Or, how a 19th century scientist nearly invented the state monad

07 Jul 2015

*UPDATE: Slides and video from my talk on this topic*

*Warning! This post contains gruesome topics, strained analogies, discussion of monads*

For generations, we have been captivated by the tragic story of Dr Frankenfunctor. The fascination with vital forces, the early experiments with electricity and galvanism, and finally the breakthrough culminating in the bringing to life of a collection of dead body parts – the Monadster.

But then, as we all know, the creature escaped and the free Monadster rampaged through computer science conferences, bringing fear to the hearts of even the most seasoned programmers.

*CAPTION: The terrible events at the 1990 ACM Conference on LISP and Functional Programming.*

I will not repeat the details here; the story is still too terrible to recall.

But in all the millions of words devoted to this tragedy, one topic has never been satisfactorily addressed.

*How was the creature assembled and brought to life?*

We know that Dr Frankenfunctor built the creature from dead body parts, and then animated them in a single instant, using a bolt of lightning to create the vital force.

But the various body parts had to be assembled into a whole, and the vital force had to be transmitted through the assembly in the appropriate manner, and all this done in a split second, in the moment that the lightning struck.

I have devoted many years of research into this matter, and recently, at great expense, I have managed to obtain Dr Frankenfunctor’s personal laboratory notebooks.

So at last, I can present Dr Frankenfunctor’s technique to the world. Use it as you will. I do not make any judgements as to its morality, after all, it is not for mere developers to question the real-world effects of what we build.

To start with, you need to understand the fundamental process involved.

First, you must know that no whole body was available to Dr Frankenfunctor. Instead, the creature was created from an assemblage of body parts – arms, legs, brain, heart – whose provenances were murky and best left unspoken.

Dr Frankenfunctor started with a dead body part, and infused it with some amount of vital force. The result was two things: a now live body part, and the remaining, diminished, vital force, because of course some of the vital force was transferred to the live part.

Here is a diagram demonstrating the principle:

But this creates only *one* body part. How can we create more than one? This is the challenge that faced Dr Frankenfunctor.

The first problem is that we only have a limited quantity of the vital force. This means that when we need to animate a second body part, we have available only the remaining vital force from a previous step.

How can we connect the two steps together so that the vital force from the first step is fed into the input of the second step?

Even if we have chained the steps correctly, we need to take the various live body parts and combine them somehow. But we only have access to *live* body parts during the moment of creation.
How can we combine them in that split second?

It was Dr Frankenfunctor’s genius that led to an elegant approach that solved both of these problems, the approach that I will present to you now.

Before discussing the particulars of assembling the body parts, we should spend a moment on common functionality that is required for the rest of the procedure.

First, we need a label type. Dr Frankenfunctor was very disciplined in labeling the source of every part used.

```
type Label = string
```

The vital force we will model with a simple record type:

```
type VitalForce = {units:int}
```

Since we will be using vital force frequently, we will create a function that extracts one unit and returns a tuple of the unit and remaining force.

```
let getVitalForce vitalForce =
let oneUnit = {units = 1}
let remaining = {units = vitalForce.units-1} // decrement
oneUnit, remaining // return both
```

With the common code out of the way, we can return to the substance.

Dr Frankenfunctor’s notebooks record that the lower extremities were created first. There was a left leg lying around in the laboratory, and that was the starting point.

```
type DeadLeftLeg = DeadLeftLeg of Label
```

From this leg, a live leg could be created with the same label and one unit of vital force.

```
type LiveLeftLeg = LiveLeftLeg of Label * VitalForce
```

The type signature for the creation function would thus look like this:

```
type MakeLiveLeftLeg =
DeadLeftLeg * VitalForce -> LiveLeftLeg * VitalForce
```

And the actual implementation like this:

```
let makeLiveLeftLeg (deadLeftLeg,vitalForce) =
// get the label from the dead leg using pattern matching
let (DeadLeftLeg label) = deadLeftLeg
// get one unit of vital force
let oneUnit, remainingVitalForce = getVitalForce vitalForce
// create a live leg from the label and vital force
let liveLeftLeg = LiveLeftLeg (label,oneUnit)
// return the leg and the remaining vital force
liveLeftLeg, remainingVitalForce
```

As you can see, this implementation matched the earlier diagram precisely.

