&pg=PA356|year=1997|publisher=Academic Press|isbn=978-0-12-622760-4|page=356}}</ref> but this word can have many other meanings in mathematics. In logic and philosophy, arity is also called ''adicity'' and ''degree''.<ref name="DetlefsenBacon1999">{{cite book|author1=Michael Detlefsen|author2=David Charles McCarty|author3=John B. Bacon|title=Logic from A to Z|url=http://books.google.com/books?id=pc3E4Xdf2GgC&pg=PP17|year=1999|publisher=Routeledge|isbn=978-0-415-21375-2|page=7}}</ref><ref name="CocchiarellaFreund2008">{{cite book|author1=Nino B. Cocchiarella|author2=Max A. Freund|title=Modal Logic: An Introduction to its Syntax and Semantics|url=http://books.google.com/books?id=zLmxqytfLhgC&pg=PA121|year=2008|publisher=Oxford University Press|isbn=978-0-19-536658-7|page=121}}</ref> In [[linguistics]], arity is usually named ''[[valency (linguistics)|valency]]''.<ref name="Crystal2008">{{cite book|author=David Crystal|title=Dictionary of Linguistics and Phonetics|year=2008|publisher=John Wiley & Sons|isbn=978-1-405-15296-9|page=507|edition=6th}}</ref>
| |
| | |
| In [[computer programming]], there is often a [[Syntax (programming languages)|syntactical]] distinction between [[Operator (programming)|operators]] and [[Function (computer science)|functions]]; syntactical operators usually have arity 0, 1, or 2. Functions vary widely in the number of arguments, though large numbers can become unwieldy. Some programming languages also offer support for [[variadic functions]], i.e. functions syntactically accepting a variable number of arguments.
| |
| | |
| ==Examples==
| |
| {{Expand section|more math and logic examples|date=March 2012}}
| |
| The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the [[addition]] operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation".
| |
| In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based [[numeral system]]s such as [[Binary numeral system|binary]] and [[hexadecimal]]. One combines a [[Latin]] prefix with the -ary ending; for example:
| |
| * A nullary function takes no arguments.
| |
| * A [[unary operation|unary function]] takes one argument.
| |
| * A [[binary operation|binary function]] takes two arguments.
| |
| * A [[ternary operation|ternary function]] takes three arguments.
| |
| * An ''n''-ary function takes ''n'' arguments.
| |
| | |
| ===Nullary===
| |
| Sometimes it is useful to consider a [[constant (mathematics)|constant]] to be an operation of arity 0, and hence call it ''nullary''.
| |
| | |
| Also, in non-[[functional programming]], a function without arguments can be meaningful and not necessarily constant (due to [[side effect (computer science)|side effect]]s). Often, such functions have in fact some ''hidden input'' which might be [[global variable]]s, including the whole state of the system (time, free memory, ...). The latter are important examples which usually also exist in "purely" functional programming languages.
| |
| | |
| ===Unary===
| |
| Examples of [[unary operator]]s in mathematics and in programming include the unary minus and plus, the increment and decrement operators in [[C (programming language)|C]]-style languages (not in logical languages), and the [[factorial]], [[Multiplicative inverse|reciprocal]], [[floor function|floor]], [[ceiling function|ceiling]], [[fractional part]], [[sign (mathematics)|sign]], [[absolute value]], [[complex conjugate]], and [[Norm (mathematics)|norm]] functions in mathematics. The [[two's complement]], [[Reference (computer science)|address reference]] and the [[logical NOT]] operators are examples of unary operators in math and programming. According to [[Willard Van Orman Quine|Quine]], a more suitable term is "singulary".<ref>{{Citation
| |
| | last = Quine
| |
| | first = W. V. O.
| |
| | title = Mathematical logic
| |
| | year = 1940
| |
| | place = Cambridge, MA
| |
| | publisher = Harvard University Press
| |
| | page=13
| |
| }}</ref>
| |
| | |
| All functions in [[lambda calculus]] and in some [[functional programming language]]s (especially those descended from [[ML (programming language)|ML]]) are technically unary, but see [[#n-ary|n-ary]] below.
| |
| | |
| ===Binary===
| |
| Most operators encountered in programming are of the [[binary operation|binary]] form. For both programming and mathematics these can be the [[multiplication operator]], the addition operator, the division operator. Logical predicates such as ''OR'', ''[[exclusive or|XOR]]'', ''AND'', ''IMP'' are typically used as binary operators with two distinct operands.
