A Type of Programming

by Renzo Carbonara


Computers blindly follow orders, and at some fundamental level, programming is about giving computers orders to follow. The expectation is that when a computer carries them out, it will achieve a particular goal a programmer had in mind. Coming up with these orders, however, is not easy. While computers are thorough and efficient, they are also rather limited in the vocabulary of instructions they can understand. They mostly know how to count and store data, which means our orders, as complex as they may be, can only be conveyed in those terms. Even deceptively simple programs can involve thousands or millions of these instructions, each of which needs to be correct and executed at the right time. And if we consider that computers won’t judge whether any of these instructions are right or wrong, some of which could have important consequences on our lives, it should be easy to appreciate how taking appropriate measures to prevent undesirable outcomes is the logical thing to do. In order to accomplish this, however, we need a different perspective.

Perhaps surprisingly, these computer instructions are a bad tool for reasoning about computer programs. Yet, it is only through reasoning that we can be confident about the correctness of our solutions, and more importantly, about the correctness of our problems. At the other end of the programming spectrum, far away from computers, we have our imagination. Programming is the conversation that happens while these two ends struggle to understand each other. We will explore this process and how to improve it.

This book doesn’t assume any previous programming knowledge, yet both newcomers and experienced programmers are welcome. Those approaching software development for the first time will discover a fascinating field while acquiring a good understanding of the principles and tools involved, whereas the experienced shall have their conceptions challenged by a different type of programming. We will use the Haskell programming language as our main vehicle, and while this book is not solely about Haskell but about programming, it will be very thorough at it.

But please, be patient. We prioritize the kind of understanding that stays with us for a very long time, the one that teaches questions. This means the topics in this book are presented in a rather unorthodox and entertaining fashion where immediate practicality never seems to be a goal. Don’t worry, we will get there. And when we do, we won’t need to look back.


We can’t really talk about programming without defining what a program is, which we can’t do unless we understand exactly what it is that we are trying to solve. For this, we first need to sample the world for suitable problems to tackle, a seemingly intractable endeavour unless we understand the limits and purpose of the field, for which we most certainly need to be able to talk about it. This is the very nature of new ideas, and why appreciating them is often so hard. We need to break this loop somehow. We need to acknowledge that there might be a problem to be tackled even if we can’t readily recognize it at first, and see how the proposed idea tries to approach it.

Breaking this loop is not a particularly hard thing to do in our imagination. A little dare, that’s all we need. There, boundaries and paradoxes are gone. We can fantasize allegedly impossible things, dwell on problems that they say can’t be solved. We can build, we can explore every future. To indulge in thought that challenges our understanding and that of others is not only interesting, it is a requirement. A curious and challenging mind is what it takes to make this journey.

But why would we do that? Why would we care? Well, look around. What do you see? There is a computer in our phone, there is another one in our car. There is one for talking to our families and another one for knowing how much money we have. There are computers that spy, there are computers that judge. There are computers that help businesses grow, and others that take jobs away. We even have computers saying whether we matter at all. Computers are our civilization, we have put them up there quite prominently. We care about programming because it is our voice, and only those who know how to speak stand a chance. It’s our power, freedom and responsibility to decide where to go next. We care because civilization is ours to build, ours to change, ours to care for.


Let’s picture ourselves in a not so distant past looking at our landline phone, looking at the elevator in our building, looking at our clock. Besides their usefulness, these machines have in common a single purpose. Not a shared single purpose, but a single different one each of them. The phone phones, the elevator elevates, and that is all they will ever do. But these machines, however disparate, at some fundamental level are essentially just interacting with their environment. Somehow they get inputs from it, and somehow they observably react according to pre-established logical decisions. We press a key and a phone rings. Another key and we are on a different floor. Yet, we rarely see these machines and their complex and expensive electronic circuitry repurposed to solve a different problem if necessary.

