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Programming languages don't have to have a built-in notion of null values. Missing or optional values can be created from first principles. An introduction for object-oriented programmers.

This article is part of a series of articles about Church encoding. In this series, you'll learn how to re-create various programming language features from first principles. In previous articles, you learned how to implement Boolean logic without Boolean primitives, as well as how to model natural numbers. Through these examples, you'll learn how to model sum types without explicit language support.

The billion-dollar mistake #

All mainstream programming languages have a built-in notion of null: a value that isn't there. There's nothing wrong with the concept; you often run into situations where you need to return a value, but in certain cases, you'll have nothing to return. Division by zero would be one example. Attempting to retrieve the first element from an empty collection would be another.

Unfortunately, for fifty years, we've been immersed in environments where null references have been the dominant way to model the absence of data. This, despite the fact that even Sir Antony Hoare, the inventor of null references, has publicly called it his billion-dollar mistake.

You can, however, model the potential absence of data in saner ways. Haskell, for example, has no built-in null support, but it does include a built-in Maybe type. In Haskell (as well as in F#, where it's called option), Maybe is defined as a sum type:

data Maybe a = Nothing | Just a deriving (Eq, Ord)

If you're not familiar with Haskell syntax, this is a type declaration that states that the parametrically polymorphic (AKA generic) data type Maybe is inhabited by Just values that contain other values, plus the constant Nothing.

This article series, however, examines how to implement sum types with Church encoding.

Lambda calculus maybe #

Church encoding is based on the lambda calculus, which defines a universal model of computation based entirely on functions (lambda expressions) and recursion. In lambda calculus, the contract of Maybe is defined as an expression that takes two arguments. There's two fundamental 'implementations' of the contract:

nothing =    λn.λj.n
   just = λx.λn.λj.j x

The contract is that the first function argument (n) represents the nothing case, whereas the second argument (j) represents the just case.

The nothing function is a lambda expression that takes two arguments (n and j), and always returns the first, left-most argument (n).

The just function is a lambda expression that takes three arguments (x, n, and j), and always returns j x. Recall that in the lambda calculus, everything is a function, including j, so j x means that the function j is called with the argument x.

A few paragraphs above, I wrote that the contract of maybe is modelled as an expression that takes two arguments, yet just takes three arguments. How does that fit?

In the lambda calculus, expressions are always curried, so instead of viewing just as a function with three arguments, you can view it as a function that takes a single element (x) and returns a function that takes two arguments. This agrees with Haskell's Just data constructor:

Prelude> :t Just
Just :: a -> Maybe a

Haskell tells us that Just is a function that takes an a value (corresponding to x in the above just lambda expression) and returns a Maybe a value.

Church-encoded Maybe in C# #

Both lambda calculus and Haskell rely on currying and partial application to make the contract fit. In C#, as you've previously seen, you can instead define an interface and rely on class fields for the 'extra' function arguments. Since Church-encoded Maybe is represented by a function that takes two arguments, we'll once again define an interface with a single method that takes two arguments:

public interface IMaybe<T>
{
    TResult Match<TResult>(TResult nothing, Func<T, TResult> just);
}

In the first article, about Church-encoded Boolean values, you saw how two mutually exclusive values could be modelled as a method that takes two arguments. Boolean values are simply constants (true and false), where the next example (natural numbers) included a case where one case (successor) contained data. In that example, however, the data was statically typed as another INaturalNumber value. In the current IMaybe<T> example, the data contained in the just case is generic (it's of the type T).

Notice that there's two levels of generics in play. IMaybe<T> itself is a container of the generic type T, whereas Match enables you to convert the container into the rank-2 polymorphic type TResult.

Once more, the contract of IMaybe<T> is that the first, left-hand argument represents the nothing case, whereas the second, right-hand argument represents the just case. The nothing implementation, then, is similar to the previous ChurchTrue and Zero classes:

public class Nothing<T> : IMaybe<T>
{
    public TResult Match<TResult>(TResult nothing, Func<T, TResult> just)
    {
        return nothing;
    }
}

Again, the implementation unconditionally returns nothing while ignoring just. You may, though, have noticed that, as is appropriate for Maybe, Nothing<T> has a distinct type. In other words, Nothing<string> doesn't have the same type as Nothing<int>. This is not only 'by design', but is a fundamental result of how we define Maybe. The code simply wouldn't compile if you tried to remove the type argument from the class. This is in contrast to C# null, which has no type.

You implement the just case like this:

public class Just<T> : IMaybe<T>
{
    private readonly T value;
 
    public Just(T value)
    {
        this.value = value;
    }
 
    public TResult Match<TResult>(TResult nothing, Func<T, TResult> just)
    {
        return just(value);
    }
}

According to the contract, Just<T> ignores nothing and works exclusively with the just function argument. Notice that the value class field is private and not exposed as a public member. The only way you, as a caller, can potentially extract the value is by calling Match.

Here are some examples of using the API:

> new Nothing<Guid>().Match(nothing: "empty", just: g => g.ToString())
"empty"
> new Just<int>(42).Match(nothing: "empty", just: i => i.ToString())
"42"
> new Just<int>(1337).Match(nothing: 0, just: i => i)
1337

Notice that the third example shows how to extract the value contained in a Nothing<int> object without changing the output type. All you have to do is to supply a 'fall-back' value that can be used in case the value is nothing.

Maybe predicates #

You can easily implement the standard Maybe predicates IsNothing and IsJust:

public static IChurchBoolean IsNothing<T>(this IMaybe<T> m)
{
    return m.Match<IChurchBoolean>(
        nothing :   new ChurchTrue(), 
        just : _ => new ChurchFalse());
}
 
public static IChurchBoolean IsJust<T>(this IMaybe<T> m)
{
    return m.Match<IChurchBoolean>(
        nothing :   new ChurchFalse(),
        just : _ => new ChurchTrue());
}

Here, I arbitrarily chose to implement IsJust 'from scratch', but I could also have implemented it by negating the result of calling IsNothing. Once again, notice that the predicates are expressed in terms of Church-encoded Boolean values, instead of the built-in bool primitives.

Functor #

From Haskell (and F#) we know that Maybe is a functor. In C#, you turn a container into a functor by implementing an appropriate Select method. You can do this with IMaybe<T> as well:

public static IMaybe<TResult> Select<T, TResult>(
    this IMaybe<T> source,
    Func<T, TResult> selector)
{
    return source.Match<IMaybe<TResult>>(
        nothing:   new Nothing<TResult>(),
        just: x => new Just<TResult>(selector(x)));
}

Notice that this method turns an IMaybe<T> object into an IMaybe<TResult> object, using nothing but the Match method. This is possible because Match has a generic return type; thus, among other types of values, you can make it return IMaybe<TResult>.

When source is a Nothing<T> object, Match returns the object in the nothing case, which here becomes a new Nothing<TResult> object.

When source is a Just<T> object, Match invokes selector with the value contained in the just object, packages the result in a new Just<TResult> object, and returns it.

Because the Select method has the correct signature, you can use it with query syntax, as well as with normal method call syntax:

IMaybe<int> m = new Just<int>(42);
IMaybe<string> actual = from i in m
                        select i.ToString();

This example simply creates a just value containing the number 42, and then maps it to a string. Another way to write the same expression would be with method call syntax:

IMaybe<int> m = new Just<int>(42);
IMaybe<string> actual = m.Select(i => i.ToString());

In both cases, the result is a just case containing the string "42".

Summary #

In this article, you saw how it's possible to define the Maybe container from first principles, using nothing but functions (and, for the C# examples, interfaces and classes in order to make the code easier to understand for object-oriented developers).

The code shown in this article is available on GitHub.

Church-encoding enables you to model sum types as functions. So far in this article series, you've seen how to model Boolean values, natural numbers, and Maybe. Common to all three examples is that the data type in question consists of two mutually exclusive cases. There's at least one more interesting variation on that pattern.

Next: Church-encoded Either.

Comments

public class MaybeSelected<T, TSelectResult> : IMaybe<TSelectResult>
{
    private readonly IMaybe<T> source;
    private readonly Func<T, TSelectResult> selector;
		 
    public MaybeSelected(
        IMaybe<T> source,
        Func<T, TSelectResult> selector)
    {
        this.source = source;
        this.selector = selector;
    }
 
    public TResult Match<TResult>(TResult nothing, Func<TSelectResult, TResult> just)
    {
        return source.Match(
            nothing: nothing,
            just: x => just(selector(x)));
    }
}

public static IMaybe<TResult> Select<T, TResult>(
    this IMaybe<T> source,
    Func<T, TResult> selector)
{
    return new MaybeSelected<T, TResult>(source, selector);
}

public class MaybeFlattened<T> : IMaybe<T>
{
    private readonly IMaybe<IMaybe<T>> source;
		 
    public MaybeFlattened(IMaybe<IMaybe<T>> source)
    {
        this.source = source;
    }
 
    public TResult Match<TResult>(TResult nothing, Func<T, TResult> just)
    {
        return source.Match(
            nothing: nothing,
            just: x => x.Match(
                nothing: nothing,
                just: just));
    }
}

Natural numbers don't have to be built into programming languages. An introduction for object-oriented programmers.

This article is part of a series of articles about Church encoding. The previous article, about Church-encoding of Boolean values, concluded with the question: how do you determine whether an integer is even or odd?

That sounds easy, but turns out to be more complicated that you might think at first glance.

Built-in options #

How would you normally check whether a number is even? In some languages, like Haskell, it's built into the base library:

Prelude> even 1337
False
Prelude> even 42
True

In C#, surprisingly, I don't think it's built-in, but it's easy to implement a method to answer the question:

public static bool IsEven(this int i)
{
    return i % 2 == 0;
}

You could implement an IsOdd method either by using the != operator instead of ==, but otherwise copy the implementation of IsEven; or, alternatively, call IsEven and negate the result.

This works fine in normal C# code, but in this article, the agenda is different. We're investigating how programming with the previous article's IChurchBoolean API would look. The above built-in options use Boolean language primitives, so that's not really instructive.

Boolean conversions #

It's easy to convert between Church-encoded Boolean values and built-in Boolean values. For reasons I'll explain shortly, I still don't think that's instructive in this particular context, but for good measure I'll cover how to do it.

A method like the above IsEven returns bool. If you, instead, want an IChurchBoolean, you can use this simple conversion method:

public static IChurchBoolean ToChurchBoolean(this bool b)
{
    if (b)
        return new ChurchTrue();
    else
        return new ChurchFalse();
}

Alternatively, you can also use the ternary operator, but an ugly cast is necessary to make the C# compiler happy:

public static IChurchBoolean ToChurchBoolean(this bool b)
{
    return b ? (IChurchBoolean)new ChurchTrue() : new ChurchFalse();
}

Regardless of which implementation you choose, you'd be able to interact with the result as an IChurchBoolean values, as this small interactive session demonstrates:

> 42.IsEven().ToChurchBoolean().Match("Even", "Odd")
"Even"
> 1337.IsEven().ToChurchBoolean().Match("Even", "Odd")
"Odd"

Still, converting from bool to IChurchBoolean doesn't address the underlying question: is it possible to write programs without built-in Boolean primitives?

