Movatterモバイル変換

[0]ホーム

Jump to content

Arity

Edit links

From Wikipedia, the free encyclopedia

Number of arguments required by a function

"Adicity" redirects here. Not to be confused withAcidity.

Inlogic,mathematics, andcomputer science,arity (/ˈærɪti/ ^ⓘ) is the number ofarguments oroperands taken by afunction,operation orrelation. 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 calledadicity anddegree.^[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 asbinary andhexadecimal. ALatin prefix is combined with the -ary suffix. For example:

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

Nullary

[edit]

Aconstant can 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 variables or the whole state of the system (time, free memory, etc.).

Unary

[edit]

Examples ofunary operators in mathematics and in programming include theunary minus and 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), andnorm functions in mathematics. In programming thetwo's complement,address reference, and thelogical NOT operators are examples of unary operators.

All functions inlambda calculus and in somefunctional programming languages (especially those descended fromML) are technically unary, but seen-ary below.

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

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

Binary

[edit]

Most operators encountered in programming and mathematics are of thebinary form. For both programming and mathematics, these include themultiplication operator, the radix operator, the often omittedexponentiation operator, thelogarithm operator, theaddition operator, and thedivision operator. Logical predicates such asOR,XOR,AND,IMP are typically used as binary operators with two distinct operands. InCISC architectures, it is common to have two source operands (and store result in one of them).

Ternary

[edit]

The computer programming languageC and 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. This operator has a lazy or 'shortcut'evaluation strategy that does not evaluate whichever of the second and third arguments is not used. Some functional programming languages, such asAgda, have such an evaluation strategy for all functions and consequently implementif...then...else as an ordinary function; several others, such asHaskell, can do this but for syntactic, performance or historical reasons choose to define keywords instead.

ThePython language has a ternary conditional expression,xifCelsey. InElixir the equivalent would beif(C,do:x,else:y).

TheForth language 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 calculator has 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 language instructions are ternary (as opposed to only two operands specified in CISC); or higher, such asMOV%AX,(%BX,%CX), which will load (MOV) into registerAX the contents of a calculated memory location that is the sum (parenthesis) of the registersBX andCX.

n-ary

[edit]

Thearithmetic mean ofn real numbers is ann-ary function: ${\bar {x}}={\frac {1}{n}}\left(\sum _{i=1}^{n}{x_{i}}\right)={\frac {x_{1}+x_{2}+\dots +x_{n}}{n}}$

Similarly, thegeometric mean ofnpositive real numbers is ann-ary function: $\left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}=\ {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}.$ Note that alogarithm of the geometric mean is the arithmetic mean of the logarithms of itsn arguments

From a mathematical point of view, a function ofn arguments 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 type such 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]

Latinate names are commonly used for specific arities, primarily based on Latindistributive numbers meaning "in group ofn", though some are based on Latincardinal numbers orordinal numbers. For example, 1-ary is based on cardinalunus, rather than from distributivesingulī that 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	logicalNOT operator
2-ary	binary	dyadic	addition	logicalOR,XOR,AND operators
3-ary	ternary	triadic	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

n-ary means havingn 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 achess variant with an 11×11 board, or theMillenary Petition of 1603).

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

Incomputer programming, there is often asyntactical distinction betweenoperators andfunctions; 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]

Look upAppendix:English arities and adicities in Wiktionary, the free dictionary.

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.

Mathematical logic

General

Theorems (list)
and paradoxes

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive