# One Reason Typeclasses Are Useful

In this post, we explore one way that typeclasses are useful. We do so by first painting ourselves into a corner while building a toy Common Lisp program, and then seeing how Coalton’s typeclasses can ameliorate the issues.

## A Graphics Library

Let’s write a graphics library. Or, a tad less ambitiously, let’s write some routines for performing transformations on 2D points, which might serve as the foundation of a graphics library. We’ll be interested in:

1. translations,
2. dilations, and
3. rotations.

All of these transformations have the effect of taking an $(x,y)$ point, and producing a new $(x',y')$ point. First, let’s go through what these mean in the language of coordinate pairs, and then how we’d approach implementing it in plain old Common Lisp.

### Translations

Translations represent “sliding” a point along a straight line. They indicate a change of position in a particular direction by a particular distance. We represent this by supplying a $d_x$ and a $d_y$ which tell us how much to slide in the $x$- and $y$-direction respectively. So, starting with a point $(x,y)$, and our new point would have coordinates $$x' = x + d_x$$ and $$y' = y + d_y.$$ In Lisp, we could write

(defun easy-translate (dx dy x y)
(values (+ x dx) (+ y dy)))


### Dilations

Dilation is a general term that typically refers to enlarging (or shrinking) something. But points can’t get bigger or smaller: they’re points. So dilation, in the context of a point, means moving closer or farther away from a fixed origin point. We move the point by scaling its distance from the origin by a certain factor. We could use dilations to implement features like pinch-to-zoom.

If we have a dilation factor of $z$ (for zoom), then our $x' = zx$ and our $y' = zy$.

(defun easy-dilate (z x y)
(values (* z x) (* z y)))


### Rotations

Rotations are pretty tricky. They usually require trigonometry to calculate. Suppose I have a point $(x,y)$, and I want to rotate it by $\theta$ radians (one radian is about $57.3^{\circ}$) about the origin in a counterclockwise direction. Using Math (TM), the new point $(x',y')$ will have coordinates $$x' = x\cos\theta-y\sin\theta$$ and $$y'=x\sin\theta+y\cos\theta.$$

It’s pretty gnarly, and we definitely prefer to bake this in a library so we don’t have to remember it.

If all we’re going to do is rotate points about the origin, that’s easy to do in Lisp.

(defun easy-rotate (theta x y)
(let ((new-x (- (* x (cos theta)) (* y (sin theta))))
(new-y (+ (* x (sin theta)) (* y (cos theta)))))
(values new-x new-y)))


Next, we’ll see how to rotate about an arbitrary point. We actually have all the tools ready to do that.

## Assembling Transformations

So are we done? Have we implemented a complete, if a bit spartan, library for 2D graphics? Well, sort of. Spartan is the operative word. We could assemble complicated animations and graphical transformations by stringing these functions together in various ways. For instance, let’s suppose we want to rotate a point $(x,y)$ about some center point $(C_x, C_y)$. We can accomplish this as follows.

(defun rotate-around-point (theta cx cy x y)
(setf (values x y) (easy-translate (- cx) (- cy) x y))
(setf (values x y) (easy-rotate theta x y))
(setf (values x y) (easy-translate cx cy x y))
(values x y))


Can you see what’s going on here? We first translate so our frame of reference is centered at the origin, then we do the rotation, then we translate back. In math, this “do X, do Y, do the opposite of X” is called a conjugation. Conjugations happen all the time in graphics programming, because sometimes one frame of reference is easier to work in compared to another. How could we encode a conjugation in Lisp? Well, first, let’s give some things names.

A conjugation is a recipe:

1. A transformation to set up the frame of reference; let’s call it $F$.
2. An action of interest we want to do in our set-up frame; let’s call it $A$.
3. An inverse transformation of our set-up. In math we usually write this as $F^{-1}$, which means “the inverse of $F$”.

So, in a sort of Lisp pseudocode, we’d want something like:

(defun conjugate (F A x y)
(setf (values x y) (transform F x y))
(setf (values x y) (transform A x y))
(setf (values x y) (inverse-transform F x y))
(values x y))


How in the world do we implement this? Well, we can get part of the way there. One approach is to be very functional about it. That is, we turn our transformations into transformers. What do I mean?