The conversion function ToChurchBoolean uses built-in Boolean values and functions, so it doesn't show whether or not it would be possible to make do without those.

Before we abandon that line of inquiry, however, I think it's only fair to share a conversion method that goes the other way:

public static bool ToBool(this IChurchBoolean b)
{
    return b.Match(true, false);
}

This function enables you to convert an IChurchBoolean value into a primitive C# bool, because when b represents true, the first argument (i.e. true) is returned, and when b represents false, the second argument (i.e. false) is returned.

Peano numbers #

If we can't use built-in primitives or operators that return them (e.g. ==), we may not be able to move forward with built-in numbers, either. What we can do, however, is to follow the lambda calculus to implement natural numbers using Church encoding. This will enable us to determine if a natural number is even or odd.

Lambda calculus models natural numbers according to Peano's model. In short, a natural number is either zero (or one, depending on the specific interpretation), or a successor to another natural number. As an example, using the Successor class that I'll develop later in this article, the number three can be represented as new Successor(new Successor(new Successor(new Zero()))) - it's the number after the number after the number after zero.

Like Church-encoded Boolean values, a Church-encoded natural number is a function that takes two arguments, corresponding to zero, and a successor function:

zero = λf.λx.x
one = λf.λx.f x
two = λf.λx.f (f x)
three = λf.λx.f (f (f x))
...

Each of these functions takes an initial value x, as well as a function f. In the lambda calculus, neither x nor f have any implied interpretation; it's the number of applications of f that defines the number.

In most translations into programming languages that I've encountered, however, x is usually interpreted as zero, and f as the successor function. In Haskell, for example, a common way to model Peano numbers is to use a sum type:

data Peano = Zero | Succ Peano deriving (Eq, Show)

Basically, this means that a value of the Peano type can either be the atom Zero, or a Succ value. Notice that Succ contains another Peano value; the data type is recursive.

You can write Haskell values like these:

*Peano> zero = Zero
*Peano> one = Succ Zero
*Peano> two = Succ (Succ Zero)
*Peano> three = Succ (Succ (Succ Zero))

Alternatively, you can also define the numbers based on previous definitions:

*Peano> zero = Zero
*Peano> one = Succ zero
*Peano> two = Succ one
*Peano> three = Succ two

This variation of Peano numbers uses an explicit sum type, but as the lambda calculus representation suggests, you can also use Church encoding to represent the two cases.

Church-encoded natural numbers #

If you recall Church-encoded Boolean values, you may remember that they are functions that take two values: a value to be used in case of true, and a value to be used in the case of false. You can do something similar with natural numbers. Zero is like true and false, in the sense that it's nothing but a label without any associated data. Succ, on the other hand, contains another Peano value. The way to do that is to turn the successor case into a function. Doing that, you'll arrive at an interface like this:

public interface INaturalNumber
{
    T Match<T>(T zero, Func<INaturalNumber, T> succ);
}

The first argument, on the left-hand side, is the case to use when an object represents zero. The second argument, on the right-hand side, is a function that will ultimately produce the value associated with a successor. The implied contract here is that the INaturalNumber passed as input to succ is the predecessor to 'the current value'. This may seem counter-intuitive, but hopefully becomes clearer when you see the Successor class below. The crucial insight is that a successor value has no intrinsic value; it's entirely defined by how many predecessors it has.

The zero implementation is similar to how Church-encoding implements true:

public class Zero : INaturalNumber
{
    public T Match<T>(T zero, Func<INaturalNumber, T> succ)
    {
        return zero;
    }
}

Notice that the Zero class implements INaturalNumber by always returning zero, and consequently always ignoring succ.

Another class, Successor, handles the right-hand side of the Match method:

public class Successor : INaturalNumber
{
    private readonly INaturalNumber predecessor;
 
    public Successor(INaturalNumber n)
    {
        this.predecessor = n;
    }
 
    public T Match<T>(T zero, Func<INaturalNumber, T> succ)
    {
        return succ(predecessor);
    }
}

Notice that Successor composes its predecessor via Constructor Injection, and unconditionally calls succ with its predecessor when Match is invoked.

Working with natural numbers #

What can you do with this INaturalNumber API, then?

Initially, you can define some numbers, like the above Haskell examples:

public static class NaturalNumber
{
    public static INaturalNumber  Zero = new Zero();
    public static INaturalNumber   One = new Successor(Zero);
    public static INaturalNumber   Two = new Successor(One);
    public static INaturalNumber Three = new Successor(Two);
    public static INaturalNumber  Four = new Successor(Three);
    public static INaturalNumber  Five = new Successor(Four);
    public static INaturalNumber   Six = new Successor(Five);
    public static INaturalNumber Seven = new Successor(Six);
    public static INaturalNumber Eight = new Successor(Seven);
    public static INaturalNumber  Nine = new Successor(Eight);
 
    // More memmbers go here...
}

Here, I arbitrarily chose to define the numbers from zero to nine, but you could go on for as long as you care.

You can also convert these Church-encoded numbers to primitive int values, like this:

public static int Count(this INaturalNumber n)
{
    return n.Match(
        0,
        p => 1 + p.Count());
}

Here are some examples from a C# Interactive session:

> NaturalNumber.Zero.Count()
0
> NaturalNumber.One.Count()
1
> NaturalNumber.Seven.Count()
7

The implementation of Count is recursive. When n is a Zero instance, it'll return the first argument (0), but when it's a Successor, it'll invoke the lambda expression p => 1 + p.Count(). Notice that this lambda expression recursively calls Count on p, which is the Successor's predecessor. It'll keep doing that until it reaches a Zero instance.

Recursion is a central part of the lambda calculus; you can't do anything useful without it. If you're a C# or Java programmer, you may be concerned, because recursion tends to be problematic in such languages. Deeply recursive functions will sooner or later crash because of a stack overflow.

You shouldn't, however, be concerned. First, I'm not trying to convince you to write all your future C# or Java code using Church-encoded numbers and Boolean values. The point of this article series is to investigate the fundamentals of computations, and to gain a better understanding of sum types. As such, the code examples presented here are only demonstrations of the underlying principles. Lambda calculus itself serves the same purpose: it's a universal model of computation; it wasn't intended to be a practical programming language - in fact, there were no programmable computers in 1936.

Furthermore, the problem with recursion causing stack overflow isn't universal. Languages like F# and Haskell support tail recursion, thereby enabling recursive functions to run to arbitrary depths.

Pattern matching #

In the previous article, I hinted that there's a reason I decided to name the interface method Match. This is because it looks a lot like pattern matching. In F#, you could write count like this:

type Peano = Zero | Succ of Peano
 
// Peano -> int
let rec count n =
    match n with
    | Zero -> 0
    | Succ p -> 1 + count p

This implementation, by the way, isn't tail-recursive, but you can easily refactor to a tail-recursive implementation like this:

// Peano -> int
let count n =
    let rec countImp acc n =
        match n with
        | Zero -> acc
        | Succ p -> countImp (1 + acc) p
    countImp 0 n

Both variations use the match keyword to handle both the Zero and the Succ case for any Peano value n. That's already close to the above C# code, but using the optional C# language feature of named arguments, you can rewrite the implementation of Count to this:

public static int Count(this INaturalNumber n)
{
    return n.Match(
        zero: 0,
        succ: p => 1 + p.Count());
}

This starts to look like pattern matching of sum types in F#. The argument names aren't required, but using them makes it clearer which cases the Match method handles.

Addition #

You can now start to add features and capabilities to the natural numbers API. An obvious next step is to implement addition:

public static INaturalNumber Add(this INaturalNumber x, INaturalNumber y)
{
    return x.Match(
        zero: y,
        succ: p => new Successor(p.Add(y)));
}

Again, the implementation is recursive. When x is zero, you simply return y, because zero + y is y. When x is a Successor, you recursively add y to its predecessor, and put the result in a new Successor. You can think of the predecessor p as one less than the successor. By recursively subtracting one from any Successor object, you'll eventually match the zero case, which will then return y. When the stack unrolls, each stack puts the previous result into a new Successor. This happens exactly the correct number of times corresponding to the value of x, because that's the size of the stack when Add hits zero.

Here are some examples:

> NaturalNumber.One.Add(NaturalNumber.Two).Count()
3
> NaturalNumber.Four.Add(NaturalNumber.Three).Count()
7
> NaturalNumber.Seven.Add(NaturalNumber.Six).Count()
13

You can also implement multiplication, but that's a bit more complicated, and not relevant to the topic of this article (which is how to determine if a number is even or odd).

Testing for zero #

In addition to basic arithmetic, you can also define functions that tell you something about a natural number. We'll start gently with a function that tells us whether or not a number is zero:

public static IChurchBoolean IsZero(this INaturalNumber n)
{
    return n.Match<IChurchBoolean>(
        zero: new ChurchTrue(),
        succ: _ => new ChurchFalse());
}

The IsZero method simply returns a ChurchTrue object when n is a Zero instance, and a ChurchFalse object for all other numbers.

You can see that this works in this C# Interactive session:

> NaturalNumber.Two.IsZero()
ChurchFalse { }
> NaturalNumber.Zero.IsZero()
ChurchTrue { }
> NaturalNumber.Three.IsZero()
ChurchFalse { }

You can also Match on the returned Boolean value to return e.g. a string:

> NaturalNumber.Nine.IsZero().Match(trueCase: "Zero", falseCase: "Not zero")
"Not zero"
> NaturalNumber.Zero.IsZero().Match(trueCase: "Zero", falseCase: "Not zero")
"Zero"

This already demonstrates that you can implement predicates and branching logic from first principles, without resorting to built-in Boolean primitives or operators.

Detecting even numbers #

Testing whether a natural number is even or uneven requires a bit more work. It's probably easiest to understand if we first consider an F# implementation:

// Peano -> ChurchBoolean
let rec isEven n =
    match n with
    | Zero -> ChurchTrue
    | Succ Zero -> ChurchFalse
    | Succ (Succ p) -> isEven p

Zero is even, so when n matches Zero, isEven returns ChurchTrue. Conversely, when the input is Succ Zero (i.e. one), the return value is ChurchFalse because one is odd.

The zero and one cases serve as exit cases for the recursive algorithm. Since we've handled Zero and Succ Zero (that is, zero and one), we know that any other case must be at least twice nested. This means that the Succ (Succ p) pattern matches all other cases. You can think of p as n - 2.

The algorithm proceeds to recursively call isEven with p (i.e. n - 2). Sooner or later, these recursive function calls will match either the Zero or the Succ Zero case, and exit with the appropriate return value.