| |
| | |
| ===Ternary===
| |
| From [[C (programming language)|C]], [[C++]], [[C Sharp (programming language)|C#]], [[Java (programming language)|Java]], [[Perl]] and variants comes the [[ternary operator]] <code>[[?:]]</code>, which is a so-called [[conditional operator]], taking three parameters.
| |
| [[Forth (programming language)|Forth]] also contains a ternary operator, <code>*/</code>, which multiplies the first two (one-cell) numbers, dividing by the third, with the intermediate result being a double cell number. This is used when the intermediate result would overflow a single cell.
| |
| [[Python (programming language)|Python]] has a ternary conditional statement, <code>x if C else y</code>.
| |
| The [[dc (computer program)|dc calculator]] has several ternary operators, such as <tt>|</tt>, which will pop three values from the stack and efficiently compute <math>x^y \mod z</math> with arbitrary precision.
| |
| Additionally, many [[assembly language]] instructions are ternary or higher, such as <tt>MOV %AX, (%BX,%CX)</tt>, which will load (MOV) into register <tt>AX</tt> the contents of a calculated memory location that is the sum (parenthesis) of the registers <tt>BX</tt> and <tt>CX</tt>.
| |
| <!-- examples section needs complete rewrite, with links and subsection on math, logic and programming -->
| |
| | |
| ===''n''-ary===
| |
| From a mathematical point of view, a function of ''n'' arguments can always be considered as a function of one single argument which is an element of some [[product space]]. However, it may be convenient for notation to consider ''n''-ary functions, as for example [[multilinear map]]s (which are not linear maps on the product space, if ''n''≠1).
| |
| | |
| The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some [[object composition|composite type]] such as a [[tuple]], or in languages with [[higher-order function]]s, by [[currying]].
| |
| | |
| === Variable arity===
| |
| In computer science, a function accepting a variable number of arguments is called ''[[variadic function|variadic]]''. In logic and philosophy, predicates or relations accepting a variable number of arguments are called ''[[multigrade predicate|multigrade]]'', ''[[anadic predicate|anadic]]'', or ''[[variably polyadic predicate|variably polyadic]]''.<ref>{{Cite journal | doi = 10.1093/mind/113.452.609 | last1 = Oliver | first1 = Alex | year = 2004 | title = Multigrade Predicates | url = | journal = Mind | volume = 113 | issue = | pages = 609–681 }}</ref>
| |
| | |
| ==Other names==
| |
| * ''Nullary'' means 0-ary.
| |
| * ''[[Unary operation|Unary]]'' means 1-ary.
| |
| * ''[[Binary operation|Binary]]'' means 2-ary.
| |
| * ''[[Ternary operation|Ternary]]'' means 3-ary.
| |
| * ''Quaternary'' means 4-ary.
| |
| * ''Quinary'' means 5-ary.
| |
| * ''Senary'' means 6-ary.
| |
| * ''Septenary'' means 7-ary.
| |
| * ''Octary'' means 8-ary.
| |
| * ''Nonary'' means 9-ary.
| |
| * ''Polyadic'', ''multary'' and ''multiary'' mean 2 or more operands (or parameters).
| |
| * ''n''-''ary'' means ''n'' operands (or parameters), but is often used as a synonym of "polyadic".
| |
| | |
| An alternative nomenclature is derived in a similar fashion from the corresponding [[Greek language|Greek]] roots; for example, ''niladic'' (or ''medadic''), ''monadic'', ''dyadic'', ''triadic'', ''polyadic'', and so on. Thence derive the alternative terms ''adicity'' and ''adinity'' for the Latin-derived ''arity''.
| |
| | |
| These words are often used to describe anything related to that number (e.g., [[undenary chess]] is a chess variant with an 11×11 board, or the [[Millenary Petition]] of 1603).
| |
| | |
| ==See also==
| |
| {{Portal|Mathematics|Logic}}
| |
| <div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
| |
| *[[Logic of relatives]]
| |
| *[[Binary relation]]
| |
| *[[Triadic relation]]
| |
| *[[Theory of relations]]
| |
| *[[Signature (logic)]]
| |
| *[[Parameter]]
| |
| *[[Variadic]]
| |
| *[[Valency (linguistics)|Valency]]
| |
| *[[n-ary code|''n''-ary code]]
| |
| *[[n-ary group|''n''-ary group]]
| |
| </div>
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| A monograph available free online:
| |
| * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2. Especially pp. 22–24.
| |
| | |
| [[Category:Abstract algebra]]
| |
| [[Category:Universal algebra]]
| |
| | |
| [[cs:Operace (matematika)#Arita operace]]
| |