(defun translator (dx dy)
(lambda (x y)
(easy-translate dx dy x y)))

(defun horizontal-dilator (z)
(lambda (x y)
(easy-dilate z x y)))

(defun rotator (theta)
(lambda (x y)
(easy-rotate theta x y)))

(defun transform (transformer x y)
(funcall transformer x y))


This may look sort of silly if you’re not used to functional programming, but now we can create functions (more specifically, closures) that themselves represent a transformation. For example, we could call our fabled conjugate function like so:

(conjugate (translator -5 -5)
(rotator (/ pi 2))
x
y)


This would rotate a point $(x,y)$ about the center point $(5,5)$ at an angle of $\pi/2$ radians (which is $90^{\circ}$). But, we have a big issue. How do we write inverse-transform?

In this style of functional programming, we can’t. Closures don’t ordinarily1 let us peek at the data underneath, so we have no way to negate or reverse any numbers.

## Objects to the Rescue!

Every enterprising Lisp programmer knows the next step is to get everything represented as objects2.

(defclass translation ()

(defclass dilation ()

(defclass rotation ()


Now that we have objects, we can implement a generic function for handling the actual act of transforming points. We can even re-use all of our easy- functions.

(defgeneric transform (xform x y))

(defmethod transform ((xform translation) x y)
(easy-translate (translation-dx xform)
(translation-dy xform)
x
y))

(defmethod transform ((xform dilation) x y)
(easy-dilate (dilation-factor xform) x y))

(defmethod transform ((xform rotation) x y)
(easy-rotate (rotaton-angle xform) x y))


We’ve created some objects and have implemented a simple protocol for applying those translations to points. We can add to that protocol, like making a generic function to invert a transformation. This was critical functionality we couldn’t do with closures.

(defgeneric inverse (xform))

(defmethod inverse ((xform translation))
(make-instance 'translation :dx (- (translation-dx xform))
:dy (- (translation-dy xform))))

(defmethod inverse ((xform dilation))
(make-instance 'dilation :factor (/ (dilation-factor xform))))

(defmethod inverse ((xform rotation))
(make-instance 'rotation :angle (- (rotation-angle xform))))


Now we can confidently write our conjugate function!

(defun inverse-transform (xform x y)
(transform (inverse xform) x y))

(defun conjugate (F A x y)
(setf (values x y) (transform F x y))
(setf (values x y) (transform A x y))
(setf (values x y) (inverse-transform F x y))
(values x y))


And just like that, our pseudocode has become real, working code.

At this point, we can go down the rabbit hole deep. For example, why not have a conjugation transformation? It could be a class like

(defclass conjugation ()
((frame-transformation ...)
(action-transformation ...)))


Or perhaps we generalize even more and create a class composite-transformation to encode a sequence of transformations. As long as we implement our protocol methods transform and inverse, we should be good to go.

## Combining Transformations

Another protocol function that seems obvious to implement is a way to combine two transformations. It’s pretty easy; the compound effect of two of the same classes of transformation is not too mysterious. Rotation angles add. Translation distances add. Dilation factors multiply.

(defgeneric combine (a b))

(defmethod combine ((a translation) (b translation))
(make-instance 'translation :dx (+ (translation-dx a)
(translation-dx b))
:dy (+ (translation-dy a)
(translation-dy b))))

(defmethod combine ((a dilation) (b dilation))
(make-instance 'dilation :factor (* (dilation-factor a)
(dilation-factor b))))

(defmethod combine ((a rotation) (b rotation))
(make-instance 'rotation :angle (+ (rotation-angle a)
(rotation-angle b))))


One nice aspect of the methods above is that they’re relatively efficient; we are eagerly combining transformations into new objects instead of somehow “queueing up” the transformations in a list. If we took a closure-based approach of combining, under the hood, all sorts of pointers would be stored. With enough calls to combine, in a closure-based implementation, we would be generating large, inefficient trees of closures that consume lots of time to execute and memory to store.

As discussed in the last section, if we had some sort of composite-transformation, then we could easily handle combine receiving different classes. But, it won’t be necessary for our discussion.

## A Transformation That Does Nothing

I’m not joking when I say this: One of the most important transformations is one that doesn’t do anything. Everyone agrees 0 is a useful number, which does nothing when you add; and 1 is a useful number, which does nothing when you multiply. A transformation that keeps a point put is also useful.

There are both practical and mathematical reasons for this. One super practical reason for a do-nothing transformation is that it’s a sensible default. Maybe somebody is using your graphics library to implement electrical circuit design software. This design software has settings for the viewport, like where we are centered and how zoomed in we are. “Where are we centered?” is answered by a translation transformation. Nominally we are centered at the origin, which would be a translation of nothing:

(defvar *identity-translation* (make-instance 'translation :dx 0 :dy 0))


We call these “do nothing” transformations identities, because the input to the transformation is identical to the output of it. We can make identities for the other ones too.

