layout: post
title: “How types work with functions”
description: “Understanding the type notation”
nav: thinking-functionally
seriesId: “Thinking functionally”
seriesOrder: 4

categories: [Types, Functions]

Now that we have some understanding of functions, we’ll look at how types work with functions, both as domains and ranges. This is just an overview; the series “understanding F# types” will cover types in detail.

First, we need to understand the type notation a bit more. We’ve seen that the arrow notation “->“ is used to show the domain and range. So that a function signature always looks like:

val functionName : domain -> range

Here are some example functions:

let intToString x = sprintf "x is %i" x  // format int to string
let stringToInt x = System.Int32.Parse(x)

If you evaluate that in the F# interactive window, you will see the signatures:

val intToString : int -> string
val stringToInt : string -> int

This means:

• intToString has a domain of int which it maps onto the range string.
• stringToInt has a domain of string which it maps onto the range int.

Primitive types

The possible primitive types are what you would expect: string, int, float, bool, char, byte, etc., plus many more derived from the .NET type system.

Here are some more examples of functions using primitive types:

let intToFloat x = float x // "float" fn. converts ints to floats
let intToBool x = (x = 2)  // true if x equals 2
let stringToString x = x + " world"

and their signatures are:

val intToFloat : int -> float
val intToBool : int -> bool
val stringToString : string -> string

Type annotations

In the previous examples, the F# compiler correctly determined the types of the parameters and results. But this is not always the case. If you try the following code, you will get a compiler error:

let stringLength x = x.Length         
   => error FS0072: Lookup on object of indeterminate type

The compiler does not know what type “x” is, and therefore does not know if “Length” is a valid method. In most cases, this can be fixed by giving the F# compiler a “type annotation” so that it knows which type to use. In the corrected version below, we indicate that the type of “x” is a string.

let stringLength (x:string) = x.Length

The parens around the x:string param are important. If they are missing, the compiler thinks that the return value is a string! That is, an “open” colon is used to indicate the type of the return value, as you can see in the example below.

let stringLengthAsInt (x:string) :int = x.Length

We’re indicating that the x param is a string and the return value is an int.

Function types as parameters

A function that takes other functions as parameters, or returns a function, is called a higher-order function (sometimes abbreviated as HOF). They are used as a way of abstracting out common behavior. These kinds of functions are extremely common in F#; most of the standard libraries use them.

Consider a function evalWith5ThenAdd2, which takes a function as a parameter, then evaluates the function with the value 5, and adds 2 to the result:

let evalWith5ThenAdd2 fn = fn 5 + 2     // same as fn(5) + 2

The signature of this function looks like this:

val evalWith5ThenAdd2 : (int -> int) -> int

You can see that the domain is (int->int) and the range is int. What does that mean? It means that the input parameter is not a simple value, but a function, and what’s more is restricted only to functions that map ints to ints. The output is not a function, just an int.

Let’s try it:

let add1 x = x + 1      // define a function of type (int -> int)
evalWith5ThenAdd2 add1  // test it

gives:

val add1 : int -> int
val it : int = 8

“add1“ is a function that maps ints to ints, as we can see from its signature. So it is a valid parameter for the evalWith5ThenAdd2 function. And the result is 8.

By the way, the special word “it“ is used for the last thing that was evaluated; in this case the result we want. It’s not a keyword, just a convention.

Here’s another one:

let times3 x = x * 3      // a function of type (int -> int)
evalWith5ThenAdd2 times3  // test it

gives:

val times3 : int -> int
val it : int = 17

“times3“ is also a function that maps ints to ints, as we can see from its signature. So it is also a valid parameter for the evalWith5ThenAdd2 function. And the result is 17.

Note that the input is sensitive to the types. If our input function uses floats rather than ints, it will not work. For example, if we have:

let times3float x = x * 3.0  // a function of type (float->float)  
evalWith5ThenAdd2 times3float

Evaluating this will give an error:

error FS0001: Type mismatch. Expecting a int -> int but 
              given a float -> float

meaning that the input function should have been an int->int function.

