The Marvels of Monads

wesdyer — Fri, 11 Jan 2008 03:06:13 GMT

If the word "continuation" causes eyes to glaze over, then the word "monad" induces mental paralysis. Perhaps, this is why some have begun inventing more benign names for monads.

These days, monads are the celebrities of programming language theory. They gloss the cover of blogs and have been compared to everything from boxes of fruit to love affairs. Nerds everywhere are exclaiming that the experience of understanding monads causes a pleasantly painful mental sensation.

Like continuations, monads are simpler than they sound and are very useful in many situations. In fact, programmers write code in a variety of languages that implicitly use common monads without even breaking a sweat.

With all of the attention that monads get, why am I writing yet another explanation of monads? Not to compare them to some everyday occurrence or to chronicle my journey to understanding. I explain monads because I need monads. They elegantly solve programming problems in a number of languages and contexts.

Introducing Monads

Monads come from category theory. Moggi introduced them to computer scientists to aid in the analysis of the semantics of computations. In an excellent paper, The Essence of Functional Programming, Wadler showed that monads are generally useful in computer programs to compose together functions which operate on amplified values rather than values. Monads became an important part of the programming language Haskell where they tackle the awkward squad: IO, concurrency, exceptions, and foreign-function calls.

Monads enjoy tremendous success in Haskell, but like an actor who does well in a particular role, monads are now stereotyped in the minds of most programmers as useful only in pure lazy functional languages. This is unfortunate, because monads are more broadly applicable.

Controlling Complexity

Composition is the key to controlling complexity in software. In The Structure and Interpretation of Computer Programs, Abelson and Sussman argue that composition beautifully expresses complex systems from simple patterns.

In our study of program design, we have seen that expert programmers control the complexity of their designs with the same general techniques used by designers of all complex systems. They combine primitive elements to form compound objects, they abstract compound objects to form higher-level building blocks, and they preserve modularity by adopting appropriate large-scale views of system structure.

One form of composition, function composition, succinctly describes the dependencies between function calls. Function composition takes two functions and plumbs the result from the second function into the input of the first function, thereby forming one function.

public static Func<T, V> Compose<T, U, V>(this Func<U, V> f, Func<T, U> g)
{
    return x => f(g(x));
}

For example, instead of applying g to the value x and then applying f to the result, compose f with g and then apply the result to the value x. The key difference is the abstraction of the dependency between f and g.

var r = f(g(x));         // without function composition
var r = f.Compose(g)(x); // with function composition

Given the function Identity, function composition must obey three laws.

public static T Identity<T>(this T value)
{
    return value;
}

1. Left identity

     Identity.Compose(f) = f

2. Right identity

     f.Compose(Identity) = f

3. Associative

     f.Compose(g.Compose(h)) = (f.Compose(g)).Compose(h)

Often, values are not enough. Constructed types amplify values. The type IEnumerable<T> represents a lazily computed list of values of type T. The type Nullable<T> represents a possibly missing value of type T. The type Func<Func<T, Answer>, Answer> represents a function, which returns an Answer given a continuation, which takes a T and returns an Answer. Each of these types amplifies the type T.

Suppose that instead of composing functions which return values, we compose functions which take values and return amplified values. Let M<T> denote the type of the amplified values.

public static Func<T, M<V>> Compose<T, U, V>(this Func<U, M<V>> f, Func<T, M<U>> g)
{
    return x => f(g(x)); // error, g(x) returns M<U> and f takes U
}

Function composition fails, because the return and input types do not match. Composition with amplified values requires a function that accesses the underlying value and feeds it to the next function. Call that function "Bind" and use it to define function composition.

public static Func<T, M<V>> Compose<T, U, V>(this Func<U, M<V>> f, Func<T, M<U>> g)
{
    return x => Bind(g(x), f);
}

The input and output types determine the signature of Bind. Therefore, Bind takes an amplified value, M<U>, and a function from U to M<V>, and returns an amplified value, M<V>.

public static M<V> Bind<U, V>(this M<U> m, Func<U, M<V>> k)

The body of Bind depends on the mechanics of the amplified values, M<T>. Each amplified type will need a distinct definition of Bind.

In addition to Bind, define a function which takes an unamplified value and amplifies it. Call this function "Unit".

public static M<T> Unit<T>(this T value)

Together the amplified type, M<T>, the function Bind, and the function Unit enable function composition with amplified values.

Meet the Monads

Viola, we have invented monads.

Monads are a triple consisting of a type, a Unit function, and a Bind function. Furthermore, to be a monad, the triple must satisfy three laws:

1. Left Identity

     Bind(Unit(e), k) = k(e)

2. Right Identity

     Bind(m, Unit) = m

3. Associative

     Bind(m, x => Bind(k(x), y => h(y)) = Bind(Bind(m, x => k(x)), y => h(y))

The laws are similar to those of function composition. This is not a coincidence. They guarantee that the monad is well behaved and composition works properly.

To define a particular monad, the writer supplies the triple, thereby specifying the mechanics of the amplified values.