What happens is that the wires and electronics in these machines are connected in such a way that their accomplishments are mostly irrelevant to the wider electronic community. Like those steampunk contraptions, they each represent a solution to a finite understanding of one particular problem, and not more. But, is sending electricity to the engine that lifts the elevator really that different from sending it to the one that moves the hand of the clock? No, it is not. The logical decisions however, of when to do what and how, are. We say that the purpose of these machines is hardwired into their very existence, and it cannot be altered without physically modifying the machine itself. And yes, perhaps it is possible to adjust the time in a clock, but subtle variations like that one are considered and allowed upfront by the grand creators of the machine, who leave enough cranks, switches and wires inside that we can alter this behaviour somehow within a known limited set of possibilities.

But what about the unknown? If interacting with the environment is the essence of these machines, and moreover, if they are built with many of the same electronic components: Isn’t there some underlying foundation that would allow them to do more than what their upbringing dictates? Luckily for us, and for the steampunk fantasies that can now characterize themselves as nostalgic, there is. And this is what computers, otherwise known as general purpose machines, are all about.


What differentiates computers from other machines is that their purpose, instead of being forever imprinted in their circuitry by the manufacturer, is programmed into them afterwards using some programming language such as that Haskell thing we mentioned before. The general purpose machine is a canvas and a basic set of instructions that we can combine in order to achieve a purpose of our own choosing while interacting with the outside environment through peripherals such as keyboards, microphones or display screens. A computer program, in other words, is something that takes into account some inputs from the environment where it runs, and after doing some calculations, provides some kind of output in return. We can introduce some notation to represent this idea.

program :: input -> output

Prose is an excellent medium for conveying ideas to other human beings. However, prose can be ambiguous, it wastes words talking about non-essential things, and it allows for the opportunity to express a same idea in a myriad of different ways. These features, while appealing to the literary explorer, just hinder the way of the programmer. Computers can’t tell whether we ordered them to do the right thing, so we can’t afford to have ambiguities in our programs. We need to be precise lest we accomplish the wrong things. Irrelevant stuff may entertain us as great small talk material, but computers just don’t care, so we avoid that altogether. This is why a precise notation like the one above becomes necessary.

Perhaps surprisingly, our notation, also known as syntax, is part of a valid program written in the Haskell programming language, a language that we use to program computers, to tell them what to do. This is why among many other fascinating reasons we’ll come to cherish, this book uses Haskell as its main vehicle. The distance between our thoughts and our Haskell programs is just too small to ignore. We can read program :: input → output out loud as “a program from inputs into outputs”, and as far as Haskell is concerned, it would be alright.

Computers themselves don’t understand this Haskell notation directly, though. We use this notation for writing our programs because it brings us clarity, but computers don’t care about any of that. For this, we have compilers. Compilers are programs that take as input the description of our program, called source code, here specified using the Haskell programming language, and convert it to an executable sequence of instructions that computers can actually understand and obey. That is, we could say a compiler is itself a program :: source code → executable, if we wanted to reuse the same notation we used before.

Compiling is a complex, time-consuming and error prone operation, so it’s usually done just once and the resulting executable code is reused as many times as necessary. When we install a new program on our computer, for example, chances are we are installing its executable version, not its source code. Quite likely the author already compiled the program for us, so that we can start using our program right away rather than spend our time figuring out how to compile it ourselves first. This is not different from what happens when we buy a new car, say. The car has already been assembled for us beforehand, and from then on we can drive it whenever we want without having to re-assemble it.


What are our input and output, though, concretely? We don’t yet know exactly, but it turns out it doesn’t really matter for our very first program, the simplest possible one. Can we imagine what such program could possibly do? A program that, given some input, any input, will produce some output in return? If we think carefully about this for a minute, we will realize that there’s not much it could do. If we were told that the input was an apple, the output of the program could be apple slices or apple juice. If we knew that the input was a number, then the output could be that number times three. Or times five perhaps. But what if we were told nothing about the input? What could we ever do with it? Essentially, we wouldn’t be able do anything. We just wouldn’t know what type of input we are dealing with, so the only sensible thing left for us to do would be nothing. We give up and return the input to whomever gave it to us. That is, our input becomes our output as well.

program :: input -> input

Our program, the simplest possible one, can now be described as simply returning the input that is provided to it. But of course, we can also look at this from the opposite perspective, and say that the output of our program is also its input.