Functions as output

A function value can also be the output of a function. For example, the following function will generate an “adder” function that adds using the input value.

let adderGenerator numberToAdd = (+) numberToAdd

The signature is:

val adderGenerator : int -> (int -> int)

which means that the generator takes an int, and creates a function (the “adder”) that maps ints to ints. Let’s see how it works:

let add1 = adderGenerator 1
let add2 = adderGenerator 2

This creates two adder functions. The first generated function adds 1 to its input, and the second adds 2. Note that the signatures are just as we would expect them to be.

val add1 : (int -> int)
val add2 : (int -> int)

And we can now use these generated functions in the normal way. They are indistinguishable from functions defined explicitly

add1 5    // val it : int = 6
add2 5    // val it : int = 7

Using type annotations to constrain function types

In the first example, we had the function:

let evalWith5ThenAdd2 fn = fn 5 +2
    => val evalWith5ThenAdd2 : (int -> int) -> int

In this case F# could deduce that “fn“ mapped ints to ints, so its signature would be int->int

But what is the signature of “fn” in this following case?

let evalWith5 fn = fn 5

Obviously, “fn“ is some kind of function that takes an int, but what does it return? The compiler can’t tell. If you do want to specify the type of the function, you can add a type annotation for function parameters in the same way as for a primitive type.

let evalWith5AsInt (fn:int->int) = fn 5
let evalWith5AsFloat (fn:int->float) = fn 5

Alternatively, you could also specify the return type instead.

let evalWith5AsString fn :string = fn 5

Because the main function returns a string, the “fn“ function is also constrained to return a string, so no explicit typing is required for “fn”.

The “unit” type

When programming, we sometimes want a function to do something without returning a value. Consider the function “printInt“, defined below. The function doesn’t actually return anything. It just prints a string to the console as a side effect.

let printInt x = printf "x is %i" x        // print to console

So what is the signature for this function?

val printInt : int -> unit

What is this “unit“?

Well, even if a function returns no output, it still needs a range. There are no “void” functions in mathematics-land. Every function must have some output, because a function is a mapping, and a mapping has to have something to map to!

So in F#, functions like this return a special range called “unit“. This range has exactly one value in it, called “()“. You can think of unit and () as somewhat like “void” (the type) and “null” (the value) in C#. But unlike void/null, unit is a real type and () is a real value. To see this, evaluate:

let whatIsThis = ()

and you will see the signature:

val whatIsThis : unit = ()

Which means that the value “whatIsThis“ is of type unit and has been bound to the value ()

So, going back to the signature of “printInt“, we can now understand it:

val printInt : int -> unit

This signature says: printInt has a domain of int which it maps onto nothing that we care about.

Parameterless functions

Now that we understand unit, can we predict its appearance in other contexts? For example, let’s try to create a reusable “hello world” function. Since there is no input and no output, we would expect it to have a signature unit -> unit. Let’s see:

let printHello = printf "hello world"        // print to console

The result is:

hello world
val printHello : unit = ()

Not quite what we expected. “Hello world” is printed immediately and the result is not a function, but a simple value of type unit. As we saw earlier, we can tell that this is a simple value because it has a signature of the form:

val aName: type = constant

So in this case, we see that printHello is actually a simple value with the value (). It’s not a function that we can call again.

Why the difference between printInt and printHello? In the printInt case, the value could not be determined until we knew the value of the x parameter, so the definition was of a function. In the printHello case, there were no parameters, so the right hand side could be determined immediately. Which it was, returning the () value, with the side effect of printing to the console.

We can create a true reusable function that is parameterless by forcing the definition to have a unit argument, like this:

let printHelloFn () = printf "hello world"    // print to console

The signature is now:

val printHelloFn : unit -> unit

and to call it, we have to pass the () value as a parameter, like so:

printHelloFn ()

Forcing unit types with the ignore function

In some cases the compiler requires a unit type and will complain. For example, both of the following will be compiler errors:

do 1+1     // => FS0020: This expression should have type 'unit'
let something = 
  2+2      // => FS0020: This expression should have type 'unit'
  "hello"

To help in these situations, there is a special function ignore that takes anything and returns the unit type. The correct version of this code would be:

do (1+1 |> ignore)  // ok
let something = 
  2+2 |> ignore     // ok
  "hello"

Generic types

In many cases, the type of the function parameter can be any type, so we need a way to indicate this. F# uses the .NET generic type system for this situation.

