## Tuesday, October 13, 2015

### The uninteresting monoids of certain monads

Suppose there is some structure from which arises a monad. Let’s call one Sem.

data Sem a = ... -- doesn't matter

In the spirit of defining every typeclass instance you can think of—a spirit that I share, believe me—you discover a monoid, and suggest that it be included with Sem.

instance ??? => Monoid (Sem a) where
-- definition here

But then, you are surprised to encounter pessimism and waffling, from me!

I’m so skeptical of your monoid because it is “common”; many monoids simply fall out of numerous monads, to greater or lesser degree, but that doesn’t make them “good” monoids. Having rediscovered a common, uninteresting monoid, you need to provide more justification of why it should be “the” monoid for this data type.

## The lifted monoid

Every applicative functor gives rise to a monoid that lifts their arguments’ monoid.

instance Monoid a => Monoid (Sem a) where
mempty = pure mempty
mappend = liftA2 mappend

This is “the” monoid for (->) r and Maybe. It is decidedly not the monoid for []. For in that universe,

> [Sum 2] `mappend` [Sum 3, Sum 7]
[Sum 5, Sum 9]
> [Sum 42] `mappend` []
[]

Maybe you reaction is “but that’s not a legal monoid!” Sure it is. The mappend is based on combination, just as Applicative []’s <*> is. And, in the example above, the left and right identity is [Sum 0], not [].

It’s just not the monoid you’re used to.

Moreover, it isn’t quite right for Maybe! The constraint generalizes to Semigroup a. It is an unfortunate accident of history that the constraint on Haskell Maybe’s monoid is also Monoid.

Even the choice for (->) r makes many people unhappy, though we’re not quite ready to explore the reason for that.

So, what makes you think this is a good choice for Sem? It’s not enough justification that it can be written; that is always the case. There must be something that makes Sem like (->) r or Maybe, and not like [].

To be entirely modern, this would be the Alternative monoid. Despite the possibilities for equivocation, this monoid is just as good as any other.

Simply: every Alternative (a subclass of Applicative and a superclass of the more well-known MonadPlus) gives rise to a monoid that is universal over the argument, no Monoid constraint required.

-- supposing Alternative Sem,
instance Monoid (Sem a) where
mempty = empty
mappend = (<|>)

You would not be surprised at this having prepared by reading the haddock for Alternative: “a monoid on applicative functors”, it says.

[] is Alternative, and indeed this is the monoid of choice for []. But Maybe is also Alternative. Why is this one good for [], but not Maybe? Let’s take a peek through the looking glass.

> Just 1 `mappend` Just 4
Just 1
> Nothing `mappend` Just 3
Just 3

I happen to agree with the monoid of choice for Maybe. But I’m sure many have been surprised it’s not “just take the leftmost Just, or give Nothing”.

Except where phantom Const-style functors are involved, the two preceding monoids always have incompatible behavior. One sums the underlying values, the other never touchs them, only rearranging them. So, if both are available to Sem, to define a monoid, we must give up at least one of these.

Alternatively, we could put off the decision until someone comes up with a convincing argument for “the” monoid.

## The category endomorphism monoid

This monoid hasn’t let the lack of a pithy name handicap it; despite the stunning blow of losing the prized (->) to the lifted monoid (the commit), this one probably has even more fans eager for a rematch today than it did back then.

I’m referring to this one, still thought of as “the” monoid for (->) by some.

instance Monoid (a -> a) where
mempty = id
mappend = (.)

The elegance of this kind of “summing” of functions is undeniable. Moreover, it applies to every Category, not just (->). Even more, it works for anything sufficiently Category-ish, such as ReaderT.

Its fatal flaw is that twin appearance of a; it requires FlexibleInstances, so can’t be written in portable Haskell 2010. As such, it will probably remain in the minor leagues of newtypes like Endo.

Moreover, should you discover it for Sem, its applicability to any category-ish thing should still give you pause.

## The burden of proof

In Haskell, hacking until it compiles is a great way to work. It is tempting to rely on its conclusions in ever more cases, once you have discovered its effectiveness. However, in the cases above, it is very easy to be led astray by the facile promises of the typechecker.

Introducing one of these monoids is risky. It precludes the later introduction of the “right” monoid for a datatype, for want of compatibility. If you really must offer one of these monoids as “the” monoid for a datatype, the responsibility falls to you: demonstrate that this is a good monoid, not just an easy one.

#### 1 comment:

Edward Kmett said...

The issue with the

instance Monoid (a -> a)

is that it has _terrible_ type inference. The FlexibleInstances requirement is the compiler telling you not just that "hey this isn't Haskell 98" but that it may not be able to help you without type annotations.

Simple code like `const 1 <> (+) 2` will fail to work with that instance. Why? Well, they are too polymorphic for the instance to resolve.

instance a ~ b => Monoid (a -> b)

would resolve that issue at least, but now drags you even _farther_ into exotic type extensions land.