arities

Arity is the number of arguments that a function, operation, relation, or symbol accepts. It is a fundamental notion in mathematics, logic, computer science, and database theory, used to classify and distinguish expressions by the number of inputs they require. In formal languages, the arity of a symbol determines how it combines with terms to form larger expressions.

Arity categories include nullary (0-arity), unary (1-arity), binary (2-arity), ternary (3-arity), and polyadic for higher numbers. In databases, a relation’s arity equals the number of attributes it involves. In logic, n-ary predicates and n-ary operators specify how many terms they relate or operate on. In programming, arity describes the number of parameters accepted by a function or method. Some functions are variadic, accepting a variable number of arguments, while others are fixed in arity and may be overloaded to provide multiple arities. In functional programming, currying can transform a function of arity n into a sequence of unary functions, one argument at a time.

The concept of arity influences type systems, evaluation strategies, and abstraction mechanisms. It affects how expressions are parsed, type-checked, and composed, and it constrains the construction of terms in algebraic specifications and term rewriting. Across domains, arities provide a concise, shared way to describe the input structure of symbols, guiding both theory and implementation.