At this point Dr Frankenfunctor had two important insights.

The first insight was that, thanks to currying, the function could be converted from a function taking a tuple to a two parameter function, with each parameter passed in turn.

And the code now looked like this:

```
type MakeLiveLeftLeg =
DeadLeftLeg -> VitalForce -> LiveLeftLeg * VitalForce
let makeLiveLeftLeg deadLeftLeg vitalForce =
let (DeadLeftLeg label) = deadLeftLeg
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveLeftLeg = LiveLeftLeg (label,oneUnit)
liveLeftLeg, remainingVitalForce
```

The second insight was that this *same* code can be interpreted as a function that in turn returns a “becomeAlive” function.

That is, we have the dead part on hand, but we won’t have any vital force until the final moment, so why not process the dead part right now and return a function that can be used when the vital force becomes available.

In other words, we pass in a dead part, and we get back a function that creates a live part when given some vital force.

These “become alive” functions can then be treated as “steps in a recipe”, assuming we can find some way of combining them.

The code looks like this now:

```
type MakeLiveLeftLeg =
DeadLeftLeg -> (VitalForce -> LiveLeftLeg * VitalForce)
let makeLiveLeftLeg deadLeftLeg =
// create an inner intermediate function
let becomeAlive vitalForce =
let (DeadLeftLeg label) = deadLeftLeg
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveLeftLeg = LiveLeftLeg (label,oneUnit)
liveLeftLeg, remainingVitalForce
// return it
becomeAlive
```

It may not be obvious, but this is *exactly the same code* as the previous version, just written slightly differently.

This curried function (with two parameters) can be interpreted as a normal two parameter function,
or it can be interpreted as a *one parameter* function that returns *another* one parameter function.

If this is not clear, consider the much simpler example of a two parameter `add`

function:

```
let add x y =
x + y
```

Because F# curries functions by default, that implementation is exactly the same as this one:

```
let add x =
fun y -> x + y
```

Which, if we define an intermediate function, is also exactly the same as this one:

```
let add x =
let addX y = x + y
addX // return the function
```

Looking ahead, we can see that we can use a similar approach for all the functions that create live body parts.

All those functions will return a function that has a signature like: `VitalForce -> LiveBodyPart * VitalForce`

.

To make our life easy, let’s give that function signature a name, `M`

, which stands for “Monadster part generator”,
and give it a generic type parameter `'LiveBodyPart`

so that we can use it with many different body parts.

```
type M<'LiveBodyPart> =
VitalForce -> 'LiveBodyPart * VitalForce
```

We can now explicitly annotate the return type of the `makeLiveLeftLeg`

function with `:M<LiveLeftLeg>`

.

```
let makeLiveLeftLeg deadLeftLeg :M<LiveLeftLeg> =
let becomeAlive vitalForce =
let (DeadLeftLeg label) = deadLeftLeg
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveLeftLeg = LiveLeftLeg (label,oneUnit)
liveLeftLeg, remainingVitalForce
becomeAlive
```

The rest of the function is unchanged because the `becomeAlive`

return value is already compatible with `M<LiveLeftLeg>`

.

But I don’t like having to explicitly annotate all the time. How about we wrap the function in a single case union – call it “M” – to give it its own distinct type? Like this:

```
type M<'LiveBodyPart> =
M of (VitalForce -> 'LiveBodyPart * VitalForce)
```

That way, we can distinguish between a “Monadster part generator” and an ordinary function returning a tuple.

To use this new definition, we need to tweak the code to wrap the intermediate function in the single case union `M`

when we return it, like this:

```
let makeLiveLeftLegM deadLeftLeg =
let becomeAlive vitalForce =
let (DeadLeftLeg label) = deadLeftLeg
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveLeftLeg = LiveLeftLeg (label,oneUnit)
liveLeftLeg, remainingVitalForce
// changed!
M becomeAlive // wrap the function in a single case union
```

For this last version, the type signature will be correctly inferred without having to specify it explicitly: a function that takes a dead left leg and returns an “M” of a live leg:

```
val makeLiveLeftLegM : DeadLeftLeg -> M<LiveLeftLeg>
```

Note that I’ve renamed the function `makeLiveLeftLegM`

to make it clear that it returns a `M`

of `LiveLeftLeg`

.

