Introduction to Programming in ATS:
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Function Arity

The arity of a function is the number of arguments the function takes. Functions of arity 0, 1, 2 and 3 are often called nullary, unary, binary and ternary functions, respectively. For example, the following function sqrsum1 is a binary function such that its two arguments are of the type int:

fn sqrsum1 (x: int, y: int): int = x * x + y * y

We can define a unary function sqrsum2 as follows:

typedef int2 = (int, int)
fn sqrsum2 (xy: int2): int =
  let val x = xy.0 and y = xy.1 in x * x + y * y end
// end of [sqrsum2]

The keyword typedef introduces a binding between the name int2 and the tuple type (int, int). In other words, int2 is treated as an abbreviation or alias for (int, int). The function sqrsum2 is unary as it takes only one argument, which is a tuple of the type int2. When applying sqrsum2 to a tuple consisting of 1 and ~1, we need to write sqrsum2 @(1, ~1). If we simply write sqrsum2 (1, ~1), then the typechecker is to report an error of arity mismatch as it assumes that sqrsum2 is applied to two arguments (instead of one that is a tuple).

Many functional languages (e.g., Haskell and ML) only allow unary functions. A function of multiple arguments is encoded in these languages as a unary function taking a tuple as its only argument or it is curried into a function that takes these arguments sequentially. ATS, however, provides direct support for functions of multiple arguments. There is even some limited support in ATS for variadic functions, that is, functions of indefinite number of arguments (e.g., the famous printf function in C). This is a topic I will cover elsewhere.