program :: output -> output

Celebrate if you can’t tell the difference between these two descriptions of our program, because they are effectively the same. When we said that our input becomes our output, we really meant it. What is important to understand here is that whether we say input or output doesn’t matter at all. What matters is that both of these words appearing around the arrow symbol , respectively describing the type of input and the type of output, are the same. This perfectly conveys the idea that anything we provide to this program as input will be returned to us. We give it an orange, we get an orange back. We give it a horse, we get a horse back. In fact, we can push this to the extreme and stop using the words input or output altogether, seeing how the notation we use already conveys the idea that the thing to the left of the arrow symbol is the input type, and the thing to its right the output type. That is, we know where the input is, we know where the output is, we know they are the same type of thing, and we don’t care about anything else. So we’ll just name it x, for mystery.

program :: x -> x

This seemingly useless program is one of the very few important enough to deserve a name of its own. This program is called identity, and it is a fundamental building block of both programming and mathematics. It might be easier to appreciate this name from a metaphysical point of view. Who am I? It is me. Indeed, philosophers rejoice, our program addresses the existential question about the identity of beings at last, even if in a tautological manner. In Haskell we call this program id, short for identity.

id :: x -> x

Actually, we could have named the identity program anything else. Nothing mandates the name id. As we saw before when we arbitrarily named our mystery type x, names don’t really matter to the computer. However, they do matter to us humans, so we choose them according to their purpose.

There is one last thing we should know about naming. Usually, we don’t really call programs like these programs, we call them functions. The term program does exist, but generally we use it to refer to functions complex enough that they can be seen as final consumer products, or things that do something yet they don’t exactly fit the shape of a function. Text editors, web browsers or music players are examples of what we would often call a program. In other words, we can say we just learned about id, the identity function. Generally speaking, though, we use these terms more or less interchangeably, as we have been doing so far.


We also have useful programs. They are less interesting, much more complicated, but deserve some attention too. We were able to discover the identity function because we didn’t know much about the types of things we were dealing with, and this ignorance gave us the freedom to unboundedly reason about our problem and derive truth and understanding from it. What would happen if we knew something about our types, though? Let’s pick a relatively simple and familiar problem to explore: The addition of natural numbers.

As a quick reminder, natural numbers are the zero and all the integer numbers greater than it. For example, 12, 0 and 894298731 are all natural numbers, but 4.68, π or -58 are not.

We said that our identity function, when given something as input, will return that same thing as output. This is supposed to be true for all possible types of inputs and outputs, which of course include the natural numbers. In Haskell, we can refer to this type of numbers as Natural. That is, if for example we wanted to use the identity function with a natural number like 3, then our id function would take the following type:

id :: Natural -> Natural

That is, this function takes a Natural number as input, and returns a Natural number as output. In other words, the type of this function is Natural → Natural. Yes, just like numbers, functions have their own types as well, and they are easy to recognize because of the that appears in them, with their input type appearing to the left of this arrow, and their output type to its right as we’ve seen multiple times.

Presumably, if we ask for the id of the Natural number 3, we will get 3 as our answer. Indeed, this would be the behaviour of the identity function. However, we also said that names can be arbitrarily chosen. Usually we choose them in alignment with their purpose, but what if we didn’t? What would happen if we named our id function for Natural numbers something else instead? Let’s try.

add_one :: Natural -> Natural

Interesting. At first glance, if we didn’t know that we are dealing with our funnily named identity function, we would expect a completely different behaviour from this function. The name add_one and the type of this function suggest that given a Natural number, we will add one to that number. That is, given 3 we would get 4. Very interesting indeed.

And what if we were told that one of the following functions is our identity function, renamed once again, and the other one adds as many numbers as it states? Look carefully. Can we tell which is which?

add_two :: Natural -> Natural

add_three :: Natural -> Natural

Despair. Sorrow. We can’t tell. We can’t tell without knowing the answer beforehand. The more we know about the things we work with, the less we know about the work itself. When we knew nothing about x, we knew that all we could ever do was nothing. That is, a function x → x, for all possible types x, could only ever have one behaviour: Returning x unchanged. But now we know about natural numbers. We know we can count them, add them, multiply them and more, so learning that a function has a type Natural → Natural is not particularly helpful anymore. A function of this type could process the Natural number given as input in any way it knows how, return the resulting Natural number as output, and it would be technically correct, even if possibly a fraud.

