# Option

By Jacob Neumann, June 2021

option is a SML datatype for handling potential undefined values. Formally, the option type is parametrized by a single polymorphic type variable, so for every SML type t, there is a type t option.

The type t option represents the construction of "either a value of t, or nothing". For instance, a value of type bool option is either true, false, or neither. A value of type int option is either some integer, or none. A function will return t option to represent the possibility that no acceptable value of type t can be identified to return, and have a value to return to signal this circumstance. See the "Options-as-partiality" HOFs section below for some functions elaborating on this understanding of options.

Options also function as a kind of "container": a value x : t option either "contains" a value of type t (x = SOME(z)) or is an "empty container" with no values (x = NONE). We can therefore view t option as a degenerate version of t list, where the "list" is constrained to be length at most one. The notion of a "container" is made precise in functional programming with the idea of a "monad". In Haskell (which makes much more explicit use of monads), options are known as the "Maybe monad".

## Values

The type t option has exactly one more value than the type t itself. Formally,

datatype 'a option = NONE | SOME of 'a


So for each value v : t, SOME(v) : t option. And NONE is a value of type t option for each type t, as required by the surrounding context. Options can be nested, e.g. SOME(SOME(3)) : int option option.

SOME and NONE are the two constructors of the t option type, so, in addition to the constructs generally available in pattern matching (e.g. wildcards and identifier binding), we can pattern match against SOME and NONE.

fun defaultToThree (NONE : int option):int = 3
| defaultToThree (SOME x) = x

val 2 = defaultToThree(SOME 2)
val 3 = defaultToThree(NONE)

fun searchForEven [] = NONE
| searchForEven (x::xs) = if (x mod 2)=0 then SOME(x) else searchForEven xs

val (SOME _) = searchForEven [1,2,3,4]
val NONE = searchForEven [1,3,5]


If t is pretty-printed by the smlnj REPL (like int,bool,string list, etc.), so too is t option. This is demonstrated by the following smlnj REPL snippet.

- val k = SOME(SOME 5);
val k = SOME (SOME 5) : int option option


## Production

There are some basic SML functions which produce options:

    Int.fromString : string -> int option
Bool.fromString : string -> bool option


Both of these functions are partial inverses to their respective toString functions (e.g. Int.fromString(Int.toString(7)) == SOME 7), but return an option so they can return NONE on strings which do not encode an int or bool, respectively.

## Elimination

Another option for casing on options is the function (provided in the Option structure - see below)

    Option.getOpt : 'a option * 'a -> 'a


which behaves as follows: Option.getOpt(SOME x,y) will evaluate to x, and Option.getOpt(NONE,y) will evaluate to y. When writing functions operating on options, it is still generally preferable that you use clausal pattern matching to break into the "SOME case" and "NONE case", but there are situations where Option.getOpt is an elegant solution.

The Option structure also provides the "join" function

    Option.join : 'a option option -> 'a option


which sends SOME(X) to X and NONE to NONE.

## From the Structure

The Option structure (part of the SMLNJ basis) provides a number of useful utilities for working with options.

In addition to the datatype option itself being available at top-level, the exception

    exception Option


is available at top-level as well.

### Basic Functions

The Option structure provides boolean-valued functions for detecting whether a given option value is NONE or SOME, and for extracting values from SOME.

    Option.isSome : 'a option -> bool
Option.isNone : 'a option -> bool
Option.valOf : 'a option -> 'a


Option.valOf NONE raises the exception Option. NOTE: Do not use these fuctions in place of pattern-matching on an option value. Expressions like if Option.isSome(X) then Option.valOf(X) else e2 are bad style.

### "Options-as-containers" HOFs

Options are an instance of a more general structure in functional programming known as a monad. Accordingly, there are a number of higher-order functions which we can define on options. In particular, option comes equipped with a map operation:

    Option.map : ('a -> 'b) -> 'a option -> 'b option


which does what you might expect, given its type: Option.map f (SOME x) evaluates to SOME(f(x)), and Option.map f NONE produces NONE. As usual with "map" functions, we generally require f to be total.

Options, as a "container" of data, also admit a filtering operation:

    Option.filter : ('a -> bool) -> 'a option -> 'a option


Which is implemented as

fun filter p = fn NONE => NONE | (SOME x) => if p(x) then SOME x else NONE


i.e. it "filters" out the value x if x does not "satisfy" p (p(x) == false). As usual with "filter" functions, we generally require p to be total.

The Option structure also provides a utility for "folding" an option:

    Option.fold : ('a * 'b -> 'b) -> 'b -> 'a option -> 'b


which we might implement as

fun fold g z NONE = z
| fold g z (SOME x) = g(x,z)


We often assume that g is total. In certain situations, we also want the two arguments of g to be the same type, and might assume that g is "associative" (in the sense that g(x,g(y,z)) == g(g(x,y),z) for all x,y,z) or that z is a "unit" for g (i.e. g(x,z) == x == g(z,x) for all x).

### "Options-as-partiality" HOFs

As mentioned above, options provide a way to represent partial functions: a function f : t1 -> t2 option can be thought of as a function which is defined on some inputs (f(x) == SOME(z)) and not on others (f(y) == NONE). The Option structure provides several utilities consistent with this interpretation. First is composition of these "partial functions".

    Option.compose : ('a -> 'c) -> ('b -> 'a option) -> 'b -> 'c option
Option.composePartial : ('a -> 'c option) -> ('b -> 'a option) -> 'b -> 'c option


So (Option.compose (g,f))(x) will return NONE if f(x) == NONE, and will return SOME(g(y)) if f(x) == SOME(y). Option.composePartial will behave similarly, except it will return g(y) in the f(x) == SOME(y) case, since g returns an option.

It also provides a "partial" version of its map function:

    Option.mapPartial : ('a -> 'b option) -> 'a option -> 'b option