View source for Arity

{{Short description|Number of arguments required by a function}}
{{redirects here|Adicity|text=Not to be confused with [[Acidity]].}}
In [[logic]], [[mathematics]], and [[computer science]], '''arity''' ({{IPAc-en|audio=en-us-arity.ogg|ˈ|ær|ᵻ|t|i}}) is the number of [[argument of a function|arguments]] or [[operand]]s taken by a [[function (mathematics)|function]], [[operation (mathematics)|operation]] or [[relation (mathematics)|relation]]. In mathematics, arity may also be called rank,<ref name="Hazewinkel2001">{{cite book|author-link=Michiel Hazewinkel|first=Michiel|last=Hazewinkel|title=Encyclopaedia of Mathematics, Supplement III|url=https://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|first=Eric|last=Schechter|title=Handbook of Analysis and Its Foundations|url=https://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 logic and [[philosophy]], arity may also be called '''adicity''' and '''degree'''.<ref name="DetlefsenBacon1999">{{cite book|first1=Michael |last1=Detlefsen|first2=David Charles|last2=McCarty|first3=John B.|last3=Bacon|title=Logic from A to Z|url=https://archive.org/details/logicfromtoz0000detl|url-access=registration |year=1999|publisher=Routledge|isbn=978-0-415-21375-2|page=[https://archive.org/details/logicfromtoz0000detl/page/7 7]}}</ref><ref name="CocchiarellaFreund2008">{{cite book|first1=Nino B.|last1=Cocchiarella|first2=Max A.|last2=Freund|title=Modal Logic: An Introduction to its Syntax and Semantics|url=https://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]], it is usually named '''[[valency (linguistics)|valency]]'''.<ref name="Crystal2008">{{cite book|first=David|last=Crystal|title=Dictionary of Linguistics and Phonetics|year=2008|publisher=John Wiley & Sons|isbn=978-1-405-15296-9|page=507|edition=6th}}</ref>

== Examples ==
In general, functions or operators with a given arity follow the naming conventions of ''n''-based [[numeral system]]s, such as [[Binary numeral system|binary]] and [[hexadecimal]]. A [[Latin]] prefix is combined with the -ary suffix. For example:
* A nullary function takes no arguments.
** Example: <math>f()=2</math>
* A [[unary operation|unary function]] takes one argument.
** Example: <math>f(x)=2x</math>
* A [[binary operation|binary function]] takes two arguments.
** Example: <math>f(x,y)=2xy</math>
* A [[ternary operation|ternary function]] takes three arguments.
** Example: <math>f(x,y,z)=2xyz</math>
* An ''n''-ary function takes ''n'' arguments.
** Example: <math display="inline">f(x_1, x_2, \ldots, x_n)=2\prod_{i=1}^n x_i</math>

=== Nullary ===
A [[constant (mathematics)|constant]] can be treated as the output of an operation of arity 0, called a ''nullary operation''.

Also, outside of [[functional programming]], a function without arguments can be meaningful and not necessarily constant (due to [[side effect (computer science)|side effect]]s). Such functions may have some ''hidden input'', such as [[global variable]]s or the whole state of the system (time, free memory, etc.).

=== 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 [[Successor function|successor]], [[factorial]], [[Multiplicative inverse|reciprocal]], [[floor function|floor]], [[ceiling function|ceiling]], [[fractional part]], [[sign function|sign]], [[absolute value]], [[square root]] (the principal square root), [[complex conjugate]] (unary of "one" [[complex number]], that however has two parts at a lower level of abstraction), and [[Norm (mathematics)|norm]] functions in mathematics.  In programming the [[two's complement]], [[Reference (computer science)|address reference]], and the [[logical NOT]] operators are examples of unary operators.

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.

According to [[Willard Van Orman Quine|Quine]], the Latin distributives being ''singuli'', ''bini'', ''terni'', and so forth, the term "singulary" is the correct adjective, rather than "unary".<ref>
{{Citation
  | last = Quine
  | first = W. V. O.
  | title = Mathematical logic
  | year = 1940
  | place = Cambridge, Massachusetts
  | publisher = Harvard University Press
  | page=13
}}</ref> [[Abraham Robinson]] follows Quine's usage.<ref>
{{Citation
  | last = Robinson
  | first = Abraham
  | title = Non-standard Analysis
  | year = 1966
  | place = Amsterdam
  | publisher = North-Holland
  | page=19
}}</ref>

In philosophy, the adjective ''monadic'' is sometimes used to describe a [[monadic predicate calculus|one-place relation]] such as 'is square-shaped' as opposed to a [[binary relation|two-place relation]] such as 'is the sister of'.

=== Binary ===
Most operators encountered in programming and mathematics are of the [[binary operation|binary]] form. For both programming and mathematics, these include the [[multiplication operator]], the radix operator, the often omitted [[exponentiation]] operator, the [[logarithm]] operator, the [[addition]] operator, and the [[Division (mathematics)|division]] operator. Logical predicates such as ''[[Logical disjunction|OR]]'', ''[[exclusive or|XOR]]'', ''[[Logical conjunction|AND]]'', ''IMP'' are typically used as binary operators with two distinct operands. In [[Complex instruction set computing|CISC]] architectures, it is common to have two source operands (and store result in one of them).