C# doesn't have as sophisticated pattern matching features as F#, so we're going to have to figure out how implement this algorithm without relying on a nested pattern like Succ (Succ p). As an initial step, we can rewrite the function in F#, using two matches instead of one:

// Peano -> ChurchBoolean
let rec isEven n =
    match n with
    | Zero -> ChurchTrue
    | Succ p1 ->
        match p1 with
        | Zero -> ChurchFalse
        | Succ p2 -> isEven p2

This isn't as elegant as the previous implementation, but on the other hand, it's straightforward to translate to C#:

public static IChurchBoolean IsEven(this INaturalNumber n)
{
    return n.Match(
        zero: new ChurchTrue(),        // 0 is even, so true
        succ: p1 => p1.Match(          // Match previous
            zero: new ChurchFalse(),   // If 0 then successor was 1
            succ: p2 => p2.IsEven())); // Eval previous' previous
}

Like in the F# example, when n is a Zero object, it'll return the value associated with the zero case. Since zero is even, it returns a ChurchTrue object.

In all other cases, a Match on the predecessor p1 is required. If that nested match is zero, then we know that n must have been one, since the predecessor turned out to be zero. In that case, then, return a ChurchFalse object, because one isn't even.

The nested Match considers the predecessor p1. In the succ case of the nested Match, then, we can consider p2; that is, the predecessor to the predecessor to n - in other words: n - 2. The function recursively calls itself with n - 2, and it'll keep doing so until it matches either the zero or the one case.

The implementation works:

> NaturalNumber.Two.IsEven()
ChurchTrue { }
> NaturalNumber.Three.IsEven()
ChurchFalse { }

IsEven is implemented from first principles. The only language features we need are lambda expressions and recursion, although in order to make these examples slightly more idiomatic, I've also used interfaces and classes.

Detecting odd numbers #

You could implement a corresponding IsOdd method similarly to IsEven, but it's easier to use the Boolean operators already in place from the previous article:

public static IChurchBoolean IsOdd(this INaturalNumber n)
{
    return new ChurchNot(n.IsEven());
}

IsOdd is simply the Boolean negation of IsEven. Like IsEven it also works correctly:

> NaturalNumber.Six.IsOdd().Match(trueCase: "Odd", falseCase: "Even")
"Even"
> NaturalNumber.Seven.IsOdd().Match(trueCase: "Odd", falseCase: "Even")
"Odd"

You can implement other operators (like multiplication) and predicates from the building blocks shown here, but I'm not going to cover that here (see the accompanying GitHub repository for more code). I hope that this article gave you a sense of how a programming language can be designed from the low-level building blocks defined by the lambda calculus.

Summary #

Giuseppe Peano described natural numbers as an initial number (zero) and successors to that number. Church formulated Peano numbers in the lambda calculus. Using Church encoding, you can translate this representation to various programming languages, including, as you've seen in this article, C#.

In the previous article, you saw how to model Boolean values as a set of functions with two arguments. In this article, you saw how to model natural numbers with another set of functions that take two arguments. In the next article, you'll see another data type modelled as a set of functions with two arguments. It looks like a patterns is starting to appear.

Next: Church-encoded Maybe.

Boolean values, and logical branching, don't have to be built into programming languages. An introduction for object-oriented programmers.

This article is part of a series of articles about Church encoding.

Years ago, the so-called Anti-IF Campaign made the rounds on various social media (back then, IIRC, mostly known as 'the blogosphere'). The purpose of the campaign was never to eradicate every single use of if statements or expressions in source code, but rather to educate people about alternatives to the Arrow anti-pattern.

One easy way to deal with arrow code is to Replace Nested Conditionals with Guard Clauses, but that's not always possible. Another way is to encapsulate some if blocks in helper methods. Yet another way would be to use polymorphic dispatch, but how does that even work? Don't you, deep down, need at least a few if keywords here and there?

It turns out that the answer, surprisingly, is no.

Untyped Boolean functions #

if/then/else expressions are based on Boolean values (true and false): if some Boolean value is true, then something happens; otherwise, something else happens. Most programming languages, including C, C++, Java, C#, and JavaScript, have a ternary operator, which in C# looks like this:

isEven ? "Probably not a prime." : "Could be a prime.";

You can think of an expression like that as a function that takes a Boolean value and two potential return values: one for the true case, and one for the false case.

In lambda calculus, the only primitive building blocks are functions. There's no built-in Boolean values, but you can define them with functions. Boolean values are functions that take two arguments. By conventions, the first argument (the one to the left) represents the true case, whereas the second argument (to the right) signifies the false case - just like the ternary operator. In the lambda calculus, functions are curried, but we know from uncurry isomorphisms that we can also represent a two-argument function as a function that takes a two-tuple (a pair) as a single argument. Furthermore, we know from function isomorphisms that we can represent a function as an instance method. Therefore, we can declare a Boolean value in C# to be an object that implements this interface:

public interface IChurchBoolean
{
    object Match(object trueCase, object falseCase);
}

You'll notice that I've chosen to call the method Match, for reasons that should hopefully become clear as we go along.

The intent with such a Church-encoded Boolean is that any object that represents true should return the left argument (trueCase), whereas an object that represents false should return the right argument (falseCase).

In other words, true is an interface implementation:

public class ChurchTrue : IChurchBoolean
{
    public object Match(object trueCase, object falseCase)
    {
        return trueCase;
    }
}

Notice that this implementation always returns trueCase while ignoring falseCase. No explicit if statement is required.

Likewise, false is implemented the same way:

public class ChurchFalse : IChurchBoolean
{
    public object Match(object trueCase, object falseCase)
    {
        return falseCase;
    }
}

So far, this doesn't offer much capability, but it does already give you the ability to choose between two values, as this little C# Interactive session demonstrates:

> var b = new ChurchTrue();
> b.Match("foo", "bar")
"foo"
> var b = new ChurchFalse();
> b.Match("foo", "bar")
"bar"

When 'the Boolean value' is a ChurchTrue instance, then the left argument is returned; otherwise, when b is a ChurchFalse object, the return value is the right-hand value - just like the ternary operator.

Boolean And #

You can now define the standard Boolean operators and, or, and not. Starting with and:

public class ChurchAnd : IChurchBoolean
{
    private readonly IChurchBoolean x;
    private readonly IChurchBoolean y;
 
    public ChurchAnd(IChurchBoolean x, IChurchBoolean y)
    {
        this.x = x;
        this.y = y;
    }
 
    public object Match(object trueCase, object falseCase)
    {
        return x.Match(y.Match(trueCase, falseCase), falseCase);
    }
}

The ChurchAnd class is an implementation of IChurchBoolean that composes two other IChurchBoolean values, x and y. You can use it like this:

var b = new ChurchAnd(new ChurchTrue(), new ChurchFalse());

In this case, b represents false, because it'll always return the right-hand argument when Match is invoked.

Notice that the implementation of ChurchAnd.Match first matches on x. Only if x itself is true can the expression passed as the first argument be returned; otherwise, falseCase will be returned. Thus, if x is true, the expression y.Match(trueCase, falseCase) will be returned, and only if that as well evaluates to true is the final result true. The trueCase value is only returned if y represents true, as well as x.

In the lambda calculus, Boolean and is defined like this:

and = λx.λy.λt.λf.x (y t f) f

The way to read this is that Boolean and is a function that takes four arguments:

• x, a Boolean value
• y, another Boolean value
• t, the value to return if the expression is true; the trueCase argument in the above C# implementation.
• f, the value to return if the expression is false; the falseCase argument in the above C# implementation.

This is exactly the same implementation as the above C# code.

Wait a minute, though, didn't I write that Boolean values are functions that take two arguments? And isn't and a function that takes four arguments?

Yes, indeed. That's how currying works. You can view and as a function that takes four arguments, but you can also view it as a function that takes two arguments (x and y) and returns another function that takes two arguments. This becomes clearer with partial application. When translating to C#, the 'contract' (that a Boolean value is a function that takes two arguments) is modelled as the interface IChurchBoolean, while the 'extra arguments' x and y become class fields, injected via the class' constructor.

Boolean Or #

In the lambda calculus, Boolean or is defined like this:

or = λx.λy.λt.λf.x t (y t f)

Translated to C#, this becomes:

public class ChurchOr : IChurchBoolean
{
    private readonly IChurchBoolean x;
    private readonly IChurchBoolean y;
 
    public ChurchOr(IChurchBoolean x, IChurchBoolean y)
    {
        this.x = x;
        this.y = y;
    }
 
    public object Match(object trueCase, object falseCase)
    {
        return x.Match(trueCase, y.Match(trueCase, falseCase));
    }
}

You can see that this is another direct translation. Boolean or only requires (at least) one of the Boolean values to be true, so if x is true, you can immediately return trueCase. Otherwise, in the case where x is false, there's still a chance that the entire expression could be true, so you'll have to evaluate y as well. When y represents true, you can still return trueCase. Only when y is also false should you return falseCase.

You can use ChurchOr like this:

var b = new ChurchOr(new ChurchTrue(), new ChurchFalse());

Here, b is true because true or false is true.

Boolean Not #

Finally, you can also define Boolean negation. In lambda calculus it's:

not = λx.λt.λf.x f t

Notice how this simply swaps the arguments passed to x. In C#, this translates to:

public class ChurchNot : IChurchBoolean
{
    private readonly IChurchBoolean b;
 
    public ChurchNot(IChurchBoolean b)
    {
        this.b = b;
    }
 
    public object Match(object trueCase, object falseCase)
    {
        return b.Match(falseCase, trueCase);
    }
}

You can combine all the Boolean operators like this:

var b = new ChurchOr(new ChurchFalse(), new ChurchNot(new ChurchTrue()));

Here, b is false because false or (not true) is false.

Typed Boolean functions #

So far, the IChurchBoolean interface has been untyped, in the sense that it took object arguments and had an object return type. You can, however, easily make the interface strongly typed, using generics:

public interface IChurchBoolean
{
    T Match<T>(T trueCase, T falseCase);
}

This doesn't really change the rest of the code you've seen in this article. The method signatures chance, but the implementations remain as shown. You can see the change in this commit.

Semigroups and monoids #

The strongly typed signature accentuates that the Match method is a binary operation; it takes two values of the type T and returns a single T value. Is it a monoid, then?

It's not a single monoid, but rather a collection of semigroups, some of which are monoids as well. The implementation of ChurchTrue corresponds to the first semigroup, and ChurchFalse to the last semigroup. You can make this explict in Haskell:

import Data.Semigroup
 
churchTrue :: a -> a -> a
churchTrue t f = getFirst (First t <> First f)

If you compare this implementation of churchTrue to the Travis Whitaker's true function, his is much simpler. I'm not suggesting that using First is better; I'm only trying to illustrate the connection.

If you aren't familiar with how things are done in Haskell, <> is the 'generic semigroup binary operator'. What it does depends on the type of expressions surrounding it. By wrapping both t and f in First containers, the <> operator becomes the operator that always returns the first argument (i.e. First t). Since the result is a First value, you have to unwrap it again by applying getFirst.