Is this the end? Have we lost? I am afraid that in a sense we have. This is part of the struggle we talked about. Suddenly we can’t reason anymore, and are instead left at the mercy of optional meaningful naming. If we name something add_one, then we better mean add one, because we have no way to tell what’s what.

A Type of Programming

Those were the first few words of A Type of Programming, a book emphasizing sound reasoning, a playful mastery of our skills and a thirst for meaning. We start from the very beginning, and barely noticing it, as we learn Haskell, functional programming and types, we become well-versed in every topic we touch. This book, ultimately, aims to improve the ways and quality of the software we produce and demand as individuals and as civilization. If you think that's a worthy goal, if you are here to excel, then this book is for you.

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See how we deal gracefully with more advanced topics later in the book.

It's really an amazing book. I am not overreacting here. Explaining how types work to non-programmers is really challenging. But the author, with an enjoyable story pace and nice examples, manages to do that with great success.


When we say Functor f, we are saying that f is a covariant functor, which essentially means that for each function a → b, there is a corresponding function f a → f b that is the result of lifting a → b into f. The name “covariant”, with the co prefix this time meaning with or jointly, evokes the idea that as the a in f a varies through a → b, the entirety of f a varies to f b with it, in the same direction.

But as with many precious things in life, it’s hard to appreciate why this is important, or at all interesting, until we’ve lost it. So let’s lose it. Let’s look at this matter from the other side, from its dual, a dual we find by just flipping arrows. So let’s try that and see what happens.

contramap :: (  a ->   b)
          -> (g a <- g b)

A covariant functor f, we said, was one that could lift a function a → b into a function f a → f b. So, presumably, the dual of said covariant functor f —let’s call it “g the contravariant”— is one that lifts a function a → b into g a ← g b instead, arrow flipped. Conceptually, this is perfect. In practice, in Haskell, function arrows always go from left to right , never from right to left , so we need to take care of that detail. Let’s write the function arrow in the expected direction, flipping the position of the arguments instead.

contramap :: (  a ->   b)
          -> (g b -> g a)

In Haskell, contramap exists as the sole method of the typeclass called Contravariant, which is like Functor but makes our minds bend.

class Contravariant (g :: Type -> Type) where
  contramap :: (a -> b) -> (g b -> g a)

Bend how? Why, pick a g and see. Maybe perhaps? Sure, that’s simple enough and worked for us before as a Functor.

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

Here, contramap is saying that given a way to convert as into bs, we will be granted a tool for converting Maybe bs into Maybe as. That is, a lifted function with the positions of its arguments flipped, exactly what we wanted. Let’s try to implement this by just following the types, as we’ve done many times before. Let’s write the Contravariant instance for Maybe.

instance Contravariant Maybe where
  contramap = \f yb ->
    case yb of
      Nothing -> Nothing
      Just b -> ‽

What a pickle. We know that f is of type a → b, we know that yb is a Maybe b, and we know that we must return a value of type Maybe a somehow. Furthermore, we know that a and b could be anything, for they’ve been universally quantified. There is an implicit in there, always remember that. When yb is Nothing, we just return a new Nothing of the expected type Maybe a. And when yb is Just b? Why, we die of course, for we need a way to turn that b into an a that we can put in a new Just, and we have none.

But couldn’t we just return Nothing? It is a perfectly valid expression of type Maybe a, isn’t it? Well, kind of. It type-checks, sure, but a vestigial tingling sensation tells us it’s wrong. Or, well, maybe it doesn’t, but at least we have laws that should help us judge. So let’s use these laws to understand why this behaviour would be described as evil, lest we hurt ourselves later on. Here is the simplified version of the broken instance we want to check, the one that always returns Nothing.

instance Contravariant Maybe where
  contramap = \_ _ -> Nothing

Like the identity law for fmap, the identity law for contramap says that contramapping id over some value x should result in that same x.

contramap id x == x

A broken contramap for Maybe that always returns Nothing would blatantly violate this law. Applying contramap id (Just 5), for example, would result in Nothing rather than Just 5. This should be enough proof that ours would be a broken Contravariant instance, but for completeness, let’s take a look at the second contramap law as well.