So what does this “M” type mean exactly? How can we make sense of it?

One helpful way is to think of a `M<T>`

as a *recipe* for creating a `T`

. You give me some vital force and I’ll give you back a `T`

.

But how can an `M<T>`

create a `T`

out of nothing?

That’s where functions like `makeLiveLeftLegM`

are critically important. They take a parameter and “bake” it into the result.
As a result, you will see lots of “M-making” functions with similar signatures, all looking something like this:

Or in code terms:

```
DeadPart -> M<LivePart>
```

The challenge now will be how to combine these in an elegant way.

Ok, let’s test what we’ve got so far.

We’ll start by creating a dead leg and use `makeLiveLeftLegM`

on it to get an `M<LiveLeftLeg>`

.

```
let deadLeftLeg = DeadLeftLeg "Boris"
let leftLegM = makeLiveLeftLegM deadLeftLeg
```

What is `leftLegM`

? It’s a recipe for creating a live left leg, given some vital force.

What’s useful is that we can create this recipe *up front*, *before* the lightning strikes.

Now let’s pretend that the storm has arrived, the lightning has struck, and 10 units of vital force are now available:

```
let vf = {units = 10}
```

Now, inside the `leftLegM`

is a function which we can apply to the vital force.
But first we need to get the function out of the wrapper using pattern matching.

```
let (M innerFn) = leftLegM
```

And then we can run the inner function to get the live left leg and the remaining vital force:

```
let liveLeftLeg, remainingAfterLeftLeg = innerFn vf
```

The results look like this:

```
val liveLeftLeg : LiveLeftLeg =
LiveLeftLeg ("Boris",{units = 1;})
val remainingAfterLeftLeg : VitalForce =
{units = 9;}
```

You can see that a `LiveLeftLeg`

was created successfully and that the remaining vital force is reduced to 9 units now.

This pattern matching is awkward, so let’s create a helper function that both unwraps the inner function and calls it, all in one go.

We’ll call it `runM`

and it looks like this:

```
let runM (M f) vitalForce = f vitalForce
```

So the test code above would now be simplified to this:

```
let liveLeftLeg, remainingAfterLeftLeg = runM leftLegM vf
```

So now, finally, we have a function that can create a live left leg.

It took a while to get it working, but we’ve also built some useful tools and concepts that we can use moving forwards.

Now that we know what we are doing, we should be able to use the same techniques for the other body parts now.

How about a right leg then?

Unfortunately, according to the notebook, Dr Frankenfunctor could not find a right leg in the laboratory. The problem was solved with a hack… but we’ll come to that later.

Next, the arms were created, starting with the left arm.

But there was a problem. The laboratory only had a *broken* left arm lying around. The arm had to be healed before it could be used in the final body.

Now Dr Frankenfunctor, being a doctor, *did* know how to heal a broken arm, but only a live one. Trying to heal a dead broken arm would be impossible.

In code terms, we have this:

```
type DeadLeftBrokenArm = DeadLeftBrokenArm of Label
// A live version of the broken arm.
type LiveLeftBrokenArm = LiveLeftBrokenArm of Label * VitalForce
// A live version of a heathly arm, with no dead version available
type LiveLeftArm = LiveLeftArm of Label * VitalForce
// An operation that can turn a broken left arm into a heathly left arm
type HealBrokenArm = LiveLeftBrokenArm -> LiveLeftArm
```

The challenge was therefore this: how can we make a live left arm out the material we have on hand?

First, we have to rule out creating a `LiveLeftArm`

from a `DeadLeftUnbrokenArm`

, as there isn’t any such thing. Nor can we convert a `DeadLeftBrokenArm`

into a healthy `LiveLeftArm`

directly.

But what we *can* do is turn the `DeadLeftBrokenArm`

into a *live* broken arm and then heal the live broken arm, yes?