Likewise, you can define false:

churchFalse :: a -> a -> a
churchFalse t f = getLast (Last t <> Last f)

This still uses the <> operator, but now with the Last container, which gives it all the behaviour of the last semigroup.

The any and all monoids are implemented as compositions of these two fundamental semigroups. In the C# code in this article, they're implemented by ChurchAnd and ChurchOr, although in neither case have I defined an explicit identity value. This is, however, possible, so let's continue with the Haskell code to see what that would look like. First, you can define the 'naked' operations:

churchAnd x y t f = x (y t f) f
 
churchOr x y t f = x t (y t f)

I have here omitted the type signatures on purpose, as I believe they might confuse rather than help. In both cases, the logic is the same as in the above ChurchAnd and ChurchOr classes, although, as you can see, Haskell code is much terser.

These two functions already work as desired, but we can easily turn both into their respective monoids. First, the all monoid:

newtype ChurchAll = ChurchAll { runAll :: forall a. a -> a -> a }
 
instance Semigroup ChurchAll where
    ChurchAll x <> ChurchAll y = ChurchAll (churchAnd x y)
 
instance Monoid ChurchAll where
    mempty = ChurchAll churchTrue
    mappend = (<>)

In order for this code to compile, you must enable the RankNTypes language extension, which I did by adding the {-# LANGUAGE RankNTypes #-} pragma to the top of my code file. The forall a declaration corresponds to the <T> type annotation on the C# Match method. You can think of this as that the type argument is scoped to the function instead of the type.

The Semigroup instance simply delegates its behaviour to churchAnd, and the Monoid instance returns churchTrue as the identity (mempty).

Similarly, you can implement the any monoid:

newtype ChurchAny = ChurchAny { runAny :: forall a. a -> a -> a }
 
instance Semigroup ChurchAny where
    ChurchAny x <> ChurchAny y = ChurchAny (churchOr x y)
 
instance Monoid ChurchAny where
    mempty = ChurchAny churchFalse
    mappend = (<>)

As is also the case with ChurchAll, the ChurchAny instance of Semigroup simply delegates to a 'naked' function (in this case churchOr), and the Monoid instance again delegates mappend to <> and returns churchFalse as the identity.

The following brief GHCi session demonstrates that it all works as intended:

λ> runAny (ChurchAny churchTrue <> ChurchAny churchFalse) "foo" "bar"
"foo"
λ> runAny (ChurchAny churchFalse <> ChurchAny churchFalse) "foo" "bar"
"bar"
λ> runAll (ChurchAll churchFalse <> ChurchAll churchTrue) "foo" "bar"
"bar"
λ> runAll (ChurchAll churchTrue <> ChurchAll churchTrue) "foo" "bar"
"foo"

Recall that a Church-encoded Boolean is a function that takes two values - in all the four above examples "foo" and "bar". When the expression represents true it returns the left-hand value ("foo"); otherwise, it returns the right-hand value ("bar").

In summary, the Church-encoded Boolean values true and false correspond to the first and last semigroups. You can compose the well-known monoids over Boolean values using these two basic building blocks.

Summary #

You'd normally think of Boolean values as language primitives. True and false are built into most languages, as well as common operators like and, or, and not. While this is convenient, it doesn't have to be like this. Even in languages that already have built-in support for Boolean values, like Haskell or C#, you can define Church-encoded Boolean values from first principles.

In the lambda calculus, a Boolean value is function that takes two arguments and returns the left-hand argument when true, and the right-hand argument when false.

At this point, it may seem like you can't do much with the IChurchBoolean API. How could you, for instance, determine whether an integer is even or odd?

This innocuous-looking question is harder to answer than you may think, so that's worthy of its own article.

Next: Church-encoded natural numbers.

Church encoding is a unified way to model data and functions. An introduction for object-oriented developers.

This article series is part of an even larger series of articles about the relationship between design patterns and category theory.

When asked why I like functional programming so much, I often emphasise the superior modelling ability that I get from algebraic data types. Particularly, languages like F# and Haskell have sum types in addition to the product types that most statically typed languages seem to have.

In short, a sum type gives you the ability to declare, as part of the type system, that a particular data type must be exactly one of a finite list of mutually exclusive options. This differs from common object-oriented sub-typing because class inheritance or interface implementation offers conceptually infinite extensibility. Sometimes, unconstrained extensibility is exactly what you need, but in other cases, the ability to define a closed set of cases can be an effective modelling tool. If you need an easy-to-read introduction to algebraic data types, I recommend Tomas Petricek's fine article Power of mathematics: Reasoning about functional types.

Interestingly, TypeScript has sum types, so they don't have to belong exclusively in the realm of functional programming. In this article series, you'll see an alternative way to represent sum types in C# using Church encoding.

Lambda calculus #

In the 1930s, several mathematicians were investigating the foundations of mathematics. One of them, Alonzo Church, developed lambda calculus as a universal model of computation. In a sense, you can think of lambda calculus as a sort of hypothetical programming language, although it was never designed to be a practical programming language. Even so, you can learn a lot from it.

In the untyped lambda calculus, the only primitive data type is a function. There are no primitive numbers, Boolean values, branching instructions, loops, or anything else you'd normally consider as parts of a programming language. Instead, there's only functions, written as lambda expressions:

λf.λx.f x

This looks opaque and mathematical, but most modern programmers should be familiar with lambda (λ) expressions. The above expression is an anonymous function that takes a single argument: f. The body of the function is the return value; here, another lambda expression: λx.f x. This lambda expression also takes a single argument: x.

In the untyped lambda calculus, everything is a function, so that includes f and x. The return value of the entire expression is f x, which means that the function f is applied to the value (in fact: function) x. The entire expression is therefore a higher-order function.

In C#, the corresponding lambda expression would be:

f => x => f(x)

This is a lambda expression that returns another lambda expression, which again returns the result of calling the function f with the value x.

In F#, it would be:

fun f -> fun x -> f x

and in Haskell, it would be:

\f -> \x -> f x

In both Haskell and F#, functions are already curried, so you can shorten that Haskell lambda expression to:

\f x -> f x

and the F# lambda expression to:

fun f x -> f x

This looks more like a function that takes two arguments, so alternatively, via uncurry isomorphisms, you can also write the C# representation like this:

(f, x) => f(x)

Those six lambda expressions, however, are statically typed, even though they're generic (or, as Haskellers would put it, parametric polymorphic). This means that they're not entirely equal to λf.λx.f x, but it should give you a sense of what a lambda expression is.

It turns out that using nothing but lambda expressions, one can express any computation; lambda calculus is Turing-complete.

Church encoding #

Since languages like C#, F#, Haskell, and others, include lambda expressions, you can reproduce as much of the lambda calculus as you'd like. In this article series, I'll mainly use it to show you how to represent sum types in C#. Later, you'll see how it relates to design patterns.

• Church-encoded Boolean values
• Church-encoded natural numbers
• Church-encoded Maybe
• Church-encoded Either
• Church-encoded payment types
• Church-encoded rose tree

These articles give you examples in C#. For Haskell examples, I found Travis Whitaker's article Scrap Your Constructors: Church Encoding Algebraic Types useful.

All C# code for these articles is available on GitHub.

Summary #

You can use lambda expressions to define all sorts of data types and computations. Because lambda calculus is a universal model of computation, you can learn about fundamental representations of computation. Particularly, lambda calculus offers a model of logical branching, which again teaches us how to model sum types.

Next: Church-encoded Boolean values.

Comments

Composites are monoids. An example in C#, F#, and Haskell.

Towards the end of the first decade of the third millennium, I'd been writing object-oriented code for about ten years, and I'd started to notice some patterns in my code. I'd read Design Patterns 6-7 years earlier, but I noticed that I tended to use only a small subset of the patterns from the book - particularly Composite, Decorator, Chain of Responsibility, and a few others.

In particular, I noticed that modelling seemed to be easier, and the code better structured, when I could apply the Composite design pattern. It was also clear, however, that I couldn't always use the Composite pattern, so I started to speculate on what could be the distinguishing factors. In 2010, I made a first attempt at identifying when a Composite is possible, and when it isn't. Unfortunately, while it was a fine attempt (which I'll return to later), it didn't lead anywhere. Ultimately, I gave up on the subject and moved on to other things.

A revelation #

One of my interests in the next decade became functional programming. One day in late 2016 I came across this Code Review question by Scott Nimrod. It was an solution to the Business Rules kata, which, briefly told, is about implementing changing business rules in a sustainable manner.

In my answer to the question, I gave an outline (repeated below) of how I would address the problem in F#. As a comment to my answer, Scott wrote:

"Feels like the Decorator Pattern..."

I responded,

"Not really; it's the Composite pattern..."

A few days later, as I was doing something else, it suddenly dawned on me that not only was a few lines of F# code equivalent to the Composite design pattern, but those lines of code were also manifestations of fundamental abstractions from category theory. Originally, I thought Composite was a combination of applicative functors and monoids, but as I investigated, I discovered that Composites are simply monoids.

This article shows a concrete example of that discovery, starting with my original F# code, subsequently translating it to C# to demonstrate that it's a Composite, and concluding with a translation to Haskell in order to demonstrate that it all fits with the formalisation of Monoid there.

Original F# solution outline #

The kata is about modelling volatile business rules in a sustainable manner. Particularly, you must implement various business rules associated with payments for products and services. Making a rough outline of a model, I started by introducing some types in F#:

type Membership = Basic | Gold
 
type Good =
| PhysicalProduct of string
| Book of string
| Video of string
| Membership of Membership
| Upgrade
 
type Command =
| Slip of string * (Good list)
| Activate of Membership
| Upgrade
| PayAgent

This basically states that there's a closed hierarchy of goods, and a closed hierarchy of business commands, as well as a Membership enumeration. A good can be a physical product with a name, a book with a name, a membership or upgrade, and so on. A command can be a packing slip, a membership activation, and so on.

Since I was only interested in a rough outline of a solution, I only sketched four business rules, all implemented as functions. The first creates a packing slip for certain goods:

// Good -> Command list
let slipForShipping = function
| PhysicalProduct name -> [Slip ("Shipping", [PhysicalProduct name])]
| Book name            -> [Slip ("Shipping", [Book name])]
| Video name           -> [Slip ("Shipping", [Video name])]
| _                    -> []

This function takes a Good value as input and returns a list of Command values as output. If the Good is a PhysicalProduct, Book, or Video, it returns a packing slip command; otherwise, it returns an empty list of commands.

The next business rule is a similar function:

// Good -> Command list
let slipForRoyalty = function
| Book name -> [Slip ("Royalty", [Book name])]
| _         -> []

This business rule generates a royalty slip for any Book, but does nothing for any other Good.

The third business rule activates a membership:

// Good -> Command list
let activate = function | Membership x -> [Activate x] | _ -> []

If the Good is a Membership, the activate function returns a list containing a single Activate command; otherwise, it returns an empty list.