The Identity Monad

The simplest monad is the Identity monad. The type represents a wrapper containing a value.

class Identity<T>
{
    public T Value { get; private set; }
    public Identity(T value) { this.Value = value; }
}

The Unit function takes a value and returns a new instance of Identity, which wraps the value.

static Identity<T> Unit<T>(T value)
{
    return new Identity<T>(value);
}

The bind function takes an instance of Identity, unwraps the value, and invokes the delegate, k, with the unwrapped value. The result is a new instance of Identity.

static Identity<U> Bind<T,U>(Identity<T> id, Func<T,Identity<U>> k)
{
    return k(id.Value);
}

Consider a simple program that creates two Identity wrappers and performs an operation on the wrapped values. First, bind x to the value within the wrapper containing the value five. Then, bind y to the value within the wrapper containing the value six. Finally, add the values, x and y, together. The result is an instance of Identity wrapping the value eleven.

var r = Bind(Unit(5), x =>
Bind(Unit(6), y =>
Unit(x + y)));

Console.WriteLine(r.Value);

While this works, it is rather clumsy. It would be nice to have syntax for dealing with the monad. Fortunately, we do.

C# 3.0 introduced query comprehensions which are actually monad comprehensions in disguise. We can rewrite the identity monad to use LINQ. Perhaps, it should have been called LINM (Language INtegrated Monads), but it just doesn't have the same ring to it.

Rename the method Unit to ToIdentity and Bind to SelectMany. Then, make them both extension methods.

public static Identity<T> ToIdentity<T>(this T value)
{
    return new Identity<T>(value);
}

public static Identity<U> SelectMany<T, U>(this Identity<T> id, Func<T, Identity<U>> k)
{
    return k(id.Value);
}

The changes impact the calling code.

var r = 5.ToIdentity().SelectMany(
            x => 6.ToIdentity().SelectMany(
                y => (x + y).ToIdentity()));

Console.WriteLine(r.Value);

Equivalent methods are part of the standard query operators defined for LINQ. However, the standard query operators also include a slightly different version of SelectMany for performance reasons. It combines Bind with Unit, so that lambdas are not deeply nested. The signature is the same except for an extra argument that is a delegate which takes two arguments and returns a value. The delegate combines the two values together. This version of SelectMany binds x to the wrapped value, applies k to x, binds the result to y, and then applies the combining function, s, to x and y. The resultant value is wrapped and returned.

public static Identity<V> SelectMany<T, U, V>(this Identity<T> id, Func<T, Identity<U>> k, Func<T,U,V> s)
{
    return id.SelectMany(x => k(x).SelectMany(y => s(x, y).ToIdentity()));
}

Of course, we can remove some of the code from the generalized solution by using our knowledge of the Identity monad.

public static Identity<V> SelectMany<T, U, V>(this Identity<T> id, Func<T, Identity<U>> k, Func<T,U,V> s)
{
    return s(id.Value, k(id.Value).Value).ToIdentity();
}

The call-site does not need to nest lambdas.

var r = 5.ToIdentity()
         .SelectMany(x => 6.ToIdentity(), (x, y) => x + y);

Console.WriteLine(r.Value);

With the new definition of SelectMany, programmers can use C#'s query comprehension syntax. The from notation binds the introduced variable to the value wrapped by the expression on the right. This allows subsequent expressions to use the wrapped values without directly calling SelectMany.

var r = from x in 5.ToIdentity()
        from y in 6.ToIdentity()
        select x + y;

Since the original SelectMany definition corresponds directly to the monadic Bind function and because the existence of a generalized transformation has been demonstrated, the remainder of the post will use the original signature. But, keep in mind that the second definition is the one used by the query syntax.

The Maybe Monad

The Identity monad is an example of a monadic container type where the Identity monad wrapped a value. If we change the definition to contain either a value or a missing value then we have the Maybe monad.

Again, we need a type definition. The Maybe type is similar to the Identity type but adds a property denoting whether a value is missing. It also has a predefined instance, Nothing, representing all instances lacking a value.

class Maybe<T>
{
    public readonly static Maybe<T> Nothing = new Maybe<T>();
    public T Value { get; private set; }
    public bool HasValue { get; private set; }
    Maybe()
    {
        HasValue = false;
    }
    public Maybe(T value)
    {
        Value = value;
        HasValue = true;
    }
}

The Unit function takes a value and constructs a Maybe instance, which wraps the value.

public static Maybe<T> ToMaybe<T>(this T value)
{
    return new Maybe<T>(value);
}

The Bind function takes a Maybe instance and if there is a value then it applies the delegate to the contained value. Otherwise, it returns Nothing.

public static Maybe<U> SelectMany<T, U>(this Maybe<T> m, Func<T, Maybe<U>> k)
{
    if (!m.HasValue)
        return Maybe<U>.Nothing;
    return k(m.Value);
}

The programmer can use the comprehension syntax to work with the Maybe monad. For example, create an instance of Maybe containing the value five and add it to Nothing.

var r = from x in 5.ToMaybe()
        from y in Maybe<int>.Nothing
        select x + y;

Console.WriteLine(r.HasValue ? r.Value.ToString() : "Nothing");

The result is "Nothing". We have implemented the null propagation of nullables without explicit language support.

