Functional Programming

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\Z}{\mathbb{Z}}$

Table of Contents

  1. Overview
  2. Libraries
  3. Parser Combinators
  4. Curry Howard
  5. Recursion Schemes
  6. The Free Monad
  7. Lenses
  8. Continuations
  9. Category Theory
  10. Functional Reactive Programming
  11. Generics
  12. GADTs
  13. Type families

Haskell has a difficult learning curve. A familiar complaint people have with Haskell is that you learn the basics (the type system, higher order functions, pattern matching, lazy evaluation, various forms of recursion, functors, monads, typeclasses) but then are unable to actually use standard Haskell libraries, and are baffled by the apparent lack of documentation.

These are some resources aimed at an intermediate level. By that, I mean everything after Learn You a Haskell.


Extremely useful overview of important Haskell libraries and concepts

Ecosystem: Haskell is a research language, so there are a lot of half-finished or broken libraries made public. There are also a lot of good robust libraries though. Useful overview of the ecosystem here.

Documentation: libraries in Haskell tend to be less verbosely documented then you’d think would be useful. Often this is because it’s research code and the authors didn’t care if other people used it. But equally often it’s because Haskell types do a lot of the talking, so that you can work out what a library is doing by reading through the types.

Because Haskell is so expressive, it’s common for libraries to be fairly small collections of combinators, centering around some central datatype. The key to understanding the library is to grok that datatype, largely by seeing what things you can do with it via the available functions. Yampa is a nice example of a library with seemingly minimal documentation, which is in fact very easy to understand in this way.


The best way to learn how to write good Haskell is to read libraries. Some really great, well-documented libraries:

  • diagrams
  • pipes
  • megaparsec

More abstract libraries which I found pretty intelligible include:

  • recursion-schemes (again, rather than getting too hung up on how mind-bending catamorphisms are, just look at the types of functions like cata)
  • monad-coroutine

A couple of more niche libraries that I found very useful for seeing more advanced features/libraries in action (type level arithmetic, automatic differentiation, proper use of typeclasses, free monad transformers with the codensity optimization, coroutines) include:

  • hamilton
  • monad-bayes

Miscellaneous topics

Parser combinators

A tutorial for a good parser combinator library

Example of simple parser combinators written from the ground up

Types are propositions

Types are propositions and programs are constructive proofs of those propositions, i.e. a construction of an inhabitant of the type, which witnesses the truth of the proposition.

So Int -> Int is “proved” by (+5), among many other proofs.

This perspective plays well with polymorphism, which corresponds to propositions involving quantifiers. For example forall a. a -> a is “proved” by id. And in fact that’s the only possible proof.

This also makes sense for existential types, which correspond to propositions with existential quantifiers, like this example from (

{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE RankNTypes #-}

-- ∃ t. (t, t → t, t → String)
data Box = forall a. Box
  { value  :: a
  , update :: a -> a
  , print  :: a -> String

Note that you use a universal quantifier to implement the existential, but the moral type is given in the comment. This is saying: a value of type Box must be a program which proves the claim that there is some inhabitant of Box. So you get to have boxes which contain totally different types, like:

boxa :: Box
boxa = Box 1 negate show

boxb :: Box
boxb = Box "foo" reverse show

Contradictory propositions: If you have a type (forall a. a), a program of that type would be a value which is of all types at once. Which you can’t do, except with undefined or non-termination. For example, let x = fix id :: forall a. a. Similarly for forall a b. a -> b.

Recursion schemes

This is a place where the type system of Haskell really shines, because it allows you to modularize a recursive program into two parts: the recursion itself, and the non-recursive logic.

Here’s my take. I’ll proceed by example, instead of abstractly, since I think once you see how things work for one recursive datatype, you have the means to extrapolate. Consider the data type

data Tree a = Leaf a | Branch (Tree a) (Tree a)

First, before even getting into the details, it’s crucial to recognize that Leaf and Branch are data constructors, while Tree is a type constructor. Total clarity on that point is a prerequisite, which I only mention because it’s the kind of thing I got very confused by.