Finally, the last rule upgrades a membership:

// Good -> Command list
let upgrade = function | Good.Upgrade -> [Upgrade] | _ -> []

Similar to the previous functions, this one looks at the type of Good, and returns an Upgrade command when the input is an Upgrade good, and an empty list otherwise.

Notice that all four functions share the same type: Good -> Command list. I designed them like that on purpose, because this enables you to compose a list of business rules to a function that looks like a single rule:

// ('a -> 'b list) list -> 'a -> 'b list
let handle rules good = List.collect (fun r -> r good) rules

This handle function takes a list of business rules (rules) and returns a new function with the type Good -> Command list (or, actually, a function with the type 'a -> 'b list - once again I've fallen into the trap of using too descriptive names). Notice that this is the same type as the individual rules.

You can now compose the four specific business rules:

// Good -> Command list
let handleAll = handle [slipForShipping; slipForRoyalty; activate; upgrade]

This function also has the type Good -> Command list although it's a composition of four rules.

You can use it like this F# Interactive example:

> handleAll (Book "The Annotated Turing");;
val it : Command list =
  [Slip ("Shipping",[Book "The Annotated Turing"]);
   Slip ("Royalty",[Book "The Annotated Turing"])]

(Yes, I like The Annotated Turing - read my review on Goodreads.)

Notice that while each of the business rules produces only zero or one Command values, in this example handleAll returns two Command values.

This design, where a composition looks like the things it composes, sounds familiar.

Business rules in C# #

You can translate the above F# model to an object-oriented model in C#. Translating discriminated unions like Good and Command to C# always involves compromises. In order to keep the example as simple as possible, I decided to translate each of those to a marker interface, although I loathe that 'pattern':

public interface IGood { }

While the interface doesn't afford any behaviour, various classes can still implement it, like, for example, Book:

public class Book : IGood
{
    public Book(string name)
    {
        Name = name;
    }
 
    public string Name { get; }
}

Other IGood 'implementations' looks similar, and there's a comparable class hierarchy for ICommand, which is another marker interface.

The above F# code used a shared function type of Good -> Command list as a polymorphic type for a business rule. You can translate that to a C# interface:

public interface IRule
{
    IReadOnlyCollection<ICommand> Handle(IGood good);
}

The above slipForShipping function becomes a class that implements the IRule interface:

public class SlipForShippingRule : IRule
{
    public IReadOnlyCollection<ICommand> Handle(IGood good)
    {
        if (good is PhysicalProduct ||
            good is Book ||
            good is Video)
            return new[] { new SlipCommand("Shipping", good) };
 
        return new ICommand[0];
    }
}

Instead of pattern matching on a discriminated union, the Handle method examines the subtype of good and only returns a SlipCommand if the good is either a PhysicalProduct, a Book, or a Video.

The other implementations are similar, so I'm not going to show all of them, but here's one more:

public class ActivateRule : IRule
{
    public IReadOnlyCollection<ICommand> Handle(IGood good)
    {
        var m = good as MembershipGood;
        if (m != null)
            return new[] { new ActivateCommand(m.Membership) };
 
        return new ICommand[0];
    }
}

Since 'all' members of IRule return collections, which form monoids over concatenation, the interface itself gives rise to a monoid. This means that you can create a Composite:

public class CompositeRule : IRule
{
    private readonly IRule[] rules;
 
    public CompositeRule(params IRule[] rules)
    {
        this.rules = rules;
    }
 
    public IReadOnlyCollection<ICommand> Handle(IGood good)
    {
        var commands = new List<ICommand>();
        foreach (var rule in rules)
            commands.AddRange(rule.Handle(good));
        return commands;
    }
}

Notice how the implementation of Handle follows the template for monoid accumulation. It starts with the identity, which, for the collection concatenation monoid is the empty collection. It then loops through all the composed rules and updates the accumulator commands in each iteration. Here, I used AddRange, which mutates commands instead of returning a new value, but the result is the same. Finally, the method returns the accumulator.

You can now compose all the business rules and use the composition as though it was a single object:

var rule =
    new CompositeRule(
        new SlipForShippingRule(),
        new SlipForRoyaltyRule(),
        new ActivateRule(),
        new UpgradeRule());
 
var book = new Book("The Annotated Turing");
var commands = rule.Handle(book);

When the method returns, commands contains two SlipCommand objects - a packing slip, and a royalty slip.

Business rules in Haskell #

You can also port the F# code to Haskell, which is usually easy as long as the F# is written in a 'functional style'. Since Haskell has an explicit notion of monoids, you'll be able to see how the two above solutions are monoidal.

The data types are easy to translate to Haskell - you only have to adjust the syntax a bit:

data Membership = Basic | Gold deriving (Show, Eq, Enum, Bounded)
 
data Good =
    PhysicalProduct String
  | Book String
  | Video String
  | Membership Membership
  | UpgradeGood
  deriving (Show, Eq)
 
data Command =
    Slip String [Good]
  | Activate Membership
  | Upgrade
  | PayAgent
  deriving (Show, Eq)

The business rule functions are also easy to translate:

slipForShipping :: Good -> [Command]
slipForShipping pp@(PhysicalProduct _) = [Slip "Shipping" [pp]]
slipForShipping             b@(Book _) = [Slip "Shipping"  [b]]
slipForShipping            v@(Video _) = [Slip "Shipping"  [v]]
slipForShipping                     _  = []
 
slipForRoyalty :: Good -> [Command]
slipForRoyalty b@(Book _) = [Slip "Royalty" [b]]
slipForRoyalty         _  = []
 
activate :: Good -> [Command]
activate (Membership m) = [Activate m]
activate             _  = []
 
upgrade :: Good -> [Command]
upgrade (UpgradeGood) = [Upgrade]
upgrade            _  = []

Notice that all four business rules share the same type: Good -> [Command]. This is conceptually the same type as in the F# code; instead of writing Command list, which is the F# syntax, the Haskell syntax for a list of Command values is [Command].

All those functions are monoids because their return types form a monoid, so in Haskell, you can compose them without further ado:

handleAll :: Good -> [Command]
handleAll = mconcat [slipForShipping, slipForRoyalty, activate, upgrade]

mconcat is a built-in function that aggregates any list of monoidal values to a single value:

mconcat :: Monoid a => [a] -> a

Since all four functions are monoids, this just works out of the box. A Composite is just a monoid. Here's an example of using handleAll from GHCi:

*BusinessRules> handleAll $ Book "The Annotated Turing"
[Slip "Shipping" [Book "The Annotated Turing"],Slip "Royalty" [Book "The Annotated Turing"]]

The result is as you'd come to expect.

Notice that not only don't you have to write a CompositeRule class, you don't even have to write a handle helper function. Haskell already understands monoids, so composition happens automatically.

If you prefer, you could even skip the handle function too:

*BusinessRules> mconcat [slipForShipping, slipForRoyalty, activate, upgrade] $ Book "Blindsight"
[Slip "Shipping" [Book "Blindsight"],Slip "Royalty" [Book "Blindsight"]]

(Yes, you should also read Blindsight.)

It's not that composition as such is built into Haskell, but rather that the language is designed around a powerful ability to model abstractions, and one of the built-in abstractions just happens to be monoids. You could argue, however, that many of Haskell's fundamental abstractions are built from category theory, and one of the fundamental characteristics of a category is how morphisms compose.

Summary #

Composite are monoids. This article shows an example, starting with a solution in F#. You can translate the F# code to object-oriented C# and model the composition of business rules as a Composite. You can also translate the F# code 'the other way', to the strictly functional language Haskell, and thereby demonstrate that the solution is based on a monoid.

This article is a repost of a guest post on the NDC blog, reproduced here with kind permission.

Write expressive property-based test with QuickCheck and view patterns.

Recently, I was writing some QuickCheck-based tests of some business logic, and since the business logic in question involved a custom domain type called Reservation, I had to write an Arbitrary instance for it. Being a dutiful Haskell programmer, I wrapped it in a newtype in order to prevent warnings about orphaned instances:

newtype ArbReservation = ArbReservation { getReservation :: Reservation } deriving (Show, Eq)

instance Arbitrary ArbReservation where
  arbitrary = do
    (d, e, n, Positive q, b) <- arbitrary
    return $ ArbReservation $ Reservation d e n q b

This is all fine as long as you just need one Reservation in a test, because in that case, you can simply pattern-match it out of ArbReservation:

testProperty "tryAccept reservation in the past" $ \
  (Positive capacity) (ArbReservation reservation)
  ->
  let stub (IsReservationInFuture _ next) = next False
      stub (ReadReservations _ next) = next []
      stub (Create _ next) = next 0

      actual = iter stub $ runMaybeT $ tryAccept capacity reservation
      
  in  isNothing actual

Here, reservation is a Reservation value because it was pattern-matched out of ArbReservation reservation. That's just like capacity is an Int, because it was pattern-matched out of Positive capacity.

Incidentally, in the spirit of the previous article, I'm here using in-lined properties implemented as lambda expressions. The lambda expressions use non-idiomatic formatting in order to make the tests more readable (and to prevent horizontal scrolling), but the gist of the matter is that the entire expression has the type Positive Int -> ArbReservation -> Bool. This is a Testable property because all the input types have Arbitrary instances.

Discommodity creeps in #

That's fine for that test case, but for the next, I needed not only a single Reservation value, but also a list of Reservation values. Again, with QuickCheck, you can't write a property with a type like Positive Int -> [Reservation] -> ArbReservation -> Bool, because there's no Arbitrary instance for [Reservation]. Instead, you'll need a property with the type Positive Int -> [ArbReservation] -> ArbReservation -> Bool.

One way to do that is to write the property like this:

testProperty "tryAccept reservation when capacity is insufficient" $ \
  (Positive i)
  reservations
  (ArbReservation reservation)
  ->
  let stub (IsReservationInFuture _ next) = next True
      stub (ReadReservations _ next) = next $ getReservation <$> reservations
      stub (Create _ next) = next 0
      reservedSeats = sum $ reservationQuantity <$> getReservation <$> reservations
      capacity = reservedSeats + reservationQuantity reservation - i

      actual = iter stub $ runMaybeT $ tryAccept capacity reservation

  in  isNothing actual

Here, reservations has type [ArbReservation], so every time the test needs to operate on the values, it first has to pull the Reservation values out of it using getReservation <$> reservations. That seems unnecessarily verbose and repetitive, so it'd be nice if a better option was available.

View pattern #

Had I been writing F# code, I'd immediately be reaching for an active pattern, but this is Haskell. If there's one thing, though, I've learned about Haskell so far, it's that, if F# can do something, there's a very good chance Haskell can do it too - only, it may be called something else.

