What is ‘Pattern Matching’ in functional languages?

Understanding pattern matching requires explaining three parts:

  1. Algebraic data types.
  2. What pattern matching is
  3. Why its awesome.

Algebraic data types in a nutshell

ML-like functional languages allow you define simple data types called “disjoint unions” or “algebraic data types”. These data structures are simple containers, and can be recursively defined. For example:

type 'a list =
    | Nil
    | Cons of 'a * 'a list

defines a stack-like data structure. Think of it as equivalent to this C#:

public abstract class List<T>
{
    public class Nil : List<T> { }
    public class Cons : List<T>
    {
        public readonly T Item1;
        public readonly List<T> Item2;
        public Cons(T item1, List<T> item2)
        {
            this.Item1 = item1;
            this.Item2 = item2;
        }
    }
}

So, the Cons and Nil identifiers define simple a simple class, where the of x * y * z * ... defines a constructor and some data types. The parameters to the constructor are unnamed, they’re identified by position and data type.

You create instances of your a list class as such:

let x = Cons(1, Cons(2, Cons(3, Cons(4, Nil))))

Which is the same as:

Stack<int> x = new Cons(1, new Cons(2, new Cons(3, new Cons(4, new Nil()))));

Pattern matching in a nutshell

Pattern matching is a kind of type-testing. So let’s say we created a stack object like the one above, we can implement methods to peek and pop the stack as follows:

let peek s =
    match s with
    | Cons(hd, tl) -> hd
    | Nil -> failwith "Empty stack"

let pop s =
    match s with
    | Cons(hd, tl) -> tl
    | Nil -> failwith "Empty stack"

The methods above are equivalent (although not implemented as such) to the following C#:

public static T Peek<T>(Stack<T> s)
{
    if (s is Stack<T>.Cons)
    {
        T hd = ((Stack<T>.Cons)s).Item1;
        Stack<T> tl = ((Stack<T>.Cons)s).Item2;
        return hd;
    }
    else if (s is Stack<T>.Nil)
        throw new Exception("Empty stack");
    else
        throw new MatchFailureException();
}

public static Stack<T> Pop<T>(Stack<T> s)
{
    if (s is Stack<T>.Cons)
    {
        T hd = ((Stack<T>.Cons)s).Item1;
        Stack<T> tl = ((Stack<T>.Cons)s).Item2;
        return tl;
    }
    else if (s is Stack<T>.Nil)
        throw new Exception("Empty stack");
    else
        throw new MatchFailureException();
}

(Almost always, ML languages implement pattern matching without run-time type-tests or casts, so the C# code is somewhat deceptive. Let’s brush implementation details aside with some hand-waving please 🙂 )

Data structure decomposition in a nutshell

Ok, let’s go back to the peek method:

let peek s =
    match s with
    | Cons(hd, tl) -> hd
    | Nil -> failwith "Empty stack"

The trick is understanding that the hd and tl identifiers are variables (errm… since they’re immutable, they’re not really “variables”, but “values” 😉 ). If s has the type Cons, then we’re going to pull out its values out of the constructor and bind them to variables named hd and tl.

Pattern matching is useful because it lets us decompose a data structure by its shape instead of its contents. So imagine if we define a binary tree as follows:

type 'a tree =
    | Node of 'a tree * 'a * 'a tree
    | Nil

We can define some tree rotations as follows:

let rotateLeft = function
    | Node(a, p, Node(b, q, c)) -> Node(Node(a, p, b), q, c)
    | x -> x

let rotateRight = function
    | Node(Node(a, p, b), q, c) -> Node(a, p, Node(b, q, c))
    | x -> x

(The let rotateRight = function constructor is syntax sugar for let rotateRight s = match s with ....)

So in addition to binding data structure to variables, we can also drill down into it. Let’s say we have a node let x = Node(Nil, 1, Nil). If we call rotateLeft x, we test x against the first pattern, which fails to match because the right child has type Nil instead of Node. It’ll move to the next pattern, x -> x, which will match any input and return it unmodified.

For comparison, we’d write the methods above in C# as:

public abstract class Tree<T>
{
    public abstract U Match<U>(Func<U> nilFunc, Func<Tree<T>, T, Tree<T>, U> nodeFunc);

    public class Nil : Tree<T>
    {
        public override U Match<U>(Func<U> nilFunc, Func<Tree<T>, T, Tree<T>, U> nodeFunc)
        {
            return nilFunc();
        }
    }

    public class Node : Tree<T>
    {
        readonly Tree<T> Left;
        readonly T Value;
        readonly Tree<T> Right;

        public Node(Tree<T> left, T value, Tree<T> right)
        {
            this.Left = left;
            this.Value = value;
            this.Right = right;
        }

        public override U Match<U>(Func<U> nilFunc, Func<Tree<T>, T, Tree<T>, U> nodeFunc)
        {
            return nodeFunc(Left, Value, Right);
        }
    }

    public static Tree<T> RotateLeft(Tree<T> t)
    {
        return t.Match(
            () => t,
            (l, x, r) => r.Match(
                () => t,
                (rl, rx, rr) => new Node(new Node(l, x, rl), rx, rr))));
    }

    public static Tree<T> RotateRight(Tree<T> t)
    {
        return t.Match(
            () => t,
            (l, x, r) => l.Match(
                () => t,
                (ll, lx, lr) => new Node(ll, lx, new Node(lr, x, r))));
    }
}

For seriously.

Pattern matching is awesome

You can implement something similar to pattern matching in C# using the visitor pattern, but its not nearly as flexible because you can’t effectively decompose complex data structures. Moreover, if you are using pattern matching, the compiler will tell you if you left out a case. How awesome is that?

Think about how you’d implement similar functionality in C# or languages without pattern matching. Think about how you’d do it without test-tests and casts at runtime. Its certainly not hard, just cumbersome and bulky. And you don’t have the compiler checking to make sure you’ve covered every case.

So pattern matching helps you decompose and navigate data structures in a very convenient, compact syntax, it enables the compiler to check the logic of your code, at least a little bit. It really is a killer feature.

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