Arity

Inlogic,mathematics,andcomputer science,arity(/ˈærɪti/^ⓘ) is the number ofargumentsoroperandstaken by afunction,operationorrelation.In mathematics, arity may also be called rank,^[1]^[2]but this word can have many other meanings. In logic andphilosophy,arity may also be calledadicityanddegree.^[3]^[4]Inlinguistics,it is usually namedvalency.^[5]

Examples[edit]

In general, functions or operators with a given arity follow the naming conventions ofn-basednumeral systems,such asbinaryandhexadecimal.ALatinprefix is combined with the -ary suffix. For example:

A nullary function takes no arguments.
- Example: $f()=2$
Aunary functiontakes one argument.
- Example: $f(x)=2x$
Abinary functiontakes two arguments.
- Example: $f(x,y)=2xy$
Aternary functiontakes three arguments.
- Example: $f(x,y,z)=2xyz$
Ann-ary function takesnarguments.
- Example: ${\textstyle f(x_{1},x_{2},\ldots,x_{n})=2\prod _{i=1}^{n}x_{i}}$

Nullary[edit]

Aconstantcan be treated as the output of an operation of arity 0, called anullary operation.

Also, outside offunctional programming,a function without arguments can be meaningful and not necessarily constant (due toside effects). Such functions may have somehidden input,such asglobal variablesor the whole state of the system (time, free memory, etc.).

Unary[edit]

Examples ofunary operatorsin mathematics and in programming include theunary minusand plus, the increment and decrement operators inC-style languages (not in logical languages), and thesuccessor,factorial,reciprocal,floor,ceiling,fractional part,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), andnormfunctions in mathematics. In programming thetwo's complement,address reference,and thelogical NOToperators are examples of unary operators.

All functions inlambda calculusand in somefunctional programming languages(especially those descended fromML) are technically unary, but seen-arybelow.

According toQuine,the Latin distributives beingsinguli,bini,terni,and so forth, the term "singulary" is the correct adjective, rather than "unary".^[6]Abraham Robinsonfollows Quine's usage.^[7]

In philosophy, the adjectivemonadicis sometimes used to describe aone-place relationsuch as 'is square-shaped' as opposed to atwo-place relationsuch as 'is the sister of'.

Binary[edit]

Most operators encountered in programming and mathematics are of thebinaryform. For both programming and mathematics, these include themultiplication operator,the radix operator, the often omittedexponentiationoperator, thelogarithmoperator, theadditionoperator, and thedivisionoperator. Logical predicates such asOR,XOR,AND,IMPare typically used as binary operators with two distinct operands. InCISCarchitectures, it is common to have two source operands (and store result in one of them).

Ternary[edit]

The computer programming languageCand its various descendants (includingC++,C#,Java,Julia,Perl,and others) provide theternary conditional operator?:.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.

ThePythonlanguage has a ternary conditional expression,xifCelsey.InElixirthe equivalent would be,if(C,do:x,else:y).

TheForthlanguage 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.

The Unixdc calculatorhas several ternary operators, such as|,which will pop three values from the stack and efficiently compute ${\textstyle x^{y}{\bmod {z}}}$ witharbitrary precision.

Many (RISC)assembly languageinstructions are ternary (as opposed to only two operands specified in CISC); or higher, such asMOV%AX,(%BX,%CX),which will load (MOV) into registerAXthe contents of a calculated memory location that is the sum (parenthesis) of the registersBXandCX.

n-ary[edit]

From a mathematical point of view, a function ofnarguments can always be considered as a function of a single argument that is an element of someproduct space.However, it may be convenient for notation to considern-ary functions, as for examplemultilinear maps(which are not linear maps on the product space, ifn≠ 1).

The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of somecomposite typesuch as atuple,or in languages withhigher-order functions,bycurrying.