So, having defined this recursive type Tree a, we can ask about functions of the form Tree a -> a, for a given a. For example sum :: Tree Int -> Int would take a tree with integers on the leaves, and sum them all to a single integer. If you imagine how this would actually define this, you can see that it is recursive.

The core idea of recursion schemes is this: rather than writing a recursive function on a recursive type, write a non-recursive function on a non-recursive type and pass that to a higher order function. In particular, consider the following type, with an obvious functor instance:

data TreeF a x = Leaf a | Branch x x

Here, TreeF a x is a non-recursive type that is counterpart to Tree a. So the idea is that we will have a folding function of the type

  • The x parameter in TreeF a x represents the hole where the recursive nesting happens in Tree a.
  • In particular, to obtain Tree a from TreeF a x and eliminate the x in the process you do:
TreeF a (TreeF a (TreeF a...))
  • That is, Fix (TreeF a) is isomorphic to Tree a

With this datatype introduced, here is fold:

fold :: (TreeF a x -> x) -> Tree a -> a

Note that, in fold :: (TreeF a x -> x) -> Tree a -> a, you can choose x to be any type.

In the recursion-schemes library, type families are used, so that Base is a function on types defined such that Base (Tree a) equals TreeF a. This allows for a generalization to arbitrary recursive datatypes, as follows:

fold :: Recursive t => (Base t a -> a) -> t -> a

fold (or synonymously cata for “catamorphism”) is only one of many recursion schemes provided in libraries like Data.Functor.Foldable. The beauty of this approach is that much more complex recursion patterns are expressible very simply.

For example, I often want to do some kind of rather sophisticated fold of some scary datatype like FreeT f m a. Haskell can tell me exactly the type of the Base function here, and so writing a fold is easy.

The free monad

The documentation in is surprisingly illuminating.

The theory of f-algebras and recursion schemes is nicely summarized by This is a good tutorial building up to Free from scratch. This is another good tutorial on free monads

Fixpoints and monadfix

Excellent introduction to difficult topic and another blog post which builds on that.


Roughly, a way to modify subparts of a larger data structure, functionally.

Lens s t a b

means: if you can tell me how to turn an $a$ into a $b$, I’ll turn an $s$ (of which $a$ is a subpart) into a $t$ (of which $b$ is a subpart).

The most well known style of lens implementation, but not the only one, is the van Laarhoven style approach by Edward Kmett in the lens package. Great overview: . The gist is that the underlying representation of a lens is:

type Lens s t a b = forall f. Functor f => (a -> f b) -> (s -> f t)

where there is some constraint on $f$. Exactly what functor $f$ we use determines the functionality we want; we can pick any. For example, Const b gives you (a -> Const r b) -> (s -> Const r t) ~ (a -> r) -> (s -> r), which lets you view. Meanwhile Identity gives you (a -> Identity b) -> (s -> Identity t) ~ (a -> b) -> (s -> t), which lets you update.

Moreover, lenses compose by function composition. Kmett’s library extends this idea to other similar types where the constraint is e.g. Applicative not Functor, and other generalizations of this sort, which yields a family of beautifully composable “optics”.


A brief summary of some of the optics in the lens package. (Note that there is also a category theory story about optics, based on profunctors, which is somewhat separate.)


type Traversal s t a b = forall f. Applicative f => (a -> f b) -> (s -> f t)

Like a lenses but gets or sets multiple places. A characteristic usage is a ^.. t, where a is our data and t is our traversal, to obtain a list of the elemens being pointed to by the traversal.

We already have:

instance Monoid m => Applicative (Const m) where
    pure _ = Const mempty
    liftA2 _ (Const x) (Const y) = Const (x `mappend` y)

so for m a monoid, we can view it. For instance, if we have a traversal which points to every leaf in a tree, we can extract a list (the free monoid) of those leafs.

And Identity is an applicative, so no problem there.

