This time we’re going to explore how functional programs express

variance. We’ll consider the same case as we did for TDA – getting a

stored value from a Map, and storing a new value in a Map.

### Retrieval

lookup :: Ord k => k -> Map k a -> Maybe a

In Haskell, the variance is expressed in the return type.

data Maybe a = Nothing | Just a

We might express this as follows:

variance âŠ‚ return type

### Storage

We want to put a new value into the map.

Once again, the entropy of this function is two, see the previous

variance post for further discussion.

Here’s the comprehensive version:

insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

This function actually allows the caller to do more than our original

requirement, so a little explanation of that type signature is perhaps

necessary.

(k -> a -> a -> a)

The caller passes in this function, which takes the key, the new value

and the old value and produces the value that will be stored (most

implementations will simply return the new value, rather than

performing a calculation, mind).

Data.Map will only call this function if a value is already

present – this information is present in the last part of the type signature:

k -> a -> Map k a -> (Maybe a, Map k a)

This is the part we’re going to focus on. Here we pass the key, new

value, and the map to operate on. Note that the return type contains

both the possibility of an old value (the presence of which tells us

our value transforming function was called) *and* the new map.

### Diversion one: encapsulation in FP

It’s starting to look as though functional programs eschew

encapsulation in favour of type signature totality (definitely a

thing). This is not true.

- What if
`Maybe`

didn’t export its two constructors?

I fear the answer for some may be “huh”? So, a brief explanation of

how this might work.

Haskell makes it relatively simple to obscure your implementation

details. At the moment, we know that:

Maybe a = Just a | Nothing

In fact though, we only get to know this because the original author

allowed those constructors out to play. It would be just as valid to

export two functions that did the same but obscure the constructors, like so:

module Maybe (Maybe,just,nothing) where data Maybe a = Just a | Nothing just :: a -> Maybe a just = Just nothing :: Maybe a nothing = Nothing instance Functor Maybe where fmap f (Just a) = Just $ f a fmap _ Nothing = Nothing

Now, at the point of invocation, we can’t pattern match any more. We

have to use the fact that `Maybe`

is an instance of Functor in order

to do anything with the value inside; there’s no way *at all* for us

to get it out.

Example: reversing the value, if it exists.

fmap reverse $ lookup "key" map

So, it looks like the core of the FP world; `Functor`

s and `Monad`

s

are very interested in not letting us execute a function on a value

directly; you pass the function you would like executed to the

enclosing context, and it can choose (based on the value inside) how

to execute it.

In particular in Haskell, syntactic sugar (`do`

,`<-`

, etc) is provided

to make it appear to the eye that this is not the case. That’s only

for humans, though; in reality the value never escapes.

### Uh, so what now?

We used the term “return type” on the assumption that it is the last

thing we see in the type signature of a function. This is either almost

true or almost completely false.

Tony

Morris

tells us that all Haskell functions take one argument, and, once

you’ve read that link, I’ll hope you’ll agree he is right.

This means that for a function which appears to take `n`

arguments

there are in fact `n - 1`

return types. Or, perhaps, they have one

return type, and it’s everything we see after the first `->`

.

We’ll stick with:

variance âŠ‚ return type

### Totalitarianism, dogma and consequences

That’s enough pretentious subheadings -Ed

Just as we distilled OOP into a very dogmatic TDA, we’ve taken a very

dogmatic view of what FP is. Effects are delayed until the last

possible moment, and mutable state is excised.

Oddly, unlike in the TDA case, our chosen implementation language,

Haskell, is very keen on enforcing this dogma. Attempting to perform

calculations or effects outside of a function’s type signature is* a compile

error.

From here on out, we’ll assume dogmatic FP `===`

FP and that non

dogmatic usages of FP languages are charlatanic. This isn’t

necessarily true, but it will save us a few words here and there.

(*almost always is. There is always `Debug.trace`

)

### Diversion 2: A mirror image

Cast your mind back to our purified TDA map insert. What would it look

like if we translate it to Haskell?

put :: (Ord k) => Map k a -> k -> a -> (a -> IO()) -> IO() -> IO()

You can think of `a -> IO()`

as being a bit like `Consumer`

and `IO()`

as being like `Runnable`

. Haskell’s lazy predisposition allows us to

pass around `putStrLn "nothing here"`

as an expression without

executing it.

