Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product. The term springs from such words as unary, binary, ternary, etc.

The term "arity" is primarily used with reference to operations. If f is the function f : Sⁿ → S, where S is some set, then f is an operation and n is its arity.

Arities greater than 2 are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 are seldom encountered in theoretical computer science. Practical computer programming commonly define functions with many arguments, say N, but there is a matter of conventions (in the particular language as well as in the culture of the programmers), whether these are regarded as N separate arguments or as an N-tuple of (often heterogenously) typed arguments.

In mathematics, depending on the branch, arity may be called type, adicity or rank.

In computer science arity may be called adicity, a function that takes a variable number of arguments being called variadic.

In linguistics and in logic, arity is sometimes called valency, not to be confused with valency in graph theory.

Examples

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 as 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 twos 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",^{[citation needed]} though the term "unary" remains the de facto usage.

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.

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 complex type such as a tuple, or in languages with higher-order functions, by currying.

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 or multary (or multiary) means any number of 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).

References

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.