Arity
In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.
Examples
In general, functions or operators with a given arity follow the naming conventions of n-based numeral systems, such as binary and hexadecimal. A Latin prefix is combined with the -ary suffix. For example:- A nullary function takes no arguments.
- * Example:
- A unary function takes one argument.
- * Example:
- A binary function takes two arguments.
- * Example:
- A ternary function takes three arguments.
- * Example:
- An n-ary function takes n arguments.
- * Example:
Nullary
Also, outside of functional programming, a function without arguments can be meaningful and not necessarily constant. Such functions may have some hidden input, such as global variables or the whole state of the system.
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, and the successor, factorial, reciprocal, floor, ceiling, fractional part, sign, absolute value, square root, complex conjugate, and norm functions in mathematics. In programming the two's complement, address reference, and the logical NOT operators are examples of unary operators.All functions in lambda calculus and in some functional programming languages are technically unary, but see [|n-ary] below.
According to Quine, the Latin distributives being singuli, bini, terni, and so forth, the term "singulary" is the correct adjective, rather than "unary". Abraham Robinson follows Quine's usage.
In philosophy, the adjective monadic is sometimes used to describe a one-place relation such as 'is square-shaped' as opposed to a two-place relation such as 'is the sister of'.
Binary
Most operators encountered in programming and mathematics are of the 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 operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with two distinct operands. In CISC architectures, it is common to have two source operands.Ternary
The computer programming language C and its various descendants provide the ternary conditional operator?:. The first operand 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 as Agda, have such an evaluation strategy for all functions and consequently implement as an ordinary function; several others, such as Haskell, can do this but for syntactic, performance or historical reasons choose to define keywords instead.The Python language has a ternary conditional expression,. In Elixir the equivalent would be.
The Forth language also contains a ternary operator,
*/, which multiplies the first two 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 calculator has several ternary operators, such as
|, which will pop three values from the stack and efficiently compute with arbitrary precision.Many assembly language instructions are ternary ; or higher, such as
''n''-ary
The arithmetic mean of n real numbers is an n-ary function:Similarly, the geometric mean of n positive real numbers is an n-ary function: 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 maps.
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.
Varying arity
In computer science, a function that accepts 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.Terminology
ate names are commonly used for specific arities, primarily based on Latin distributive numbers meaning "in group of n", though some are based on Latin cardinal numbers or ordinal numbers. For example, 1-ary is based on cardinal unus, rather than from distributive singulī that would result in singulary.| n-ary | Arity | Adicity | Example in mathematics | Example in computer science |
| 0-ary | nullary | niladic | a constant | a function without arguments, True, False |
| 1-ary | unary | monadic | additive inverse | logical NOT operator |
| 2-ary | binary | dyadic | addition | logical OR, 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 | enneadic | ||
| 10-ary | denary | decadic | ||
| more than 2-ary | multary and multiary | polyadic | ||
| varying | variadic | sum; e.g., Σ | variadic function, reduce |
n-''ary means having n'' operands, but is often used as a synonym of "polyadic".
These words are often used to describe anything related to that number.
The arity of a relation is the dimension of the domain in the corresponding Cartesian product.
In computer programming, there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 1, 2, or 3. 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.