Let’s rename that, and fiddle with argument order:

tda_insert :: (Ord k) => k -> a -> Map k a -> (a -> IO()) -> IO() -> IO()

Let’s also consider the core of the FP signature:

fp_insert :: (Ord k) => k -> a -> Map k a -> (Maybe a, Map k a)

Now, recall our stricter, non pattern matching `Maybe`

implementation

from earlier. This told us that `Maybe`

is actually a context for

executing functions that take an `a`

. It also provides the following:

maybe :: b -> (a -> b) -> Maybe a -> b

If we just plug `b = IO()`

into that, we get:

maybe :: IO() -> (a -> IO()) -> Maybe a -> IO()

and with some further shuffling, it becomes this:

maybe_ish :: Maybe a -> IO() -> (a -> IO()) -> IO()

Now, if we elide `Map k a`

from fp_insert (thus making the state

change implicit)

fp_insert_ish :: (Ord k) => k -> a -> Map k a -> Maybe a

…and now, we *almost* have

tda_insert = maybe_ish . fp_insert_ish

This is very interesting! It feels like we’ve come across a little

piece of symmetry between the two approaches.

One way to think of this, perhaps, is that our TDA `insert`

has

*inlined* `Maybe`

. It acts both as an associative container *and* as a

context for function execution. The only difference (mind out though,

it’s a big one) is that `fp_insert`

is far stricter about what those

functions are allowed to do.

We perhaps shouldn’t be surprised at this; what we’ve really found is

an artifact of tight encapsulation, which we introduced by definition

in both cases.

#### Why `compose`

isn’t quite working

`fp_insert_ish`

, as far as `.`

is concerned, has type

fp_insert_ish :: (Ord k) => k -> (a -> Map k a -> Maybe a)

and `maybe_ish`

is really wanting a `Maybe a`

to start with. One can bind

enough of `fp_insert_ish`

‘s parameters to get this to work; we’ll not

spend the time doing that here.

Having found some symmetry (admittedly with a little bit of hand

waving), let’s consider a less friendly *a*symmetry.

### Interactions between FP and OOP

A lot of hot air has been generated about the exciting possibility of

combining the powers of OOP and FP approaches, as if they are some

sort of transformer robots that get a attack bonus, amazing music and

lens flares whenever they touch.

It’s not particularly clear whether this is true. We won’t try to

answer this question here at all. All we can do here is look at

*TDA*‘s interaction with FP.

- Can FP code call TDA code?

*No*

The TDA approach deliberately obscures some variance in its type

signatures. You’d have to do some serious

gymnastics

to express the same thing in Haskell, and, once the core is infected

with mutability, the `IO()`

virus spreads inexorably outwards.

- Can TDA code call FP code?

*Just about.*

One can start to preach the church of return type in the core of an

application and gently push it outwards (TDA style callers can update a

mutable reference to the newly returned FP core, then send out the

(also returned) events).

This will cause some dissonance, but, and this is the very important

point, *only* at changeover points. Purity, unlike mutability, isn’t

upwardly mobile.

#### Diversion 3: Not today…

Whether this has any bearing on OOP / FP interweaving is left

temporarily as an exercise to the reader.

Serendipitously, Eric Meijer just wrote

something

excellent on the

wider question of hybrid OOP/FP approaches.

### Conclusion

We found a tiny `Maybe`

hidden in our TDA code; a symmetry due to the

encapsulation shared between our two approaches.

One wonders if other exercises TDAing the living daylights out of

other data structures would uncover any other fundamental types.

We found also one asymmetry between TDA and FP; implicit mutation

simply isn’t permitted in FP, so we can nest the approaches in one

direction only.

#### Extremely satisfying summary

Once we start to mutate, anyone above must mutate with us.

So: Pick the point of no return *very* carefully.

#### Next time

A selection from:

- Do our truths about TDA tell us anything about OOP?
- Does our TDA world bring us any new problems?
- Does our TDA world bring us any benefits?
- Will we ever get to talking about the Actor model?

#### Post script

There may be a small hiatus over the next couple of months while I

develop related material into a workshop for 2014′s SPA

Conference ( This

session is

mine ).