Just like we have a composition law for fmap, we have one for contramap as well. It says that contramapping the composition of two functions f and g over some value x should achieve the same as applying, to that x, the composition of the contramapping of g with the contramapping of f.

contramap (compose f g)
  == compose (contramap g) (contramap f)

Isn’t this the same as the composition law for fmap? Nice try, but take a closer look.

fmap (compose f g)
  == compose (fmap f) (fmap g)

contramap (compose f g)
  == compose (contramap g) (contramap f)

Whereas in the case of fmap, the order in which f and g appear at opposite sides of this equality is the same, it is not the same in the case of contramap. This shouldn’t come as a big surprise, seeing how we already knew that contramap gives us a lifted function with its arguments in the opposite position. Intuitively, dealing with Contravariant values means we are going to be composing things backwards. Or, should we say, forwards? After all, compose initially shocked us with its daring, so-called backwards sense of direction, and now we are mostly just backpedaling on it.

But the important question is whether our broken Contravariant instance for Maybe violates this law. And the answer is, unsurprisingly I hope, not at all. Both sides of this equality always result in Nothing, so they are indeed equal. Many times they will be equally wrong, but that doesn’t make them any less equal. And this is, technically, sufficient for our broken instance to satisfy this law. Thankfully we had that other law we could break.

So how do we contramap our Maybes? Well, we don’t. As handy as it would be to have a magical way of going from b to a when all we know is going from a to b, we just can’t do that. Think how absurd it would be. If your imagination is lacking, just picture a being a tree and b being fire. So, to sum up, Maybes are covariant functors, as witnessed by the existence of their Functor instance, but they are not contravariant functors at all, as witnessed by the impossibility of coming up with a lawful Contravariant instance for them. So once again, like in our previous Bifunctor conundrum, we find ourselves wanting for a function b → a when all we have is a function a → b. This time, however, we are prepared. We know that we keep ending up here because we are getting the position of our arguments, the variance of our functors, wrong. And we will fix that.

Wait. Covariant functors? Contravariant functors? That’s right, both these things are functors. All that talk we had growing up about how functors are this or that? A lie. Well, not a lie, but rather, we hid the fact we were talking only about covariant functors. Now we know that there are other types of functors too. Unfortunately, the Haskell nomenclature doesn’t help here. For example, one could argue that the Functor typeclass should have been called Covariant instead, or perhaps the Contravariant typeclass should be called Cofunctor and contramap should be renamed to cofmap, etc. Anyway, not important. As we continue learning more about functors we’ll see that even these names fall short of the true nature of functors. There’s more, yes, and it’s beautiful. But despite their names, both Contravariant and Functor are typeclasses that describe, indeed, functors. And it shouldn’t surprise us, considering how we saw in detail that except for their opposite argument positions, the types of fmap and contramap, as well as their laws, are exactly the same.

In the sea of often dry Haskell learning resources, this approach is truly intuitive, loving and compassionate.

I love this book. It’s beautiful. I like it so much that I’ve given it as a gift to multiple people.

This is thoroughly enjoyable even if you are quite familiar with Haskell!

Having a good time reading A Type of Programming. A really gentle but purposed introduction to programming in a functional setting. Kudos for the great work!

Well, I must say that A Type of Programming is the smoothest monad tutorial I have ever read. It takes several hundred pages before you see the M word, but the trip is certainly worth it.

A Type of Programming really struck me how beautiful the arc develops right before the recursive List appears! It gives much to contemplate.

A Type of Programming is such a great book, relaxing and insightful. Carving out more time to read it and learn Haskell.

I love A Type of Programming. Such beautiful perspectives. And a great way to meet Haskell for the first time I bet.

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