For example, the following function converts the parameter to a string and appends some text:

let onAStick x = x.ToString() + " on a stick"

It doesn’t matter what type the parameter is, as all objects understand ToString().

The signature is:

val onAStick : 'a -> string

What is this type called 'a? That is F#’s way of indicating a generic type that is not known at compile time. The apostrophe in front of the “a” means that the type is generic. The signature for the C# equivalent of this would be:

string onAStick<a>();   
//or more idiomatically 
string OnAStick<TObject>();   // F#'s use of 'a is like 
                              // C#'s "TObject" convention

Note that the F# function is still strongly typed with a generic type. It does not take a parameter of type Object. This strong typing is desirable so that when functions are composed together, type safety is still maintained.

Here’s the same function being used with an int, a float and a string

onAStick 22
onAStick 3.14159
onAStick "hello"

If there are two generic parameters, the compiler will give them different names: 'a for the first generic, 'b for the second generic, and so on. Here’s an example:

let concatString x y = x.ToString() + y.ToString()

The type signature for this has two generics: 'a and 'b:

val concatString : 'a -> 'b -> string

On the other hand, the compiler will recognize when only one generic type is required. In the following example, the x and y parameters must be of the same type:

let isEqual x y = (x=y)

So the function signature has the same generic type for both of them:

val isEqual : 'a -> 'a -> bool

Generic parameters are also very important when it comes to lists and more abstract structures, and we will be seeing them a lot in upcoming examples.

Other types

The types discussed so far are just the basic types. These types can be combined in various ways to make much more complex types. A full discussion of these types will have to wait for another series, but meanwhile, here is a brief introduction to them so that you can recognize them in function signatures.

• The “tuple” types. These are pairs, triples, etc., of other types. For example ("hello", 1) is a tuple made from a string and an int. The comma is the distinguishing characteristic of a tuple — if you see a comma in F#, it is almost certainly part of a tuple!

In function signatures, tuples are written as the “multiplication” of the two types involved. So in this case, the tuple would have type:

string * int      // ("hello", 1)

• The collection types. The most common of these are lists, sequences, and arrays. Lists and arrays are fixed size, while sequences are potentially infinite (behind the scenes, sequences are the same as IEnumerable). In function signatures, they have their own keywords: “list“, “seq“, and “[]“ for arrays.

int list          // List type  e.g. [1;2;3]
string list       // List type  e.g. ["a";"b";"c"]
seq<int>          // Seq type   e.g. seq{1..10}
int []            // Array type e.g. [|1;2;3|]

• The option type. This is a simple wrapper for objects that might be missing. There are two cases: Some and None. In function signatures, they have their own “option“ keyword:

int option        // Some(1)

• The discriminated union type. These are built from a set of choices of other types. We saw some examples of this in the “why use F#?” series. In function signatures, they are referred to by the name of the type, so there is no special keyword.
• The record type. These are like structures or database rows, a list of named slots. We saw some examples of this in the “why use F#?” series as well. In function signatures, they are referred to by the name of the type, so again there is no special keyword.

Test your understanding of types

How well do you understand the types yet? Here are some expressions for you — see if you can guess their signatures. To see if you are correct, just run them in the interactive window!

let testA   = float 2
let testB x = float 2
let testC x = float 2 + x
let testD x = x.ToString().Length
let testE (x:float) = x.ToString().Length
let testF x = printfn "%s" x
let testG x = printfn "%f" x
let testH   = 2 * 2 |> ignore
let testI x = 2 * 2 |> ignore
let testJ (x:int) = 2 * 2 |> ignore
let testK   = "hello"
let testL() = "hello"
let testM x = x=x
let testN x = x 1          // hint: what kind of thing is x?
let testO x:string = x 1   // hint: what does :string modify?