With a vague sense that I'd seen something similar in some Haskell code somewhere, I went looking, and about fifteen minutes later I'd found what I was looking for: a little language extension called view patterns. Just add the language extension to the top of the file where you want to use it:

{-# LANGUAGE ViewPatterns #-}

You can now change the property to pattern-match reservations out of a function call, so to speak:

testProperty "tryAccept reservation when capacity is insufficient" $ \
  (Positive i)
  (fmap getReservation -> reservations)
  (ArbReservation reservation)
  ->
  let stub (IsReservationInFuture _ next) = next True
      stub (ReadReservations _ next) = next reservations
      stub (Create _ next) = next 0
      reservedSeats = sum $ reservationQuantity <$> reservations
      capacity = reservedSeats + reservationQuantity reservation - i

      actual = iter stub $ runMaybeT $ tryAccept capacity reservation

  in  isNothing actual

The function getReservation has the type ArbReservation -> Reservation, so fmap getReservation is a partially applied function with the type [ArbReservation] -> [Reservation]. In order to be able to call the overall lambda expression, the caller must supply an [ArbReservation] value to the view pattern, which means that the type of that argument must be [ArbReservation]. The view pattern then immediately unpacks the result of the function and gives you reservations, which is the return value of calling fmap getReservation with the input value(s). Thus, reservations has the type [Reservation].

The type of the entire property is now Positive Int -> [ArbReservation] -> ArbReservation -> Bool.

This removes some noise from the body of the property, so I find that this is a useful trick in this particular situation.

Summary #

You can use the view patterns GHC language extension when you need to write a function that takes an argument of a particular type, but you never care about the original input, but instead immediately wish to project it to a different value.

I haven't had much use for it before, but it seems to be useful in the context of QuickCheck properties.

Comments

pattern ArbReservations :: [Reservation] -> [ArbReservation]
pattern ArbReservations rs <- (coerce -> rs) where ArbReservations rs = coerce rs

foo :: [ArbReservation] -> IO ()
foo (ArbReservations rs) = traverse print rs
			

An alternative way to organise tests lists with HUnit.

In the previous article you saw how to write parametrised test with HUnit. While the tests themselves were elegant and readable (in my opinion), the composition of test lists left something to be desired. This article offers a different way to organise test lists.

Duplication #

The main problem is one of duplication. Consider the main function for the test library, as defined in the previous article:

main = defaultMain $ hUnitTestToTests $ TestList [
  "adjustToBusinessHours returns correct result" ~: adjustToBusinessHoursReturnsCorrectResult,
  "adjustToDutchBankDay returns correct result" ~: adjustToDutchBankDayReturnsCorrectResult,
  "Composed adjust returns correct result" ~: composedAdjustReturnsCorrectResult ]

It annoys me that I have a function with a (somewhat) descriptive name, like adjustToBusinessHoursReturnsCorrectResult, but then I also have to give the test a label - in this case "adjustToBusinessHours returns correct result". Not only is this duplication, but it also adds an extra maintenance overhead, because if I decide to rename the test, should I also rename the label?

Why do you even need the label? When you run the test, that label is printed during the test run, so that you can see what happens:

$ stack test --color never --ta "--plain"
ZonedTimeAdjustment-0.1.0.0: test (suite: ZonedTimeAdjustment-test, args: --plain)

:adjustToDutchBankDay returns correct result:
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]
:adjustToBusinessHours returns correct result:
  : [OK]
  : [OK]
  : [OK]
:Composed adjust returns correct result:
  : [OK]
  : [OK]
  : [OK]
  : [OK]
  : [OK]

         Test Cases   Total
 Passed  20           20
 Failed  0            0
 Total   20           20

I considered it redundant to give each test case in the parametrised tests their own labels, but I could have done that too, if I'd wanted to.

What happens if you remove the labels?

main = defaultMain $ hUnitTestToTests $ TestList $
      adjustToBusinessHoursReturnsCorrectResult
  ++  adjustToDutchBankDayReturnsCorrectResult
  ++  composedAdjustReturnsCorrectResult

That compiles, but produces output like this:

$ stack test --color never --ta "--plain"
ZonedTimeAdjustment-0.1.0.0: test (suite: ZonedTimeAdjustment-test, args: --plain)

: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]
: [OK]

         Test Cases   Total
 Passed  20           20
 Failed  0            0
 Total   20           20

If you don't care about the labels, then that's a fine solution. On the other hand, if you do care about the labels, then a different approach is warranted.

Inlined test lists #

Looking at an expression like "Composed adjust returns correct result" ~: composedAdjustReturnsCorrectResult, I find "Composed adjust returns correct result" more readable than composedAdjustReturnsCorrectResult, so if I want to reduce duplication, I want to go after a solution that names a test with a label, instead of a solution that names a test with a function name.

What is composedAdjustReturnsCorrectResult? It's just the name of a pure function (because its type is [Test]). Since it's referentially transparent, it means that in the test list in main, I can replace the function with its body! I can do this with all three functions, although, in order to keep things simplified, I'm only going to show you two of them:

main :: IO ()
main = defaultMain $ hUnitTestToTests $ TestList [
  "adjustToBusinessHours returns correct result" ~: do
    (dt, expected) <-
      [
        (zt (2017, 10, 2) (6, 59,  4) 0, zt (2017, 10, 2) (9,  0,  0) 0),
        (zt (2017, 10, 2) (9, 42, 41) 0, zt (2017, 10, 2) (9, 42, 41) 0),
        (zt (2017, 10, 2) (19, 1, 32) 0, zt (2017, 10, 3) (9,  0,  0) 0)
      ]
    let actual = adjustToBusinessHours dt
    return $ ZT expected ~=? ZT actual
 
  ,
  "Composed adjust returns correct result" ~: do
    (dt, expected) <-
      [
        (zt (2017, 1, 31) ( 7, 45, 55)   2 , zt (2017, 2, 28) ( 7,  0,  0) 0),
        (zt (2017, 2,  6) (10,  3,  2)   1 , zt (2017, 3,  6) ( 9,  3,  2) 0),
        (zt (2017, 2,  9) ( 4, 20,  0)   0 , zt (2017, 3,  9) ( 9,  0,  0) 0),
        (zt (2017, 2, 12) (16,  2, 11)   0 , zt (2017, 3, 10) (16,  2, 11) 0),
        (zt (2017, 3, 14) (13, 48, 29) (-1), zt (2017, 4, 13) (14, 48, 29) 0)
      ]
    let adjustments =
          reverse [adjustToNextMonth, adjustToBusinessHours, adjustToDutchBankDay, adjustToUtc]
    let adjust = appEndo $ mconcat $ Endo <$> adjustments
 
    let actual = adjust dt
 
    return $ ZT expected ~=? ZT actual
  ]

In order to keep the code listing to a reasonable length, I didn't include the third test "adjustToDutchBankDay returns correct result", but it works in exactly the same way.

This is a list with two values. You can see that the values are separated by a ,, just like list elements normally are. What's unusual, however, is that each element in the list is defined with a multi-line do block.

In C# and F#, I'm used to being able to just write new test functions, and they're automatically picked up by convention and executed by the test runner. I wouldn't be at all surprised if there was a mechanism using Template Haskell that enables something similar, but I find that there's something appealing about treating tests as first-class values all the way.

By inlining the tests, I can retain my F# and C# workflow. Just add a new test within the list, and it's automatically picked up by the main function. Not only that, but it's no longer possible to write a test that compiles, but is never executed by the test runner because it has the wrong type. This occasionally happens to me in F#, but with the technique outlined here, if I accidentally give the test the wrong type, it's not going to compile.

Conclusion #

Since HUnit tests are first-class values, you can define them inlined in test lists. For larger code bases, I'd assume that you'd want to spread your unit tests across multiple modules. In that case, I suppose that you could have each test module export a [Test] value. In the test library's main function, you'd need to manually concatenate all the exported test lists together, so a small maintenance burden remains. When you add a new test module, you'd have to add its exported tests to main.

I wouldn't be surprised, however, if a clever reader could point out to me how to avoid that as well.

Here's a way to write parametrised unit tests in Haskell.

Sometimes you'd like to execute the same (unit) test for a number of test cases. The only thing that varies is the input values, and the expected outcome. The actual test code is the same for all test cases. Among object-oriented programmers, this is known as a parametrised test.

When I recently searched the web for how to do parametrised tests in Haskell, I could only find articles that talked about property-based testing, mostly with QuickCheck. I normally prefer property-based testing, but sometimes, I'd rather like to run a test with some deterministic test cases that are visible and readable in the code itself.

Here's one way I found that I could do that in Haskell.

Testing date and time adjustments in C# #

In an earlier article, I discussed how to model date and time adjustments as a monoid. The example code was written in C#, and I used a few tests to demonstrate that the composition of adjustments work as intended:

[Theory]
[InlineData("2017-01-31T07:45:55+2", "2017-02-28T07:00:00Z")]
[InlineData("2017-02-06T10:03:02+1", "2017-03-06T09:03:02Z")]
[InlineData("2017-02-09T04:20:00Z" , "2017-03-09T09:00:00Z")]
[InlineData("2017-02-12T16:02:11Z" , "2017-03-10T16:02:11Z")]
[InlineData("2017-03-14T13:48:29-1", "2017-04-13T14:48:29Z")]
public void AccumulatedAdjustReturnsCorrectResult(
    string dtS,
    string expectedS)
{
    var dt = DateTimeOffset.Parse(dtS);
    var sut = DateTimeOffsetAdjustment.Accumulate(
        new NextMonthAdjustment(),
        new BusinessHoursAdjustment(),
        new DutchBankDayAdjustment(),
        new UtcAdjustment());
 
    var actual = sut.Adjust(dt);
 
    Assert.Equal(DateTimeOffset.Parse(expectedS), actual);
}

The above parametrised test uses xUnit.net (particularly its Theory feature) to execute the same test code for five test cases. Here's another example:

[Theory]
[InlineData("2017-10-02T06:59:04Z", "2017-10-02T09:00:00Z")]
[InlineData("2017-10-02T09:42:41Z", "2017-10-02T09:42:41Z")]
[InlineData("2017-10-02T19:01:32Z", "2017-10-03T09:00:00Z")]
public void AdjustReturnsCorrectResult(string dts, string expectedS)
{
    var dt = DateTimeOffset.Parse(dts);
    var sut = new BusinessHoursAdjustment();
 
    var actual = sut.Adjust(dt);
 
    Assert.Equal(DateTimeOffset.Parse(expectedS), actual);
}

This one covers the three code paths through BusinessHoursAdjustment.Adjust.

These tests are similar, so they share some good and bad qualities.

On the positive side, I like how readable such tests are. The test code is only a few lines of code, and the test cases (input, and expected output) are in close proximity to the code. Unless you're on a phone with a small screen, you can easily see all of it at once.

For a problem like this, I felt that I preferred examples rather than using property-based testing. If the date and time is this, then the adjusted result should be that, and so on. When we read code, we tend to prefer examples. Good documentation often contains examples, and for those of us who consider tests documentation, including examples in tests should advance that cause.