No, I’m afraid that won’t work. We can’t create live parts directly, we can only create live parts in the context of the `M`

recipe.

What we need to do then is create a special version of `healBrokenArm`

(call it `healBrokenArmM`

) that converts a `M<LiveBrokenArm>`

to a `M<LiveArm>`

.

But how do we create such a function? And how can we reuse `healBrokenArm`

as part of it?

Let’s start with the most straightforward implementation.

First, since the function will return an `M`

something, it will have the same form as the `makeLiveLeftLegM`

function that we saw earlier.
We’ll need to create an inner function that has a vitalForce parameter, and then return it wrapped in an `M`

.

But unlike the function that we saw earlier, this one has an `M`

as parameter too (an `M<LiveBrokenArm>`

). How can we extract the data we need from this input?

Simple, just run it with some vitalForce. And where are we going to get the vitalForce from? From the parameter to the inner function!

So our finished version will look like this:

```
// implementation of HealBrokenArm
let healBrokenArm (LiveLeftBrokenArm (label,vf)) = LiveLeftArm (label,vf)
/// convert a M<LiveLeftBrokenArm> into a M<LiveLeftArm>
let makeHealedLeftArm brokenArmM =
// create a new inner function that takes a vitalForce parameter
let healWhileAlive vitalForce =
// run the incoming brokenArmM with the vitalForce
// to get a broken arm
let brokenArm,remainingVitalForce = runM brokenArmM vitalForce
// heal the broken arm
let healedArm = healBrokenArm brokenArm
// return the healed arm and the remaining VitalForce
healedArm, remainingVitalForce
// wrap the inner function and return it
M healWhileAlive
```

If we evaluate this code, we get the signature:

```
val makeHealedLeftArm : M<LiveLeftBrokenArm> -> M<LiveLeftArm>
```

which is exactly what we want!

But not so fast – we can do better.

We’ve hard-coded the `healBrokenArm`

transformation in there. What happens if we want to do some other transformation, and for some other body part?
Can we make this function a bit more generic?

Yes, it’s easy. All we need to is pass in a function (“f” say) that transforms the body part, like this:

```
let makeGenericTransform f brokenArmM =
// create a new inner function that takes a vitalForce parameter
let healWhileAlive vitalForce =
let brokenArm,remainingVitalForce = runM brokenArmM vitalForce
// heal the broken arm using passed in f
let healedArm = f brokenArm
healedArm, remainingVitalForce
M healWhileAlive
```

What’s amazing about this is that by parameterizing that one transformation with the `f`

parameter, the *whole* function becomes generic!

We haven’t made any other changes, but the signature for `makeGenericTransform`

no longer refers to arms. It works with anything!

```
val makeGenericTransform : f:('a -> 'b) -> M<'a> -> M<'b>
```

Since it is so generic now, the names are confusing. Let’s rename it.
I’ll call it `mapM`

. It works with *any* body part and *any* transformation.

Here’s the implementation, with the internal names fixed up too.

```
let mapM f bodyPartM =
let transformWhileAlive vitalForce =
let bodyPart,remainingVitalForce = runM bodyPartM vitalForce
let updatedBodyPart = f bodyPart
updatedBodyPart, remainingVitalForce
M transformWhileAlive
```

In particular, it works with the `healBrokenArm`

function, so to create a version of “heal” that has been lifted to work with `M`

s we can just write this:

```
let healBrokenArmM = mapM healBrokenArm
```

One way of thinking about `mapM`

is that it is a “function converter”. Given any “normal” function, it converts it to a function where the input and output are `M`

s.

Functions similar to `mapM`

crop up in many situations. For example, `Option.map`

transforms a “normal” function into a function whose inputs and outputs are options.
Similarly, `List.map`

transforms a “normal” function into a function whose inputs and outputs are lists. And there are many other examples.

```
// map works with options
let healBrokenArmO = Option.map healBrokenArm
// LiveLeftBrokenArm option -> LiveLeftArm option
// map works with lists
let healBrokenArmL = List.map healBrokenArm
// LiveLeftBrokenArm list -> LiveLeftArm list
```

What might be new to you is that the “wrapper” type `M`

contains a *function*, not a simple data structure like Option or List. That might make your head hurt!

