Posted on October 2, 2019

Update 5/10/2019: check the bottom of this post for some links to comments and discussion.

Beginners to Haskell are often confused as to what’s so great about the language. Much of the proselytizing online focuses on pretty abstract (and often poorly defined) concepts like “purity”, “strong types”, and (god forbid) “monads”. These things are difficult to understand, somewhat controversial, and not obviously beneficial (especially when you’ve only been using the language for a short amount of time).

The real tragedy is that Haskell (and other ML-family languages) are packed with simple, decades-old features like pattern matching and algebraic data types which have massive, clear benefits and few (if any) downsides. Some of these ideas are finally filtering in to mainstream languages (like Swift and Rust) where they’re used to great effect, but the vast majority of programmers out there haven’t yet been exposed to them.

This post aims to demonstrate some of these features in a simple (but hopefully not too simple) example. I’m going to write and package up a simple sorting algorithm in both Haskell and Python, and compare the code in each. I’m choosing Python because I like it and beginners like it, but also because it’s missing most of the features I’ll be demonstrating. It’s important to note I’m not comparing Haskell and Python as languages: the Python code is just there as a reference for people less familiar with Haskell. What’s more, the comparison is unfair, as the example deliberately plays to Haskell’s strengths (so I can show off the features I’m interested in): it wouldn’t be difficult to pick an example that makes Python look good and Haskell look poor.

This post is not meant to say “Haskell is great, and your language sucks”! It’s not even really about Haskell: much of what I’m talking about here applies equally well to Ocaml, Rust, etc. I’m really writing this as a response to the notion that functional features are somehow experimental, overly complex, or ultimately compromised. As a result of that idea, I feel like these features are left out of a lot of modern languages which would benefit from them. There exists a small set of simple, battle-tested PL ideas, which have been used for nearly forty years now: this post aims to demonstrate them, and argue for their inclusion in every general-purpose programming language that’s being designed today.

# The Algorithm

We’ll be using a skew heap to sort lists in both languages. The basic idea is to repeatedly insert stuff into the heap, and then repeatedly “pop” the smallest element from the heap until it’s empty. It’s not in-place, but it is $\mathcal{O}(n \log n)$, and actually performs pretty well in practice.

# A Tree

A Skew Heap is represented by a binary tree:

data Tree a
= Leaf
| Node a (Tree a) (Tree a)
class Tree:
def __init__(self, is_node, data, lchild, rchild):
self._is_node = is_node
self._data = data
self._lchild = lchild
self._rchild = rchild

def leaf():
return Tree(False, None, None, None)

def node(data, lchild, rchild):
return Tree(True, data, lchild, rchild)

I want to point out the precision of the Haskell definition: a tree is either a leaf (an empty tree), or a node, with a payload and two children. There are no special cases, and it took us one line to write (spread to 3 here for legibility on smaller screens).

In Python, we have to write a few more lines1. This representation uses the _is_node field is False for an empty tree (a leaf). If it’s True, the other fields are filled. We write some helper functions to give us constructors like the leaf and node ones for the Haskell example.

This isn’t the standard definition of a binary tree in Python, in fact it might looks a little weird to most Python people. Let’s run through some alternatives and their issues.

1. The standard definition:

class Tree:
def __init__(self, data, lchild, rchild):
self._data = data
self._lchild = lchild
self._rchild = rchild

Instead of having a separate field for “is this a leaf or a node”, the empty tree is simply None:

def leaf():
return None

With this approach, if we define any methods on a tree, they won’t work on the empty tree!

>>> leaf().size()
AttributeError: 'NoneType' object has no attribute 'size'
2. We’ll do inheritance! Python even has a handy abc library to help us with some of this:

from abc import ABC, abstractmethod

class Tree(ABC):
@abstractmethod
def size(self):
raise NotImplemented

class Leaf(Tree):
def __init__(self):
pass

def size(self):
return 0

class Node(Tree):
def __init__(self, data, lchild, rchild):
self._data = data
self._lchild = lchild
self._rchild = rchild

def size(self):
return 1 + self._lchild.size() + self._rchild.size()

Methods will now work on an empty tree, but we’re faced with 2 problems: first, this is very verbose, and pretty complex. Secondly, we can’t write a mutating method which changes a tree from a leaf to a node. In other words, we can’t write an insert method!

3. We won’t represent a leaf as the whole tree being None, just the data!

def leaf():
return Tree(None, None, None)

This (surprisingly) pops up in a few places. While it solves the problem of methods, and the mutation problem, it has a serious bug. We can’t have None as an element in the tree! In other words, if we ask our eventual algorithm to sort a list which contains None, it will silently discard some of the list, returning the wrong answer.