On the negative side, tests like these still contain noise. Most of it relates to the problem that xUnit.net tests aren't first-class values. These tests actually ought to take a DateTimeOffset value as input, and compare it to another, expected DateTimeOffset value. Unfortunately, DateTimeOffset values aren't constants, so you can't use them in attributes, like the [InlineData] attribute.

There are other workarounds than the one I ultimately chose, but none that are as simple (that I'm aware of). Strings are constants, so you can put formatted date and time strings in the [InlineData] attributes, but the cost of doing this is two calls to DateTimeOffset.Parse. Perhaps this isn't a big price, you think, but it does make me wish for something prettier.

Comparing date and time #

In order to port the above tests to Haskell, I used Stack to create a new project with HUnit as the unit testing library.

The Haskell equivalent to DateTimeOffset is called ZonedTime. One problem with ZonedTime values is that you can't readily compare them; the type isn't an Eq instance. There are good reasons for that, but for the purposes of unit testing, I wanted to be able to compare them, so I defined this helper data type:

newtype ZonedTimeEq = ZT ZonedTime deriving (Show)
 
instance Eq ZonedTimeEq where
  ZT (ZonedTime lt1 tz1) == ZT (ZonedTime lt2 tz2) = lt1 == lt2 && tz1 == tz2

This enables me to compare two ZonedTimeEq values, which are only considered equal if they represent the same date, the same time, and the same time zone.

Test Utility #

I also added a little function for creating ZonedTime values:

zt (y, mth, d) (h, m, s) tz =
  ZonedTime (LocalTime (fromGregorian y mth d) (TimeOfDay h m s)) (hoursToTimeZone tz)

The motivation is simply that, as you can tell, creating a ZonedTime value requires a verbose expression. Clearly, the ZonedTime API is flexible, but in order to define some test cases, I found it advantageous to trade readability for flexibility. The zt function enables me to compactly define some ZonedTime values for my test cases.

Testing business hours in Haskell #

In HUnit, a test is either a Test, a list of Test values, or an impure assertion. For a parametrised test, a [Test] sounded promising. At the beginning, I struggled with finding a readable way to express the tests. I wanted to be able to start with a list of test cases (inputs and expected outputs), and then fmap them to an executable test. At first, the readability goal seemed elusive, until I realised that I can also use do notation with lists (since they're monads):

adjustToBusinessHoursReturnsCorrectResult :: [Test]
adjustToBusinessHoursReturnsCorrectResult = do
  (dt, expected) <-
    [
      (zt (2017, 10, 2) (6, 59,  4) 0, zt (2017, 10, 2) (9,  0,  0) 0),
      (zt (2017, 10, 2) (9, 42, 41) 0, zt (2017, 10, 2) (9, 42, 41) 0),
      (zt (2017, 10, 2) (19, 1, 32) 0, zt (2017, 10, 3) (9,  0,  0) 0)
    ]
  let actual = adjustToBusinessHours dt
  return $ ZT expected ~=? ZT actual

This is the same test as the above C# test named AdjustReturnsCorrectResult, and it's about the same size as well. Since the test is written using do notation, you can take a list of test cases and operate on each test case at a time. Although the test creates a list of tuples, the <- arrow pulls each (ZonedTime, ZonedTime) tuple out of the list and binds it to (dt, expected).

This test literally consists of only three expressions, so according to my normal heuristic for test formatting, I don't even need white space to indicate the three phases of the AAA pattern. The first expression sets up the test case (dt, expected).

The next expression exercises the System Under Test - in this case the adjustToBusinessHours function. That's simply a function call.

The third expression verifies the result. It uses HUnit's ~=? operator to compare the expected and the actual values. Since ZonedTime isn't an Eq instance, both values are converted to ZonedTimeEq values. The ~=? operator returns a Test value, and since the entire test takes place inside a do block, you must return it. Since this particular do block takes place inside the list monad, the type of adjustToBusinessHoursReturnsCorrectResult is [Test]. I added the type annotation for the benefit of you, dear reader, but technically, it's not required.

Testing the composed function #

Translating the AccumulatedAdjustReturnsCorrectResult C# test to Haskell follows the same recipe:

composedAdjustReturnsCorrectResult :: [Test]
composedAdjustReturnsCorrectResult = do
  (dt, expected) <-
    [
      (zt (2017, 1, 31) ( 7, 45, 55)   2 , zt (2017, 2, 28) ( 7,  0,  0) 0),
      (zt (2017, 2,  6) (10,  3,  2)   1 , zt (2017, 3,  6) ( 9,  3,  2) 0),
      (zt (2017, 2,  9) ( 4, 20,  0)   0 , zt (2017, 3,  9) ( 9,  0,  0) 0),
      (zt (2017, 2, 12) (16,  2, 11)   0 , zt (2017, 3, 10) (16,  2, 11) 0),
      (zt (2017, 3, 14) (13, 48, 29) (-1), zt (2017, 4, 13) (14, 48, 29) 0)
    ]
  let adjustments =
        reverse [adjustToNextMonth, adjustToBusinessHours, adjustToDutchBankDay, adjustToUtc]
  let adjust = appEndo $ mconcat $ Endo <$> adjustments
 
  let actual = adjust dt
 
  return $ ZT expected ~=? ZT actual

The only notable difference is that this unit test consists of five expressions, so according to my formatting heuristic, I inserted some blank lines in order to make it easier to distinguish the three AAA phases from each other.

Running tests #

I also wrote a third test called adjustToDutchBankDayReturnsCorrectResult, but that one is so similar to the two you've already seen that I see no point showing it here. In order to run all three tests, I define the tests' main function as such:

main = defaultMain $ hUnitTestToTests $ TestList [
  "adjustToBusinessHours returns correct result" ~: adjustToBusinessHoursReturnsCorrectResult,
  "adjustToDutchBankDay returns correct result" ~: adjustToDutchBankDayReturnsCorrectResult,
  "Composed adjust returns correct result" ~: composedAdjustReturnsCorrectResult ]

This uses defaultMain from test-framework, and hUnitTestToTests from test-framework-hunit.

I'm not happy about the duplication of text and test names, and the maintenance burden implied by having to explicitly add every test function to the test list. It's too easy to forget to add a test after you've written it. In the next article, I'll demonstrate an alternative way to compose the tests so that duplication is reduced.

Conclusion #

Since HUnit tests are first-class values, you can manipulate and compose them just like any other value. That includes passing them around in lists and binding them with do notation. Once you figure out how to write parametrised tests with HUnit, it's easy, readable, and elegant.

The overall configuration of the test runner, however, leaves a bit to be desired, so that's the topic for the next article.

Next: Inlined HUnit test lists.

When can you implement a Null Object? When the return types of your methods are monoids.

This article is part of a series of articles about design patterns and their category theory counterparts. In a previous article you learned how there's a strong relationship between the Composite design pattern and monoids. In this article you'll see that the Null Object pattern is essentially a special case of the Composite pattern.

I also think that there's a relationship between monoidal identity and the Null Object pattern similar to the relationship between Composite and monoids in general:

Once more, I don't claim that an isomorphism exists. You may be able to produce Null Object examples that aren't monoidal, but on the other hand, I believe that all identities are Null Objects.

Null Object #

While the Null Object design pattern isn't one of the patterns covered in Design Patterns, I consider it a structural pattern because Composite is a structural pattern, and Null Object is a special case of Composite.

Bobby Woolf's original text contains an example in Smalltalk, and while I'm no Smalltalk expert, I think a fair translation to C# would be an interface like this:

public interface IController
{
    bool IsControlWanted();
    bool IsControlActive();
    void Startup();
}

The idea behind the Null Object pattern is to add an implementation that 'does nothing', which in the example would be this:

public class NullController : IController
{
    public bool IsControlActive()
    {
        return false;
    }
 
    public bool IsControlWanted()
    {
        return false;
    }
 
    public void Startup()
    {
    }
}

Woolf calls his implementation NoController, but I find it more intent-revealing to call it NullController. Both of the Boolean methods return false, and the Startup method literally does nothing.

Doing nothing #

What exactly does it mean to 'do nothing'? In the case of the Startup method, it's clear, because it's a bona fide no-op. This is possible for all methods without return values (i.e. methods that return void), but for all other methods, the compiler will insist on a return value.

For IsControlActive and IsControlWanted, Woolf solves this by returning false.

Is false always the correct 'do nothing' value for Booleans? And what should you return if a method returns an integer, a string, or a custom type? The original text is vague on that question:

"Exactly what "do nothing" means is subjective and depends on the sort of behavior the Client is expecting."

Doing nothing as identity #

From unit isomorphisms you know that methods without return values are isomorphic to methods that return unit. You also know that unit is a monoid. What does unit and bool have in common? They both form monoids; bool, in fact, forms four different monoids, of which all and any are the best-known.

In my experience, you can implement the Null Object pattern by returning various 'do nothing' values, depending on their types:

• For bool, return a constant value. Usually, false is the appropriate 'do nothing' value, but it does depend on the semantics of the operation.
• For string, return "".
• For collections, return an empty collection.
• For numbers, return a constant value, such as 0.
• For void, do nothing, which is equivalent to returning unit.

• The identity for the any monoid is false.
• The identity for string is "".
• The identity for collections is the empty collection.
• The identity for the addition monoid is 0.
• The identity for unit is unit.

foo.Op(Foo.Identity) == foo

In other words: the identity does nothing!

As a preliminary result, then, when all return values of your interface form monoids, you can create a Null Object.

Relationship with Composite #

In the previous article, you saw how the Composite design pattern was equivalent with the Haskell function mconcat:

mconcat :: Monoid a => [a] -> a

This function, however, seems more limited than it has to be. It says that if you have a linked list of monoidal values, you can reduce them to a single monoidal value. Is this only true for linked lists?

After all, in a language like C#, you'd typically express a Composite as a container of 'some collection' of objects, typically modelled by the IReadOnlyCollection<T> interface. There are several implementations of that interface, including lists, arrays, collections, and so on.

It seems as though we ought to be able to generalise mconcat, and, indeed, we can. The Data.Foldable module defines a function called fold:

fold :: (Monoid m, Foldable t) => t m -> m

Let me decipher that for you. It says that any monoid m contained in a 'foldable' container t can be reduced to a single value (still of the type m). You can read t m as 'a foldable container that contains monoids'. In C#, you could attempt to express it as IFoldable<TMonoid>.

In other words, the Composite design pattern is equivalent to fold. Here's how it relates to the Null Object pattern:

As a first step, consider methods like those defined by the above IController interface. In order to implement NullController, all of those methods ignore their (non-existent) input and return an identity value. In other words, we're looking for a Haskell function of the type Monoid m => a -> m; that is: a function that takes input of the type a and returns a monoidal value m.

You can do that using fold:

nullify :: Monoid m => a -> m
nullify = fold (Identity $ const mempty)

Recall that the input to fold must be a Foldable container. The good old Identity type is, among other capabilities, Foldable. That takes care of the container. The value that we put into the container is a single function that ignores its input (const) and always returns the identity value (mempty). The result is a function that always returns the identity value.