In addition, the diagram above implies that `M`

could wrap *any* normal type and `mapM`

could map *any* normal function.

Let’s try it and see!

```
let isEven x = (x%2 = 0) // int -> bool
// map it
let isEvenM = mapM isEven // M<int> -> M<bool>
let isEmpty x = (String.length x)=0 // string -> bool
// map it
let isEmptyM = mapM isEmpty // M<string> -> M<bool>
```

So, yes, it works!

Again, let’s test what we’ve got so far.

We’ll start by creating a dead broken arm and use `makeLiveLeftBrokenArm`

on it to get an `M<BrokenLeftArm>`

.

```
let makeLiveLeftBrokenArm deadLeftBrokenArm =
let (DeadLeftBrokenArm label) = deadLeftBrokenArm
let becomeAlive vitalForce =
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveLeftBrokenArm = LiveLeftBrokenArm (label,oneUnit)
liveLeftBrokenArm, remainingVitalForce
M becomeAlive
/// create a dead Left Broken Arm
let deadLeftBrokenArm = DeadLeftBrokenArm "Victor"
/// create a M<BrokenLeftArm> from the dead one
let leftBrokenArmM = makeLiveLeftBrokenArm deadLeftBrokenArm
```

Now we can use `mapM`

and `healBrokenArm`

to convert the `M<BrokenLeftArm>`

into a `M<LeftArm>`

:

```
let leftArmM = leftBrokenArmM |> mapM healBrokenArm
```

What we have now in `leftArmM`

is a recipe for creating a unbroken and live left arm. All we need to do is add some vital force.

As before, we can do all these things up front, before the lightning strikes.

Now when the storm arrives, and the lightning has struck, and vital force is available, we
can run `leftArmM`

with the vital force…

```
let vf = {units = 10}
let liveLeftArm, remainingAfterLeftArm = runM leftArmM vf
```

…and we get this result:

```
val liveLeftArm : LiveLeftArm =
LiveLeftArm ("Victor",{units = 1;})
val remainingAfterLeftArm :
VitalForce = {units = 9;}
```

A live left arm, just as we wanted.

On to the right arm next.

Again, there was a problem. Dr Frankenfunctor’s notebooks record that there was no whole arm available.
However there *was* a lower arm and an upper arm…

```
type DeadRightLowerArm = DeadRightLowerArm of Label
type DeadRightUpperArm = DeadRightUpperArm of Label
```

…which could be turned into corresponding live ones:

```
type LiveRightLowerArm = LiveRightLowerArm of Label * VitalForce
type LiveRightUpperArm = LiveRightUpperArm of Label * VitalForce
```

Dr Frankenfunctor decided to do surgery to join the two arm sections into a whole arm.

```
// define the whole arm
type LiveRightArm = {
lowerArm : LiveRightLowerArm
upperArm : LiveRightUpperArm
}
// surgery to combine the two arm parts
let armSurgery lowerArm upperArm =
{lowerArm=lowerArm; upperArm=upperArm}
```

As with the broken arm, the surgery could only be done with *live* parts. Doing that with dead parts would be yucky and gross.

But also, as with the broken arm, we don’t have access to the live parts directly, only within the context of an `M`

wrapper.

In other words we need to convert our `armSurgery`

function that works with normal live parts, and convert it into a `armSurgeryM`

function that works with `M`

s.

We can use the same approach as we did before:

- create a inner function that takes a vitalForce parameter
- run the incoming parameters with the vitalForce to extract the data
- from the inner function return the new data after surgery
- wrap the inner function in an “M” and return it

