Monoids (Solutions)

## Contents

In earlier parts of the book, we have made a few passing allusions to monoids and the `Monoid` type class (most notably when discussing MonadPlus). Now it is time to give them a more detailed look, and show what makes them useful.

## Introduction

A monoid (m, mappend, mempty) is a type m together with an associative operation mappend :: m -> m -> m (also called (<>) in Haskell) that combines two elements and a zero element mempty :: m which is the neutral element of mappend: Before describing them more formally let's see monoids in action.

### Examples

As an information introduction, let's take a look at a common pattern:

```> (5 + 6) + 10 == 5 + (6 + 10)
True
> (5 * 6) * 10 == 5 * (6 * 10)
True
> ("Hello" ++ " ") ++ "world!" == "Hello" ++ (" " ++ "world!")
True
```

This property is called associativity and doesn't just hold for those selected values but for all integers under addition, all integers under multiplication and all lists under concatenation, let's look at a few more examples:

```> 255 + 0 == 255 && 0 + 255 == 255
True
> 255 * 1 == 255 && 1 * 255 == 255
True
> [1,2,3] ++ [] == [1,2,3] && [] ++ [1,2,3] == [1,2,3]
True
```

Here 0 is the identity element when adding integers, 1 is the identity when multiplying them and [] is the identity when appending two lists: This should not surprise you. At this point you may have realised the connection.

• Integers form a monoid under addition where 0 is the unit: (Integer, (+), 0)
• Integers form a monoid under multiplication where 1 is the unit: (Integer, (*), 1)
• Lists form a monoid under concatenation: ([a], (++), [])

Since Integers form monoids under two distinct operations we instead create two new datatypes: Sum for addition and Product for multiplication.

```> import Data.Monoid
> Sum 5 <> Sum 6 <> Sum 10
Sum {getSum = 21}
> mconcat [Sum 5, Sum 6, Sum 10]
Sum {getSum = 21}
> getSum \$ mconcat \$ map Sum [5, 6, 10]
21
> getProduct \$ mconcat \$ map Product [5, 6, 10]
300
> mconcat ["5", "6", "10"]
"5610"
```

But how is this useful?

### Generalizing functions

We can generalize a function that concatenates three lists:

```threeConcat :: [a] -> [a] -> [a] -> [a]
threeConcat a b c = a ++ b ++ c
```

with:

```threeConcat' :: Monoid m => m -> m -> m -> m
threeConcat' a b c = a <> b <> c

-- > threeConcat' "Hello" " " "world!"
-- "Hello world!"
-- > threeConcat' (Sum 5) (Sum 6) (Sum 10)
-- Sum {getSum = 21}
```

Other functions like fold :: (Foldable t, Monoid m) => t m -> m from Data.Foldable use properties of monoids to reduce any foldable structure containing monoids into a single monoidal value:

```> fold ["Hello", " ", "world!"]
"Hello world!"
> fold (Just (Sum 10))
Sum {getSum = 10}
> fold Nothing :: Sum Integer
Sum {getSum = 0}
```

This way if we create our own container (like trees) containing our own monoidal data type, we can immediately make use of previously defined functions like fold!

Exercises
1. There is a second possible instance of `Monoid` for `Integer`, using multiplication instead of addition as the combining operation. Implement that instance.[1]
2. It is possible to define `mempty` and `mappend` in terms of `mconcat`. Write the definition of this. Figure out whether or not your definition automatically fullfills the above laws.

The `Monoid` type class (from Data.Monoid) captures this general concept:

```class Monoid a where
mempty  :: a
mappend :: a -> a -> a
mconcat :: [a] -> a
mconcat = foldr mappend mempty
```

and lists are defined as:

```instance Monoid [a] where
mempty  = []
mappend = (++)
```

and the third method mconcat is provided as bonus with a default implementation of `mconcat = foldr (++) []` which is equivalent to `concat` for lists.

