Standard ML does not have built-in support for optional arguments. Nevertheless, using Fold, it is easy to define functions that take optional arguments.

For example, suppose that we have the following definition of a function f.

```
fun f (i, r, s) =
concat [Int.toString i, ", ", Real.toString r, ", ", s]
```

Using the OptionalArg structure described below, we can define a function f', an optionalized version of f, that takes 0, 1, 2, or 3 arguments. Embedded within f' will be default values for i, r, and s. If f' gets no arguments, then all the defaults are used. If f' gets one argument, then that will be used for i. Two arguments will be used for i and r respectively. Three arguments will override all default values. Calls to f' will look like the following.

```
f' $
f' `2 $
f' `2 `3.0 $
f' `2 `3.0 `"four" $
```

The optional argument indicator, `, is not special syntax --- it is a normal SML value, defined in the OptionalArg structure below.

Here is the definition of f' using the OptionalArg structure, in particular, OptionalArg.make and OptionalArg.D.

```
val f' =
fn z =>
let open OptionalArg in
make (D 1) (D 2.0) (D "three") $
end (fn i & r & s => f (i, r, s))
z
```

The definition of f' is eta expanded as with all uses of fold. A call to OptionalArg.make is supplied with a variable number of defaults (in this case, three), the end-of-arguments terminator, $, and the function to run, taking its arguments as an n-ary product. In this case, the function simply converts the product to an ordinary tuple and calls f. Often, the function body will simply be written directly.

In general, the definition of an optional-argument function looks like the following.

```
val f =
fn z =>
let open OptionalArg in
make (D <default1>) (D <default2>) ... (D <defaultn>) $
end (fn x1 & x2 & ... & xn =>
<function code goes here>)
z
```

Here is the definition of OptionalArg.

```
structure OptionalArg =
struct
val make =
fn z =>
Fold.fold
((id, fn (f, x) => f x),
fn (d, r) => fn func =>
Fold.fold ((id, d ()), fn (f, d) =>
let
val d & () = r (id, f d)
in
func d
end))
z
fun D d = Fold.step0 (fn (f, r) =>
(fn ds => f (d & ds),
fn (f, a & b) => r (fn x => f a & x, b)))
val ` =
fn z =>
Fold.step1 (fn (x, (f, _ & d)) => (fn d => f (x & d), d))
z
end
```

OptionalArg.make uses a nested fold. The first fold accumulates the default values in a product, associated to the right, and a reversal function that converts a product (of the same arity as the number of defaults) from right associativity to left associativity. The accumulated defaults are used by the second fold, which recurs over the product, replacing the appropriate component as it encounters optional arguments. The second fold also constructs a "fill" function, f, that is used to reconstruct the product once the end-of-arguments is reached. Finally, the finisher reconstructs the product and uses the reversal function to convert the product from right associative to left associative, at which point it is passed to the user-supplied function.

Much of the complexity comes from the fact that while recurring over a product from left to right, one wants it to be right-associative, e.g., look like

```
a & (b & (c & d))
```

but the user function in the end wants the product to be left associative, so that the product argument pattern can be written without parentheses (since & is left associative).

## Labelled optional arguments

In addition to the positional optional arguments described above, it is sometimes useful to have labelled optional arguments. These allow one to define a function, f, with defaults, say a and b. Then, a caller of f can supply values for a and b by name. If no value is supplied then the default is used.

Labelled optional arguments are a simple extension of FunctionalRecordUpdate using post composition. Suppose, for example, that one wants a function f with labelled optional arguments a and b with default values 0 and 0.0 respectively. If one has a functional-record-update function updateAB for records with a and b fields, then one can define f in the following way.

```
val f =
fn z =>
Fold.post
(updateAB {a = 0, b = 0.0},
fn {a, b} => print (concat [Int.toString a, " ",
Real.toString b, "\n"]))
z
```

The idea is that f is the post composition (using Fold.post) of the actual code for the function with a functional-record updater that starts with the defaults.

Here are some example calls to f.

```
val () = f $
val () = f (U#a 13) $
val () = f (U#a 13) (U#b 17.5) $
val () = f (U#b 17.5) (U#a 13) $
```

Notice that a caller can supply neither of the arguments, either of the arguments, or both of the arguments, and in either order. All that matter is that the arguments be labelled correctly (and of the right type, of course).

Here is another example.

```
val f =
fn z =>
Fold.post
(updateBCD {b = 0, c = 0.0, d = "<>"},
fn {b, c, d} =>
print (concat [Int.toString b, " ",
Real.toString c, " ",
d, "\n"]))
z
```

Here are some example calls.

```
val () = f $
val () = f (U#d "goodbye") $
val () = f (U#d "hello") (U#b 17) (U#c 19.3) $
```