(defvar *identity-dilation* (make-instance 'dilation :factor 1))
(defvar *identity-rotation* (make-instance 'rotation :angle 0))


These variables could be used as defaults in our circuit design program to indicate the viewport is unperturbed.

Mathematically, we are interested in identity transformations because they’re very often the “base case” of many algorithms and procedures3. For instance, how would we write a function to make an $n$-fold transformation? For a rotation, it might look like this:

(defun n-fold-rotation (n rotation)
(make-instance 'rotation :angle (* n (rotation-angle rotation))))


It seems silly that we had to implement that for a specific class, and it seems silly that we ought to implement n-fold as a protocol function. Don’t we have the tools to do it generically? Isn’t that what our generic function combine is for? Yes… at least in principle.

(defun n-fold (n xform)
(if (zerop n)
'???
(combine xform (n-fold (1- n) xform))))


This procedure4 doesn’t account for the base case ??? which we need to fill in. What are our options here?

One option is to just not support the case of $n=0$, but that seems especially error prone because most idiomatic Common Lisp operators (e.g., expt, +, and) allow for zero or empty arguments. Another is to hard-code a typecase of some sort, but that defeats the point of having a grand object-oriented system. Our last resort option is to define one of two kinds of generic functions that act as getters.

;; Option #1
(defgeneric get-identity1 (proto-object))

(defmethod get-identity1 ((proto-object translation))
*identity-translation*)

;; Option #2
(defgeneric get-identity2 (class-name))

(defmethod get-identity2 ((class-name (eql 'translation)))
*identity-translation*)


Of course, imagine we’ve written these for all of our transformation classes.

Neither of these options is particularly satisfying. The function get-identity1 requires a manifested object of our desired type to be present (what if it’s not?); and get-identity2 requires a concrete class name at run-time. (Here, we’ll use run-time introspection to find the class name.) So, n-fold would be implemented as follows:

(defun n-fold1 (n xform)
(if (zerop n)
(get-identity1 xform)
(combine xform (n-fold (1- n) xform))))

(defun n-fold2 (n xform)
(if (zerop n)
(get-identity2 (class-name (class-of xform)))
(combine xform (n-fold (1- n) xform))))


This code is starting to smell. Before we had a protocol based off of generic functions to define the behavior of our transformations. But now, we are assigning distinguished ones to global variables and providing inelegant access to them by prototype objects or class names.

Astute Lispers might argue that for n-fold, the situation isn’t so bleak, because one of the input arguments can always be repurposed as a prototype object. I agree, but it’s a first example that shows cracks. A slight generalization would be a function that takes a list of like-typed transformations and combines them.

(defun compress (xforms)
(reduce #'combine xforms :initial-value '???))


Like n-fold, we need to choose an initial value for the base case. But unlike n-fold, we have no guarantees a prototype object will exist, because empty lists can be passed in. Again, we can skirt the issue by just disallowing empty lists, but it limits the utility of the function. We can alternatively opt to thread a prototype object through:

(defun compress (proto xforms)
(reduce #'combine xforms :initial-value (get-identity1 proto)))


but now compress is made inconvenient to use by the callers. We’ve passed the buck from callee to caller. Ideally we could just write

(defun compress (xforms)
(reduce #'combine xforms :initial-value generic-identity))


that allows Lisp to (somehow) determine what specific identity the variable generic-identity should be interpreted as.

Another way to see this pain is if we revisit implementing our conjugation class. It would be nice if we could default the transformations to be identity.

(defclass conjugation ()
((frame-transformation :initform generic-identity)
(action-transformation :initform generic-identity)))


The situation is even worse than compress: we can’t do this with our generic-getter approach above and there’s no way to “thread” a prototype object in without specializing initialize-instance with extra keyword arguments. Something like generic-identity doesn’t exist5; either of the get-identity functions has to be called on something to produce an identity transformation, and that something isn’t available to us when writing initforms.

Briefly summarized, Common Lisp only permits generic functions, not generic values. This doesn’t mean Lisp is doomed; it just means Lisp is a little less effective at dealing with a certain concept. There are lots of workarounds Lispers employ to get by, but each compromises the design of the program in some way.

## Setting the Stage for Typeclasses

Typeclasses are a way to solve this problem. Typeclasses allow us to associate almost any function or value with one or multiple types. Unfortunately, Common Lisp does not have typeclasses, so we’ll have to use an embedded domain-specific language called Coalton to make sense of them. Before we dive into that, we need to recapitulate what we’ve written so far about transformations, ported to Coalton. Fortunately, it’s easy.

