What is the name for a function's "return arity"?

Question

The number of parameters a function has (and thus the number of arguments it can take) is known as arity. What is the name for the number of values a function returns? I'm not asking for the "size" of a function's return values. A function returning a tuple returns a single, composite, value.

Answer 1

The term "arity" would have been introduced to programming via formal logic and related fields (e.g. universal algebra). In the context of formal logic, there is little reason to allow function symbols to return multiple values even in the form of a tuple, and good reasons to disallow it. Mathematically, a function $X\to Y\times Z$ is equivalent to a pair of functions $X\to Y$ and $X\to Z$. So we don't lose anything by disallowing functions into cartesian products. What we gain, though, is a uniform syntax, especially in the context of single-sorted logic which is the typical one in formal logic.

In a single-sorted logic, there is only one sort (base type), call it $S$, so all functions are of the form $S\times\cdots\times S\to S$, or more compactly $S^n\to S$. By disallowing functions into products, if we have a term $f(x,y)$ and a function symobl $g$ of arity 2, say, then $g(f(x, y), z)$ is always a valid term. If we allowed functions into products, we'd need to at least introduce the projections $\pi_i : X_1\times X_2 \to X_i$. This would require us to assume the products were associated in some particular way or we would need to allow a whole family of projections or we would need some syntax to "spread" the results into multiple parameters. Basically, the syntax of terms would become significantly more complicated for no real gain. These syntactic complexities are reflected in programming languages that have real multi-valued returns (as opposed to just a single-valued return of a tuple).

Single-sortedness is what allows us to reduce the arity to a number. Combined with the restriction that function symbols can only return into the single sort, everything you need to know about a function symbol for the purposes of syntax can be reduced to a single number. In multi-sorted logics, it makes more sense to view the arity as a list of sorts, but since the return type is no longer fixed something more like a type annotation, e.g. $(X, Y)\to Z$, makes more sense. These type annotations are more relevant than arity. They are what you need to tell if a term is well-formed.

Most programming languages don't support a true notion of multi-valued return, and many don't even have a good notion of tuple. Occasionally in more theoretical work, the term "coarity" is used for a return sort. In a single-sorted context, the coarity could be represented by a number, but in a multi-sorted, i.e. statically typed, context it would be more natural for it to be a list of sorts. Really, computational models where operations can produce multiple outputs are usually thought of with a circuit metaphor. In such cases, we'd talk about things like fan-in and fan-out, or in-degree and out-degree.