It is legitimate to think of monoids as types which support appending, though some poetic license with respect to what is meant by "appending" is advisable, as the `Monoid` definition is extremely generic and not at all limited to data structures. Integer numbers, for instance, form a monoid with addition as "appending" and 0 as, well, zero:

```-- | Monoid under addition.
newtype Sum a = Sum { getSum :: a }

-- | Monoid under multiplication.
newtype Product a = Product { getProduct :: a }

instance Num a => Monoid (Sum a) where
mempty = Sum 0
Sum x `mappend` Sum y = Sum (x + y)

instance Num a => Monoid (Product a) where
mempty = Product 1
Product x `mappend` Product y = Product (x * y)
```

Monoids also have laws, which the instances of `Monoid` must follow. They are really simple, though; corresponding to the neutral element and associativity properties we mentioned:

```mempty <> x = x
x <> mempty = x
x <> (y <> z) = (x <> y) <> z
```

The `Monoid` class is defined in Data.Monoid, along with instances for many common types.

## Examples

For a class with such a pompous name, monoids might seem just too simple, or just plain boring. Do not let any such impression mislead you, though: they have many interesting applications. We will mention some of them in this section.

The `Writer` monad
A computation of type `Writer w a` returns a value of type `a` while producing accumulating output of type `w`. `w` is an instance of `Monoid`, and the bind operator of the monad uses `mappend` to accumulate the output. It is easy to picture `w` as a list type used for some kind of logging; we can use any monoid instance, though, which opens up many possibilities thanks to the generality of the class.
The `Foldable` class
`Foldable`, available from Data.Foldable, provide a simple way to generalize list folding (as well as many related `Data.List`) functions to other data structures.[2] The implementations in `Data.Foldable` take advantage of monoids by generating monoidal values from the elements of the data structure, which can then be trivially combined with `mappend`. Another role of the `Foldable` is in the definition of the `Traversable` class (in Data.Traversable), which generalizes `sequence` from `Control.Monad`.
Finger trees
Moving on from operations on data structures to data structure implementations, monoids can be used to implement finger trees, an efficient and very versatile data structure. Its implementation makes use of monoidal values as tags for the tree nodes; and different data structures - such as sequences, priority queues and search trees - can be obtained simply by changing the involved `Monoid` instance. This blog post, based on a paper by Ralf Hinze and Ross Patterson[3], contains a brief and accessible explanation on how monoids are used in finger trees.
Options and settings
In a wholly different context, monoids can be a handy way of treating application options and settings. Two examples are Cabal, the Haskell packaging system ("Package databases are monoids. Configuration files are monoids. Command line flags and sets of command line flags are monoids. Package build information is a monoid."[1]) and XMonad, a tiling window manager implemented in Haskell ("xmonad configuration hooks are monoidal."[2]).

## Homomorphisms

A function `f :: a -> b` between two monoids `a` and `b` is called a homomorphism if it preserves the monoid structure, so that:

```f mempty          = mempty
f (x `mappend` y) = f x `mappend` f y
```

For instance, `length` is an homomorphism between ([],++) and (Int,+)

```length []         = 0
length (xs ++ ys) = length x + length y
```

An interesting example "in the wild" of monoids and homomorphisms was identified by Chris Kuklewicz amidst the Google Protocol Buffers API documentation. Based on the quotes provided in the referenced comment, we highlight that the property that

```    MyMessage message;
message.ParseFromString(str1 + str2);
```

is equivalent to

```    MyMessage message, message2;
message.ParseFromString(str1);
message2.ParseFromString(str2);
message.MergeFrom(message2);
```

means that `ParseFromString` is a homomorphism. Translating it to Haskell and monoids, the following equations hold:

```parse :: String -> Message
-- these are just equations, not actual code.
parse []         = mempty
parse (xs ++ ys) = parse xs `merge` parse ys
```

Well, it doesn't hold perfectly, because parse can fail, but roughly so.

• Additional comments on `Monoid` usage in Cabal: [3]; [4].
1. If you check the documentation for `Data.Monoid`, you will find that both instances are defined for types in `Num`, through the means of two `newtype` wrappers.