=== Ternary ===
The computer programming language [[C (programming language)|C]] and its various descendants (including [[C++]], [[C Sharp (programming language)|C#]], [[Java (programming language)|Java]], [[Julia (programming language)|Julia]], [[Perl]], and others) provide the [[ternary conditional operator]] <code>?:</code>. The first operand (the condition) is evaluated, and if it is true, the result of the entire expression is the value of the second operand, otherwise it is the value of the third operand.

The [[Python (programming language)|Python]] language has a ternary conditional expression, {{code|x if C else y|python}}. In [[Elixir (programming language)|Elixir]] the equivalent would be, {{code|if(C, do: x, else: y)|elixir}}.

The [[Forth (programming language)|Forth]] language 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.

The Unix [[dc (computer program)|dc calculator]] has several ternary operators, such as <code>|</code>, which will pop three values from the stack and efficiently compute <math display="inline">x^y \bmod z</math> with [[arbitrary-precision arithmetic|arbitrary precision]].

Many ([[Reduced instruction set computing|RISC]]) [[assembly language]] instructions are ternary (as opposed to only two operands specified in CISC); or higher, such as <syntaxhighlight lang="asm" inline="">MOV %AX, (%BX, %CX)</syntaxhighlight>, which will load ({{mono|MOV}}) into register {{mono|AX}} the contents of a calculated memory location that is the sum (parenthesis) of the registers {{mono|BX}} and {{mono|CX}}.<!-- examples section needs complete rewrite, with links and subsection on math, logic and programming
-->

=== ''n''-ary ===
The [[arithmetic mean]] of ''n'' real numbers is an ''n''-ary  function: <math>\bar{x}=\frac{1}{n}\left (\sum_{i=1}^n{x_i}\right) = \frac{x_1+x_2+\dots+x_n}{n}</math>

Similarly, the [[geometric mean]] of ''n'' [[positive real numbers]] is an ''n''-ary function: <math>\left(\prod_{i=1}^n a_i\right)^\frac{1}{n} = \ \sqrt[n]{a_1 a_2 \cdots a_n} .</math> Note that a [[logarithm]] of the geometric mean is the arithmetic mean of the logarithms of its ''n'' arguments

From a mathematical point of view, a function of ''n'' arguments can always be considered as a function of a single argument that 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 {{nowrap|''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]].

=== Varying arity ===
In computer science, a function that accepts 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, or variably polyadic.<ref>{{Cite journal | doi = 10.1093/mind/113.452.609 | last1 = Oliver | first1 = Alex | year = 2004 | title = Multigrade Predicates | journal = Mind | volume = 113 | issue = 452| pages = 609–681 }}</ref>

== Terminology ==
[[Latin]]ate names are commonly used for specific arities, primarily based on Latin [[distributive number]]s meaning "in group of ''n''", though some are based on Latin [[cardinal number]]s or [[ordinal number]]s. For example, 1-ary is based on cardinal ''unus'', rather than from distributive ''singulī'' that would result in ''singulary''. 

{| class="wikitable"
|-
! ''n''-ary !! Arity (Latin based) !! Adicity (Greek based) !! Example in mathematics !! Example in computer science
|-
| 0-ary || nullary (from ''nūllus'') || niladic || a [[constant (mathematics)|constant]]  || a function without arguments, [[Truth|True]], [[Falsity|False]]
|-
| 1-ary || [[Unary operation|unary]] || monadic || [[additive inverse]] || logical [[Logical NOT|NOT]] operator
|-
| 2-ary || [[Binary operation|binary]] || dyadic || [[addition]] || logical [[Logical OR|OR]], [[Exclusive or|XOR]], [[Logical AND|AND]] operators
|-
| 3-ary || [[Ternary operation|ternary]] || triadic || [[Triple product|triple product of vectors]] || [[ternary conditional operator]]
|-
| 4-ary || quaternary || tetradic ||  ||
|-
| 5-ary || quinary  || pentadic ||  ||
|-
| 6-ary || senary || hexadic ||  || 
|-
| 7-ary || septenary || hebdomadic ||  || 
|-
| 8-ary || octonary || ogdoadic ||  || 
|-
| 9-ary || novenary (alt. nonary) || enneadic ||  || 
|-
| 10-ary || denary (alt. decenary) || decadic ||  || 
|-
| more than 2-ary || multary and multiary || polyadic ||  || 
|-
| varying || || variadic || sum; e.g., Σ || [[variadic function]], [[Reduce (parallel pattern)|reduce]]
|}

''n''-''ary'' means having ''n'' operands (or parameters), but is often used as a synonym of "polyadic".

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).

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.) 

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 1, 2, or 3 (the [[Ternary operation|ternary operator]] [[?:]] is also common).  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.

== See also ==
{{Portal|Mathematics|Philosophy}}
{{Div col|colwidth=30em}}
* [[Logic of relatives]]
* [[Binary relation]]
* [[Ternary relation]]
* [[Theory of relations]]
* [[Signature (logic)]]
* [[Parameter]]
* [[p-adic number|''p''-adic number]]
* [[Cardinality]]
* [[Valency (linguistics)]]
* [[n-ary code|''n''-ary code]]
* [[n-ary group|''n''-ary group]]
* {{annotated link|Function prototype}}
* {{annotated link|Type signature}}
* [[Univariate and multivariate]]
* [[Finitary]]
{{colend}}

== References ==
{{Reflist}}

== External links ==
{{wiktionary|Appendix:English arities and adicities}}
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.&nbsp;22–24.

{{Mathematical logic}}

[[Category:Abstract algebra]]
[[Category:Universal algebra]]

[[cs:Operace (matematika)#Arita operace]]