We can define a new data type in Coalton with define-type.

;;                   BOA constructors
;;           type names   |||||          slots
;;           vvvvvvvvvvv  vvvvv vvvvvvvvvvvvvvvvvvvvvvvvv
(define-type Point       (Pt    Single-Float Single-Float))
(define-type Translation (Trans Single-Float Single-Float))
(define-type Rotation    (Rot   Single-Float))
(define-type Dilation    (Dil   Single-Float))


These should be read as follows:

“A Point is a type that is constructed by calling Pt on two Single-Float values. A Translation is a type that is constructed by calling Trans on two Single-Float values. A Rotation is a type that is constructed by calling Rot on one Single-Float value. A Dilation is a type that is constructed by calling Dil on one Single-Float value.”

If that was boring to read, that was intentional. These define-types are extremely simple; they try to be as minimal as possible, without requiring slot names, getters, initargs, or any of that to be specified.

The define-type operator is very flexible; we are just seeing the surface. Full-blown algebraic data types with parametric type variables are possible with define-type, but we won’t get into that.

As you’d see in any Haskell or OCaml book, we can write simple functions on these types. Maybe the simplest are functions to extract the $x$- and $y$-coordinates from a Point.

(define (coord-x p)
(match p
((Pt x _) x)))

(define (coord-y p)
(match p
((Pt _ y) y)))


This doesn’t do any computation, just pulls the innards of a Point out, by using pattern matching and destructuring.

A little more complicated, let’s write a function that calculates the distance that a Translation travels.

(declare distance (Translation -> Single-Float))
(define (distance t)
(match t
((Translation dx dy) (sqrt (+ (* dx dx) (* dy dy))))))


This is similar to the last example, but we are actually computing something with the innards. In this case, and indeed almost all cases, the declare is totally optional, and is automatically inferred by the Coalton compiler.

Surprisingly, this is all we need to start talking about typeclasses.

## Typeclasses Are Structured Protocols

A typeclass is like a protocol where we have to specify our protocol functions upfront, along with their inputs and outputs. Equally importantly, we can also require that values are specified as a part of this protocol. In Coalton, we represent transformations as a typeclass like so.

(define-class (Transformation :t)
(transform (:t -> Point -> Point))
(inverse   (:t -> :t))
(combine   (:t -> :t -> :t))
(identity  (:t)))


“The generic type :t is a Transformation if it has defined methods transform, inverse, and combine; as well as a distinguished value called identity.”

In addition, as can be seen, the precise types of each of these are spelled out. The syntax

(:t -> :t -> :t)


should be read6 as “a function that needs two :t’s to produce a :t.” Similarly,

(:t -> Point -> Point)


should be read as “a function that takes a :t and a Point and produces a Point. Simplest of all,

(:t)


reads “an element of type :t.”

Maybe you’ve anticipated it, but now we can opt-in our transformation data types into this typeclass. We instantiate the typeclass with types. Let’s start with dilation, because it’s the least to type.

(define (dilation-factor d)    ; convenience getter
(match d
((Dil factor) factor)))

(define-instance (Transformation Dilation)
(define (transform d p)
(let ((f (dilation-factor d)))
(Pt (* f (coord-x p)) (* f (coord-y p)))))

(define (inverse d)
(Dil (/ 1.0 (dilation-factor d))))

(define (combine d1 d2)
(Dil (* (dilation-factor d1)
(dilation-factor d2))))

(define identity (Dil 1.0)))


We could do precisely the same for each of the other types. It leads to exceedingly compact code, while also being completely type-safe. It is not possible in Coalton to have any of your instance functions consume or produce an incorrect type. It will all be statically verified to be correct when your code compiles.

So, assuming we’ve finished writing

(define-instance (Transformation Translation) ...)
(define-instance (Transformation Rotation)    ...)


surely we should be able to write conjugate and n-fold? Let’s start with what conjugate’s formal type should be.

(declare conjugate ((Transformation :s) (Transformation :t)
=> :s -> :t -> Point -> Point))


Everything before the => are constraints. They tell us what has to be true about type variables like :s and :t. Everything after => is the ordinary type. So in this case, we read

“Given transformation types :s and :t, conjugate is a function that takes an :s, a :t, and a Point, and produces as output a new Point.”

The type of n-fold is even easier.

(declare n-fold ((Transformation :t) => Integer -> :t -> :t))


It reads similarly. What are the implementations then?

(define (conjugate F A p)
(pipe p (transform F)
(transform A)
(transform (inverse F))))

(define (n-fold n xform)
(if (== n 0)
identity
(combine xform (n-fold (- n 1) xform))))


Our compress function is equally succinct.