Varying arity[edit]

In computer science, a function that accepts a variable number of arguments is calledvariadic.In logic and philosophy, predicates or relations accepting a variable number of arguments are calledmultigrade,anadic, or variably polyadic.^[8]

Terminology[edit]

Latinatenames are commonly used for specific arities, primarily based on Latindistributive numbersmeaning "in group ofn",though some are based on Latincardinal numbersorordinal numbers.For example, 1-ary is based on cardinalunus,rather than from distributivesingulīthat would result insingulary.

n-ary	Arity (Latin based)	Adicity (Greek based)	Example in mathematics	Example in computer science
0-ary	nullary (fromnūllus)	niladic	aconstant	a function without arguments,True,False
1-ary	unary	monadic	additive inverse	logicalNOToperator
2-ary	binary	dyadic	addition	logicalOR,XOR,ANDoperators
3-ary	ternary	triadic	triple product of vectors	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

n-arymeans havingnoperands (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 achess variantwith an 11×11 board, or theMillenary Petitionof 1603).

The arity of arelation(orpredicate) is the dimension of thedomainin the correspondingCartesian product.(A function of aritynthus has arityn+1 considered as a relation.)

Incomputer programming,there is often asyntacticaldistinction betweenoperatorsandfunctions;syntactical operators usually have arity 1, 2, or 3 (theternary operator ?:is also common). Functions vary widely in the number of arguments, though large numbers can become unwieldy. Some programming languages also offer support forvariadic functions,i.e., functions syntactically accepting a variable number of arguments.

References[edit]

^Hazewinkel, Michiel(2001).Encyclopaedia of Mathematics, Supplement III.Springer. p. 3.ISBN 978-1-4020-0198-7.
^Schechter, Eric (1997).Handbook of Analysis and Its Foundations.Academic Press. p. 356.ISBN 978-0-12-622760-4.
^Detlefsen, Michael; McCarty, David Charles; Bacon, John B. (1999).Logic from A to Z.Routledge. p.7.ISBN 978-0-415-21375-2.
^Cocchiarella, Nino B.; Freund, Max A. (2008).Modal Logic: An Introduction to its Syntax and Semantics.Oxford University Press. p. 121.ISBN 978-0-19-536658-7.
^Crystal, David (2008).Dictionary of Linguistics and Phonetics(6th ed.). John Wiley & Sons. p. 507.ISBN 978-1-405-15296-9.
^ Quine, W. V. O. (1940),Mathematical logic,Cambridge, Massachusetts: Harvard University Press, p. 13
^ Robinson, Abraham (1966),Non-standard Analysis,Amsterdam: North-Holland, p. 19
^Oliver, Alex (2004). "Multigrade Predicates".Mind.113(452): 609–681.doi:10.1093/mind/113.452.609.

External links[edit]

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.

[Hazewinkel2001-1] Hazewinkel, Michiel(2001).Encyclopaedia of Mathematics, Supplement III.Springer. p. 3.ISBN 978-1-4020-0198-7.

[Schechter1997-2] Schechter, Eric (1997).Handbook of Analysis and Its Foundations.Academic Press. p. 356.ISBN 978-0-12-622760-4.

[DetlefsenBacon1999-3] Detlefsen, Michael; McCarty, David Charles; Bacon, John B. (1999).Logic from A to Z.Routledge. p.7.ISBN 978-0-415-21375-2.

[CocchiarellaFreund2008-4] Cocchiarella, Nino B.; Freund, Max A. (2008).Modal Logic: An Introduction to its Syntax and Semantics.Oxford University Press. p. 121.ISBN 978-0-19-536658-7.

[Crystal2008-5] Crystal, David (2008).Dictionary of Linguistics and Phonetics(6th ed.). John Wiley & Sons. p. 507.ISBN 978-1-405-15296-9.

[6] Quine, W. V. O. (1940),Mathematical logic,Cambridge, Massachusetts: Harvard University Press, p. 13

[7] Robinson, Abraham (1966),Non-standard Analysis,Amsterdam: North-Holland, p. 19

[8] Oliver, Alex (2004). "Multigrade Predicates".Mind.113(452): 609–681.doi:10.1093/mind/113.452.609.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]