There are yet more options (using a wrapper class), none of them ideal. Another thing to point out is that, even with our definition with a tag, we can only represent types with 2 possible states. If there was another type of node in the tree, we couldn’t simply use a boolean tag: we’d have to switch to integers (and remember the meaning of each integer), or strings! Yuck!

What Python is fundamentally missing here is algebraic data types. This is a way of building up all of your types out of products (“my type has this and this”) and sums (“my type is this or this”). Python can do products perfectly well: that’s what classes are. The tree class itself is the product of Bool, data, Tree, and Tree. However it’s missing an entire half of the equation! This is why you just can’t express binary trees as cleanly as you can in Swift, Haskell, OCaml, etc. Python, as well as a host of other languages like Go, Java, etc, will let you express one kind of “sum” type: “or nothing” (the null pointer). However, it’s clunky and poorly handled in all of those languages (the method problems above demonstrate the issues in Python), and doesn’t work for anything other than that one special case.

Again, there’s nothing about algebraic data types that makes them ill-suited to mainstream or imperative languages. Swift uses them, and people love them!

# A Function

The core operation on skew heaps is the skew merge.

merge :: Ord a => Tree a -> Tree a -> Tree a
merge Leaf ys = ys
merge xs Leaf = xs
merge xs@(Node x xl xr) ys@(Node y yl yr)
| x <= y    = Node x (merge ys xr) xl
| otherwise = Node y (merge xs yr) yl
def merge(lhs, rhs):
if not lhs._is_node:
return rhs
if not rhs._is_node:
return lhs
if lhs._data <= rhs._data:
return Tree(lhs._data,
merge(rhs, lhs._rchild),
lhs._lchild)
else:
return Tree(rhs._data,
merge(lhs, rhs._rchild),
rhs._lchild)

The standout feature here is pattern matching. In Haskell, we’re able to write the function as we might describe it: “in this case, I’ll do this, in this other case, I’ll do this, etc.”. In Python, we are forced to think of the truth tables and sequential testing. What do I mean by truth tables? Consider the following version of the Python function above:

def merge(lhs, rhs):
if lhs._is_node:
if rhs._is_node:
if lhs._data <= rhs._data:
return Tree(lhs._data,
merge(rhs, lhs._rchild),
lhs._lchild)
else:
return Tree(rhs._data,
merge(lhs, rhs._rchild),
rhs._lchild)
else:
return lhs
else:
return rhs

You may even write this version first: it initially seems more natural (because _is_node is used in the positive). Here’s the question, though: does it do the same thing as the previous version? Are you sure? Which else is connected to which if? Does every if have an else? (some linters will suggest you remove some of the elses above, since the if-clause has a return statement in it!)

The fact of the matter is that we are forced to do truth tables of every condition in our minds, rather than saying what we mean (as we do in the Haskell version).

The other thing we’re saved from in the Haskell version is accessing undefined fields. In the Python function, we know accessing lhs._data is correct since we verified that lhs is a node. But the logic to do this verification is complex: we checked if it wasn’t a node, and returned if that was true… so if it is true that lhs isn’t a node, we would have returned, but we didn’t, so…

Bear in mind all of these logic checks happened four lines before the actual access: this can get much uglier in practice! Compare this to the Haskell version: we only get to bind variables if we’re sure they exist. The syntax itself prevents us from accessing fields which aren’t defined, in a simple way.

Pattern matching has existed for years in many different forms: even C has switch statements. The added feature of destructuring is available in languages like Swift, Rust, and the whole ML family. Ask for it in your language today!

Now that we have that function, we get to define others in terms of it:

insert :: Ord a => a -> Tree a -> Tree a
insert x = merge (Node x Leaf Leaf)
def insert(element, tree):
tree.__dict__ = merge(node(element, leaf(), leaf()),
tree).__dict__.copy()

# A Word on Types

I haven’t mentioned Haskell’s type system so far, as it’s been quite unobtrusive in the examples. And that’s kind of the point: despite more complex examples you’ll see online demonstrating the power of type classes and higher-kinded types, Haskell’s type system excels in these simpler cases.

merge :: Ord a => Tree a -> Tree a -> Tree a

Without much ceremony, this signature tells us:

1. The function takes two trees, and returns a third.
2. Both trees have to be filled with the same types of elements.
3. Those elements must have an order defined on them.

# Type Inference

I feel a lot of people miss the point of this particular feature. Technically speaking, this feature allows us to write fewer type signatures, as Haskell will be able to guess most of them. Coming from something like Java, you might think that that’s an opportunity to shorten up some verbose code. It’s not! You’ll rarely find a Haskell program these days missing top-level type signatures: it’s easier to read a program with explicit type signatures, so people are advised to put them as much as possible.