This demonstrates that Null Object is a special case of Composite, because nullify is a special application of fold.

There's no reason to write nullify in such a complicated way, though. You can simplify it like this:

nullify :: Monoid m => a -> m
nullify = const mempty

Once you recall, however, that functions are monoids if their return values are monoids, you can simplify it even further:

nullify :: Monoid m => a -> m
nullify = mempty

The identity element for monoidal functions is exactly a function that ignores its input and returns mempty.

Controller identity #

Consider the IController interface. According to object isomorphisms, you can represent this interface as a tuple of three functions:

type Controller =
  (ControllerState -> Any, ControllerState -> Any, ControllerState -> ())

This is cheating a little, because the third tuple element (the one that corresponds to Startup) is pure, although I strongly suspect that the intent is that it should mutate the state of the Controller. Two alternatives are to change the function to either ControllerState -> ControllerState or ControllerState -> IO (), but I'm going to keep things simple. It doesn't change the conclusion.

Notice that I've used Any as the return type for the two first tuples. As I've previously covered, Booleans form monoids like any and all. Here, we need to use Any.

This tuple is a monoid because all three functions are monoids, and a tuple of monoids is itself a monoid. This means that you can easily create a Null Object using mempty:

λ> nullController = mempty :: Controller

The nullController is a triad of functions. You can access them by pattern-matching them, like this:

λ> (isControlWanted, isControlActive, startup) = nullController

Now you can try to call e.g. isControlWanted with various values in order to verify that it always returns false. In this example, I cheated and simply made ControllerState an alias for String:

λ> isControlWanted "foo"
Any {getAny = False}
λ> isControlWanted "bar"
Any {getAny = False}

You'll get similar behaviour from isControlActive, and startup always returns () (unit).

Relaxation #

As Woolf wrote in the original text, what 'do nothing' means is subjective. I think it's perfectly reasonable to say that monoidal identity fits the description of doing nothing, but you could probably encounter APIs where 'doing nothing' means something else.

As an example, consider avatars for online forums, such as Twitter. If you don't supply a picture when you create your profile, you get a default picture. One way to implement such a feature could be by having a 'null' avatar, which is used in place of a proper avatar. Such a default avatar object could (perhaps) be considered a Null Object, but wouldn't necessarily be monoidal. There may not even be a binary operation for avatars.

Thus, it's likely that you could encounter or create Null Objects that aren't monoids. That's the reason that I don't claim that a Null Object always is monoidal identity, although I do think that the reverse relationship holds: if it's identity, then it's a Null Object.

Summary #

Monoids are associative binary operations with identity. The identity is the value that 'does nothing' in that binary operation, like, for example, 0 doesn't 'do' anything under addition. Monoids, however, can be more complex than operations on primitive values. Functions, for example, can be monoids as well, as can tuples of functions.

The implication of that is that objects can be monoids as well. When you have a monoidal object, then its identity is the 'natural' Null Object.

The question was: when can you implement the Null Object pattern? The answer is that you can do that when all involved methods return monoids.

Next: Builder as a monoid.

Comments

A variation of the Composite design pattern uses endomorphic composition. That's still a monoid.

This article is part of a series of articles about design patterns and their category theory counterparts. In a previous article, you learned that the Composite design pattern is simply a monoid.

There is, however, a variation of the Composite design pattern where the return value from one step can be used as the input for the next step.

Endomorphic API #

Imagine that you have to implement some scheduling functionality. For example, you may need to schedule something to happen a month from now, but it should happen on a bank day, during business hours, and you want to know what the resulting date and time will be, expressed in UTC. I've previously covered the various objects for performing such steps. The common behaviour is this interface:

public interface IDateTimeOffsetAdjustment
{
    DateTimeOffset Adjust(DateTimeOffset value);
}

The Adjust method is an endomorphism; that is: the input type is the same as the return type, in this case DateTimeOffset. A previous article already established that that's a monoid.

Composite endomorphism #

If you have various implementations of IDateTimeOffsetAdjustment, you can make a Composite from them, like this:

public class CompositeDateTimeOffsetAdjustment : IDateTimeOffsetAdjustment
{
    private readonly IReadOnlyCollection<IDateTimeOffsetAdjustment> adjustments;
 
    public CompositeDateTimeOffsetAdjustment(
        IReadOnlyCollection<IDateTimeOffsetAdjustment> adjustments)
    {
        if (adjustments == null)
            throw new ArgumentNullException(nameof(adjustments));
 
        this.adjustments = adjustments;
    }
 
    public DateTimeOffset Adjust(DateTimeOffset value)
    {
        var acc = value;
        foreach (var adjustment in this.adjustments)
            acc = adjustment.Adjust(acc);
        return acc;
    }
}

The Adjust method simply starts with the input value and loops over all the composed adjustments. For each adjustment it adjusts acc to produce a new acc value. This goes on until all adjustments have had a chance to adjust the value.

Notice that if adjustments is empty, the Adjust method simply returns the input value. In that degenerate case, the behaviour is similar to the identity function, which is the identity for the endomorphism monoid.

You can now compose the desired behaviour, as this parametrised xUnit.net test demonstrates:

[Theory]
[InlineData("2017-01-31T07:45:55+2", "2017-02-28T07:00:00Z")]
[InlineData("2017-02-06T10:03:02+1", "2017-03-06T09:03:02Z")]
[InlineData("2017-02-09T04:20:00Z" , "2017-03-09T09:00:00Z")]
[InlineData("2017-02-12T16:02:11Z" , "2017-03-10T16:02:11Z")]
[InlineData("2017-03-14T13:48:29-1", "2017-04-13T14:48:29Z")]
public void AdjustReturnsCorrectResult(
    string dtS,
    string expectedS)
{
    var dt = DateTimeOffset.Parse(dtS);
    var sut = new CompositeDateTimeOffsetAdjustment(
        new NextMonthAdjustment(),
        new BusinessHoursAdjustment(),
        new DutchBankDayAdjustment(),
        new UtcAdjustment());
 
    var actual = sut.Adjust(dt);
 
    Assert.Equal(DateTimeOffset.Parse(expectedS), actual);
}

You can see the implementation for all four composed classes in the previous article. NextMonthAdjustment adjusts a date by a month into its future, BusinessHoursAdjustment adjusts a time to business hours, DutchBankDayAdjustment takes bank holidays and weekends into account in order to return a bank day, and UtcAdjustment convert a date and time to UTC.

Monoidal accumulation #

As you've learned in that previous article that I've already referred to, an endomorphism is a monoid. In this particular example, the binary operation in question is called Append. From another article, you know that monoids accumulate:

public static IDateTimeOffsetAdjustment Accumulate(
    IReadOnlyCollection<IDateTimeOffsetAdjustment> adjustments)
{
    IDateTimeOffsetAdjustment acc =
        new IdentityDateTimeOffsetAdjustment();
    foreach (var adjustment in adjustments)
        acc = Append(acc, adjustment);
    return acc;
}

This implementation follows the template previously described:

• Initialize a variable acc with the identity element. In this case, the identity is a class called IdentityDateTimeOffsetAdjustment.
• For each adjustment in adjustments, Append the adjustment to acc.
• Return acc.

[Theory]
[InlineData("2017-01-31T07:45:55+2", "2017-02-28T07:00:00Z")]
[InlineData("2017-02-06T10:03:02+1", "2017-03-06T09:03:02Z")]
[InlineData("2017-02-09T04:20:00Z" , "2017-03-09T09:00:00Z")]
[InlineData("2017-02-12T16:02:11Z" , "2017-03-10T16:02:11Z")]
[InlineData("2017-03-14T13:48:29-1", "2017-04-13T14:48:29Z")]
public void AccumulatedAdjustReturnsCorrectResult(
    string dtS,
    string expectedS)
{
    var dt = DateTimeOffset.Parse(dtS);
    var sut = DateTimeOffsetAdjustment.Accumulate(
        new NextMonthAdjustment(),
        new BusinessHoursAdjustment(),
        new DutchBankDayAdjustment(),
        new UtcAdjustment());
 
    var actual = sut.Adjust(dt);
 
    Assert.Equal(DateTimeOffset.Parse(expectedS), actual);
}

While the implementation is different, this monoidal composition has the same behaviour as the above CompositeDateTimeOffsetAdjustment class. This, again, emphasises that Composites are simply monoids.

Endo #

For comparison, this section demonstrates how to implement the above behaviour in Haskell. The code here passes the same test cases as those above. You can skip to the next section if you want to get to the conclusion.

Instead of classes that implement interfaces, in Haskell you can define functions with the type ZonedTime -> ZonedTime. You can compose such functions using the Endo newtype 'wrapper' that turns endomorphisms into monoids:

λ> adjustments = reverse [adjustToNextMonth, adjustToBusinessHours, adjustToDutchBankDay, adjustToUtc]
λ> :type adjustments
adjustments :: [ZonedTime -> ZonedTime]

λ> adjust = appEndo $ mconcat $ Endo <$> adjustments
λ> :type adjust
adjust :: ZonedTime -> ZonedTime

In this example, I'm using GHCi (the Haskell REPL) to show the composition in two steps. The first step creates adjustments, which is a list of functions. In case you're wondering about the use of the reverse function, it turns out that mconcat composes from right to left, which I found counter-intuitive in this case. adjustToNextMonth should execute first, followed by adjustToBusinessHours, and so on. Defining the functions in the more intuitive left-to-right direction and then reversing it makes the code easier to understand, I hope.

(For the Haskell connoisseurs, you can also achieve the same result by composing Endo with the Dual monoid, instead of reversing the list of adjustments.)

The second step composes adjust from adjustments. It first maps adjustments to Endo values. While ZonedTime -> ZonedTime isn't a Monoid instances, Endo ZonedTime is. This means that you can reduce a list of Endo ZonedTime with mconcat. The result is a single Endo ZonedTime value, which you can then unwrap to a function using appEndo.

adjust is a function that you can call:

λ> dt
2017-01-31 07:45:55 +0200

λ> adjust dt
2017-02-28 07:00:00 +0000

In this example, I'd already prepared a ZonedTime value called dt. Calling adjust returns a new ZonedTime adjusted by all four composed functions.

Conclusion #

In general, you implement the Composite design pattern by calling all composed functions with the original input value, collecting the return value of each call. In the final step, you then reduce the collected return values to a single value that you return. This requires the return type to form a monoid, because otherwise, you can't reduce it.

In this article, however, you learned about an alternative implementation of the Composite design pattern. If the method that you compose has the same output type as its input, you can pass the output from one object as the input to the next object. In that case, you can escape the requirement that the return value is a monoid. That's the case with DateTimeOffset and ZonedTime: neither are monoids, because you can't add two dates and times together.

At first glance, then, it seems like a falsification of the original claim that Composites are monoids. As you've learned in this article, however, endomorphisms are monoids, so the claim still stands.

Next: Null Object as identity.