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= | 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". | ||
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]] | |||
Revision as of 04:08, 23 December 2013
In logic, mathematics, and computer science, the arity Template:IPAc-en of a function or operation is the number of arguments or operands the function or operation accepts. The arity of a relation (or predicate) is the dimension of the 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".
In mathematics arity may also be named rank,[1][2] but this word can have many other meanings in mathematics. In logic and philosophy, arity is also called adicity and degree.[3][4] In linguistics, arity is usually named valency.[5]
In computer programming, there is often a syntactical distinction between operators and 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
Template:Expand section 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 systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example:
- A nullary function takes no arguments.
- A unary function takes one argument.
- A binary function takes two arguments.
- A ternary function takes three arguments.
- An n-ary function takes n arguments.
Nullary
Sometimes it is useful to consider a 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 effects). Often, such functions have in fact some hidden input which might be global variables, 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 operators in mathematics and in programming include the unary minus and plus, the increment and decrement operators in C-style languages (not in logical languages), and the factorial, reciprocal, floor, ceiling, fractional part, sign, absolute value, complex conjugate, and norm functions in mathematics. The two's complement, address reference and the logical NOT operators are examples of unary operators in math and programming. According to Quine, a more suitable term is "singulary".[6]
All functions in lambda calculus and in some functional programming languages (especially those descended from ML) are technically unary, but see n-ary below.
Binary
Most operators encountered in programming are of the binary form. For both programming and mathematics these can be the multiplication operator, the addition operator, the division operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with two distinct operands.
Ternary
From C, C++, C#, Java, Perl and variants comes the ternary operator ?:, which is a so-called conditional operator, taking three parameters.
Forth also contains a ternary operator, */, 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 has a ternary conditional statement, x if C else y.
The dc calculator has several ternary operators, such as |, which will pop three values from the stack and efficiently compute with arbitrary precision.
Additionally, many assembly language instructions are ternary or higher, such as MOV %AX, (%BX,%CX), which will load (MOV) into register AX the contents of a calculated memory location that is the sum (parenthesis) of the registers BX and CX.
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 maps (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 composite type such as a tuple, or in languages with higher-order functions, by currying.
Variable arity
In computer science, a function accepting a variable number of arguments is called variadic. In logic and philosophy, predicates or relations accepting a variable number of arguments are called multigrade, anadic, or variably polyadic.[7]
Other names
- Nullary means 0-ary.
- Unary means 1-ary.
- Binary means 2-ary.
- 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 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
Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
A monograph available free online:
- Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2. Especially pp. 22–24.
cs:Operace (matematika)#Arita operace
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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