Here’s the code:

```
/// convert a M<LiveRightLowerArm> and M<LiveRightUpperArm> into a M<LiveRightArm>
let makeArmSurgeryM_v1 lowerArmM upperArmM =
// create a new inner function that takes a vitalForce parameter
let becomeAlive vitalForce =
// run the incoming lowerArmM with the vitalForce
// to get the lower arm
let liveLowerArm,remainingVitalForce = runM lowerArmM vitalForce
// run the incoming upperArmM with the remainingVitalForce
// to get the upper arm
let liveUpperArm,remainingVitalForce2 = runM upperArmM remainingVitalForce
// do the surgery to create a liveRightArm
let liveRightArm = armSurgery liveLowerArm liveUpperArm
// return the whole arm and the SECOND remaining VitalForce
liveRightArm, remainingVitalForce2
// wrap the inner function and return it
M becomeAlive
```

One big difference from the broken arm example is that we have *two* parameters, of course.
When we run the second parameter (to get the `liveUpperArm`

), we must be sure to pass in the *remaining vital force* after the first step, not the original one.

And then, when we return from the inner function, we must be sure to return `remainingVitalForce2`

(the remainder after the second step) not any other one.

If we compile this code, we get:

```
M<LiveRightLowerArm> -> M<LiveRightUpperArm> -> M<LiveRightArm>
```

which is just the signature we are looking for.

But as before, why not make this more generic? We don’t need to hard-code `armSurgery`

– we can pass it as a parameter.

We’ll call the more generic function `map2M`

– just like `mapM`

but with two parameters.

Here’s the implementation:

```
let map2M f m1 m2 =
let becomeAlive vitalForce =
let v1,remainingVitalForce = runM m1 vitalForce
let v2,remainingVitalForce2 = runM m2 remainingVitalForce
let v3 = f v1 v2
v3, remainingVitalForce2
M becomeAlive
```

And it has the signature:

```
f:('a -> 'b -> 'c) -> M<'a> -> M<'b> -> M<'c>
```

Just as with `mapM`

we can interpret this function as a “function converter” that converts a “normal” two parameter function into a function in the world of `M`

.

Again, let’s test what we’ve got so far.

As always, we need some functions to convert the dead parts into live parts.

```
let makeLiveRightLowerArm (DeadRightLowerArm label) =
let becomeAlive vitalForce =
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveRightLowerArm = LiveRightLowerArm (label,oneUnit)
liveRightLowerArm, remainingVitalForce
M becomeAlive
let makeLiveRightUpperArm (DeadRightUpperArm label) =
let becomeAlive vitalForce =
let oneUnit, remainingVitalForce = getVitalForce vitalForce
let liveRightUpperArm = LiveRightUpperArm (label,oneUnit)
liveRightUpperArm, remainingVitalForce
M becomeAlive
```

*By the way, are you noticing that there is a lot of duplication in these functions? Me too! We will attempt to fix that later.*

Next, we’ll create the parts:

```
let deadRightLowerArm = DeadRightLowerArm "Tom"
let lowerRightArmM = makeLiveRightLowerArm deadRightLowerArm
let deadRightUpperArm = DeadRightUpperArm "Jerry"
let upperRightArmM = makeLiveRightUpperArm deadRightUpperArm
```

And then create the function to make a whole arm:

```
let armSurgeryM = map2M armSurgery
let rightArmM = armSurgeryM lowerRightArmM upperRightArmM
```

As always, we can do all these things up front, before the lightning strikes, building a recipe (or *computation* if you like) that will do everything we need when the time comes.

When the vital force is available, we can run `rightArmM`

with the vital force…

```
let vf = {units = 10}
let liveRightArm, remainingFromRightArm = runM rightArmM vf
```

…and we get this result:

```
val liveRightArm : LiveRightArm =
{lowerArm = LiveRightLowerArm ("Tom",{units = 1;});
upperArm = LiveRightUpperArm ("Jerry",{units = 1;});}
val remainingFromRightArm : VitalForce =
{units = 8;}
```

A live right arm, composed of two subcomponents, just as required.

Also note that the remaining vital force has gone down to *eight*. We have correctly used up two units of vital force.

In this post, we saw how to create a `M`

type that wrapped a “become alive” function that in turn could only be activated when lightning struck.

We also saw how various M-values could be processed and combined using `mapM`

(for the broken arm) and `map2M`

(for the arm in two parts).

*The code samples used in this post are available on GitHub*.

This exciting tale has more shocks in store for you! Stay tuned for the next installment, when I reveal how the head and body were created.