We can also generalize from -> to any profunctor p. This is how many of the indexed versions of optics works, and also gives the most general type in the package:

type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t)

A small bit of theory of lenses: “A Lens’ s a is nothing more than a witness for the fact that there exists some q where s ~ (a, q).“. Here q is existentially quantified:


Category theory

Category theory is to set theory as Haskell is to C. It is a branch of mathematics in which you can express hugely powerful succinct generalizations. It inspires a number of ideas in Haskell.

Lovely summary of category theory origins of monads and adjunctions with the beautiful diagrammatic notation that this kind of stuff allows for.

Category theory for programmers. Learn why a monad is a monoid in the categories of endofunctors, and why polymorphic functions are natural transformations.

Two places where category theory comes up fairly directly in Haskell are recursion schemes (see below) and the free monad and its surrounding optimizations which involve Kan extensions / the Yoneda lemma / the codensity monad. See

Functional Reactive Programming

FRP is a slightly elusive concept. This 2015 talk by Conal Elliott (or this version) is nice, and serves as a good introduction to denotational semantics, but doesn’t really give the full flavour.

There are two use cases I have encountered. One is to deal with continuous time properly; rather than discretizing first and having gross resolution issues, you work with continuous time with composing and building your system and only at the end, discretize. Here, the salient analogy is to vector graphics: represent your image continuously, and later discretize.

The second is to deal with interaction with an external system. For example, you might want to write a program which does something when a mouse click happens. Rather than write a function which is triggered by a mouse click, you consider, as an abstract object, the full history of all mouse clicks that ever have or will happen (represented as a function from time values to () ). This is counterintuitive at first, since future events haven’t happened yet. If you’re confused by that, the point is that you’re going to write a function which takes any such full history, and outputs a new full history of some other value. That doesn’t mean that you have the full history of clicks determined - rather it’s just saying that if you did, this is what you would do with it. You then take the system you defined in this abstract way, and “compile” into an IO action.

A connection I find useful is to electrical engineering. There, it is common to think of systems, in particular linear systems, which take a function as input, and give a new function as output. Here, people are used to the idea that all past and future time is captured by a function. See these notes. People who work with systems also quickly build up intuitions about how they are different from just plain functions, such as the possibility of feedback loops and switches.

Yampa is a good Haskell library to look into for getting a better sense of this: you can implement switches and loops, for example.

Another approach, in libraries like reactive-banana, involves the use of recursive do-notation. This example from the reactive-banana docs is a little mind-blowing:

    let price = 50 :: Int
    bAmount  <- accumB price $ unions
                  [ subtract 10 <$ eCoin
                  , const price <$ eSold ]
    let eSold = whenE ((<= 0) <$> bAmount) eCoin


Since many types in Haskell can be expressed as sums and products, along with recursion, if you can instantiate some pattern for sums and products, it should automatically be generalizable to complex types. This is the core idea of generics, but there’s a lot of fiddly detail, most of which is boring.

Generalized Algebraic Datatypes

A normal data declaration in Haskell might look like:

data Expr a n = Bar a n | Baz a n

But GADTs allow you to do:

data Expr a where
  Foo :: a -> Expr a Int
  Bar :: a -> Expr a Double
  Bar :: Num n => a -> Expr a n
  (:||:) :: Expr a Bool -> Expr a Bool -> Expr a Bool

This means that the type of the resulting expression changes from constructor to constructor. Handy for building stronger types.

Type Families

Haskell doesn’t (currently - 2020) have dependent types (very roughly, types which depend on values), but there are ways to obtain some of the capabilities of dependent types you might want. Type families are functions from types to types. A simple example:

data Zero
data Succ n

data Vect n a where
  VNil :: Vect Zero a
  VCos :: a -> Vect n a -> Vect (Succ n) a

type family Plus x y where
  Plus Zero x = x
  Plus (Succ x) y = Succ (Plus x y)

append :: Vect x a -> Vect y a -> Vect (Plus x y) a
append = ...

The point in this particular example being that you can express numerical guarantees at the type level.

Probabilistic programming

Example of how to build simple probabilistic programming DSL in Haskell

For fun

A really mind-blowing quine:

powerset = filterM (const [True,False])