The List Monad

Another important container type is the list type. In fact, the list monad is at the heart of LINQ. The type IEnumerable<T> denotes a lazily computed list.

The Unit function takes a value and returns a list, which contains only that value.

public static IEnumerable<T> ToList<T>(this T value)
{
    yield return value;
}

The Bind function takes an IEnumerable<T>, a delegate, which takes a T and returns an IEnumerable<U>, and returns an IEnumerable<U>.

public static IEnumerable<U> SelectMany<T, U>(this IEnumerable<T> m, Func<T, IEnumerable<U>> k)
{
    foreach (var x in m)
        foreach (var y in k(x))
            yield return y;
}

Now, the programmer can write the familiar query expressions with IEnumerable<T>.

var r = from x in new[] { 0, 1, 2 }
        from y in new[] { 0, 1, 2 }
        select x + y;

foreach (var i in r)
    Console.WriteLine(i);

Remember that it is the monad that enables the magic.

The Continuation Monad

The continuation monad answers the question that was posed at the end of the last post: how can a programmer write CPS code in a more palatable way?

The type of the continuation monad, K, is a delegate which when given a continuation, which takes an argument and returns an answer, will return an answer.

delegate Answer K<T,Answer>(Func<T,Answer> k);

The type K fundamentally differs from types Identity<T>, Maybe<T>, and IEnumerable<T>. All the other monads represent container types and allow computations to be specified in terms of the values rather than the containers, but the continuation monad contains nothing. Rather, it composes together continuations the user writes.

To be a monad, there must be a Unit function which takes a T and returns a K<T,Answer> for some answer type.

public static K<T, Answer> ToContinuation<T, Answer>(this T value)

What should it do? When in doubt, look to the types. The method takes a T and returns a function, which takes a function from T to Answer, and returns an Answer. Therefore, the method must return a function and the only argument of that function must be a function from T to Answer. Call the argument c.

return (Func<T, Answer> c) => ...

The body of the lambda must return a value of type Answer. Values of type Func<T,Answer> and a T are available. Apply c to value and the result is of type Answer.

return (Func<T, Answer> c) => c(value);

To be a monad, Bind must take a K<T,Answer> and a function from T to K<U, Answer> and return a K<U, Answer>.

public static K<U, Answer> SelectMany<T, U, Answer>(this K<T, Answer> m, Func<T, K<U, Answer>> k)

But what about the body? The result must be of type K<U, Answer>, but how is a result of the correct type formed?

Expand K's definition to gain some insight.

return type

     Func<Func<U, Answer>, Answer>

m's type

     Func<Func<T, Answer>, Answer>

k's type

     Func<T, Func<Func<U, Answer>, Answer>>

Applying k to a value of type T results in a value of type K<U,Answer>, but no value of type T is available. Build the return type directly by constructing a function, which takes a function from U to Answer. Call the parameter c.

return (Func<U,Answer> c) => ...

The body must be type of Answer so that the return type of Bind is K<U,Answer>. Perhaps, m could be applied to a function from T to Answer. The result is a value of type Answer.

return (Func<U,Answer> c) => m(...)

The expression inside the invocation of m must be of type Func<T,Answer>. Since there is nothing of that type, construct the function by creating a lambda with one parameter, x, of type T.

return (Func<U,Answer> c) => m((T x) => ...)

The body of this lambda must be of type Answer. Values of type T, Func<U,Answer>, and Func<T,Func<Func<U,Answer>, Answer>> haven't been used yet. Apply k to x.

return (Func<U,Answer> c) => m((T x) => k(x)...)

The result is a value of type Func<Func<U,Answer>,Answer>. Apply the result to c.

return (Func<U,Answer> c) => m((T x) => k(x)(c));

The continuation monad turns the computation inside out. The comprehension syntax can be used to construct continuations.

Construct a computation, which invokes a continuation with the value seven. Pass this computation to another computation, which invokes a continuation with the value six. Pass this computation to another computation, which invokes a continuation with the result of adding the results of the first two continuations together. Finally, pass a continuation, which replaces "1"s with "a"s, to the result.

var r = from x in 7.ToContinuation<int,string>()
        from y in 6.ToContinuation<int,string>()
        select x + y;

Console.WriteLine(r(z => z.ToString().Replace('1', 'a'))); // displays a3

The continuation monad does the heavy-lifting of constructing the continuations.

Monadic Magic

Beautiful composition of amplified values requires monads. The Identity, Maybe, and IEnumerable monads demonstrate the power of monads as container types. The continuation monad, K, shows how monads can readily express complex computation.

Stay tuned for more with monads. Until then, see what monads can do for you.

Yet Another Language Geek : Theory

The Marvels of Monads