(declare compress ((Transformation :t) => (List :t) -> :t))
(define (compress xforms)
(fold combine identity xforms))


That’s it. Notice how we could simply write identity there as a bare variable. The Coalton compiler will figure out what identity you want, and guarantee an identity exists for your type :t.

## Typeclasses in the Big Picture

Typeclasses are like formalized protocols, but in some ways they’re more flexible because Coalton does work to determine the type of everything. Just like in Common Lisp, protocols don’t have to be about just one object class, they can also stipulate relationships between object classes. One of the classic examples in Lisp is a draw protocol:

(defgeneric draw (object medium))


We can specialize draw to work with any combination of object and medium classes, thanks to multiple dispatch. Similarly in Coalton, we can make a typeclass:

(define-class (Drawable :obj-type :med-type)
(draw (:obj-type -> :med-type -> Unit)))


Drawable is a way to specify a relationship between an :obj-type and a :med-type, and we can program in as many of them as we want.

More importantly, though, and where Lisp falls short, is that typeclasses can establish relationships with and effectively dispatch on return types. For example, consider the typeclass7

(define-class (Convertible :from :to)
(convert (:from -> :to)))

(define-instance (Convertible Boolean Integer)
(define (convert b) (if b 1 0)))


This typeclass says that a type :from can be converted to a type :to via the convert function. Critically, though, we don’t have to specify what :to must be anywhere we actually call or use convert, even as a high-order function.

Here, we give a concrete example of converting Boolean to Integer. The function middle is used to get the middle value of a list. We index at half the length. Due to zero-based indexing, this overshoots by one element when it’s even, so we subtract one in that case. In the following code, convert will automatically select the right output type based on the context of the call.

(define (middle xs)
(let ((len (length xs)))
(index xs (- (floor/ len 2)
(convert (even len))))))


Here, it is deduced that convert must return an Integer, and also that even returns a Boolean, so it selects the right instance of Convertible to convert the Boolean value into an integer one. At no point did we or do we or will we need to be explicit about that, as it’s something that can be inferred.

There is no equivalent concept in Common Lisp, because all types are dynamic. The closest we have in Common Lisp is using eql specializers or similar:

1. cl:map takes a symbol as its first argument which denotes the output type of the call. So (map 'string #'code-char xs) will return a string, because we said so explicitly. Common Lisp does not allow map to deduce its return type based off of the surrounding context, and effectively cripples map from being used polymorphically.

2. cl:coerce takes a symbol as a second argument which denotes how a value should be “coerced” into another value. The maladies are equivalent to cl:map; it cannot be used polymorphically in most circumstances.

These are mere examples from the Common Lisp standard library. The same sorts of patterns show up all the time in ordinary Common Lisp code.

## Conclusion

Common Lisp is renowned for its powerful object system, CLOS. Protocol-oriented programming in Common Lisp has demonstrated very effective in lots of application areas. However, the approach is easily suffocated as soon as we have some sort of dependence on output value. We saw two examples of this:

1. The case that a variable itself is the “output value”, like the identity element of a certain data type.
2. The case where we have a function which is polymorphic on its output, and whose output depends on its relationship with its input, like the function convert or its spiritual Common Lisp equivalent cl:coerce.

As such, we can conclude that typeclasses are a powerful way to express dynamic, polymorphic code with the benefits and guarantees of static typing.

1. There’s a whole section in the book Structure and Interpretation of Computer Programs that essentially abuses closures enough to allow the programmer to access their internals. But, they’re effectively making an object system, and foregoing the “functional"ness of closures. ↩︎

2. While it’s sometimes useful to make an abstract base class, it’s not required for our illustration, so we don’t. ↩︎

3. They’re also a critical object in the field of abstract algebra. One often stipulates the existence of an identity element as the first order of business when defining a new algebraic structure. ↩︎

4. We could easily write this as a loop or as a tail-recursive procedure. ↩︎

5. Another approach is to make another class entirely called identity-transformation whose sole purpose is to express a singleton value that acts as a generic identity. If we did this, we would need to implement relationships between our concrete transformation classes and this bespoke identity-transformation class. At the end of the day, such a construct gets us no closer to being able to work with identity values of given transformation classes in a generic way. ↩︎

6. There is more nuance to this. It has to do with the fact that functions in Coalton only accept one argument, technically. ↩︎

7. In the Coalton standard library, the typeclass Into takes the role of Convertible, and the instance method into takes the role of convert↩︎