(Amusingly, I often find older Haskell code snippets which are entirely devoid of type signatures. It seems that programmers were so excited about Hindley-Milner type inference that they would put it to the test as often as they could.)

Type inference in Haskell is actually useful in a different way. First, if I write the implementation of the merge function, the compiler will tell me the signature, which is extremely helpful for more complex examples. Take the following, for instance:

f x = ((x * 2) ^ 3) / 4

Remembering precisely which numeric type x needs to be is a little difficult (Floating? Real? Fractional?), but if I just ask the compiler it will tell me without difficulty.

The second use is kind of the opposite: if I have a hole in my program where I need to fill in some code, Haskell can help me along by telling me the type of that hole automatically. This is often enough information to figure out the entire implementation! In fact, there are some programs which will use this capability of the type checker to fill in the hole with valid programs, synthesising your code for you.

So often strong type systems can make you feel like you’re fighting more and more against the compiler. I hope these couple examples show that it doesn’t have to be that way.

# When Things Go Wrong

The next function is “pop-min”:

popMin :: Ord a => Tree a -> Maybe (a, Tree a)
popMin Leaf = Nothing
popMin (Node x xl xr) = Just (x, merge xl xr)
def popMin(tree):
if tree._is_node:
res = tree._data
tree.__dict__ = merge(tree._lchild, tree._rchild).__dict__.copy()
return res
else:
raise IndexError

At first glance, this function should be right at home in Python. It mutates its input, and it has an error case. The code we’ve written here for Python is pretty idiomatic, also: other than the ugly deep copy, we’re basically just mutating the object, and using an exception for the exceptional state (when the tree is empty). Even the exception we use is the same exception as when you try and pop() from an empty list.

The Haskell code here mainly demonstrates a difference in API style you’ll see between the two languages. If something isn’t found, we just use Maybe. And instead of mutating the original variable, we return the new state in the second part of a tuple. What’s nice about this is that we’re only using simple core features like algebraic data types to emulate pretty complex features like exceptions in Python.

You may have heard that “Haskell uses monads to do mutation and exceptions”. This is not true. Yes, state and exceptions have patterns which technically speaking are “monadic”. But make no mistake: when we want to model “exceptions” in Haskell, we really just return a maybe (or an either). And when we want to do “mutation”, we return a tuple, where the second element is the updated state. You don’t have to understand monads to use them, and you certainly don’t “need” monads to do them. To drive the point home, the above code could actually equivalently have a type which mentions “the state monad” and “the maybe monad”:

popMin :: Ord a => StateT (Tree a) Maybe a

But there’s no need to!

# Gluing It All Together

The main part of our task is now done: all that is left is to glue the various bits and pieces together. Remember, the overall algorithm builds up the heap from a list, and then tears it down using popMin. First, then, to build up the heap.

listToHeap :: Ord a => [a] -> Tree a
listToHeap = foldr insert Leaf
def listToHeap(elements):
res = leaf()
for el in elements:
insert(el, res)
return res

To my eye, the Haskell code here is significantly more “readable” than the Python. I know that’s a very subjective judgement, but foldr is a function so often used that it’s immediately clear what’s happening in this example.

Why didn’t we use a similar function in Python, then? We actually could have: python does have an equivalent to foldr, called reduce (it’s been relegated to functools since Python 3 (also technically it’s equivalent to foldl, not foldr)). We’re encouraged not to use it, though: the more pythonic code uses a for loop. Also, it wouldn’t work for our use case: the insert function we wrote is mutating, which doesn’t gel well with reduce.

I think this demonstrates another benefit of simple, functional APIs. If you keep things simple, and build things out of functions, they’ll tend to glue together well, without having to write any glue code yourself. The for loop, in my opinion, is “glue code”. The next function, heapToList, illustrates this even more so:

heapToList :: Ord a => Tree a -> [a]
heapToList = unfoldr popMin
def heapToList(tree):
res = []
try:
while True:
res.append(popMin(tree))
except IndexError:
return res

Again, things are kept simple in the Haskell example. We’ve stuck to data types and functions, and these data types and functions mesh well with each other. You might be aware that there’s some deep and interesting mathematics behind the foldr and unfoldr functions going on, and how they relate. We don’t need to know any of that here, though: they just work together well.

Again, Python does have a function which is equivalent to unfoldr: iter has an overload which will repeatedly call a function until it hits a sentinel value. But this doesn’t fit with the rest of the iterator model! Most iterators are terminated with the StopIteration exception; ours (like the pop function on lists) is terminated by the IndexError exception; and this function excepts a third version, terminated by a sentinel!

Finally, let’s write sort:

sort :: Ord a => [a] -> [a]
sort = heapToList . listToHeap
def sort(elements):
return heapToList(listToHeap(elements))

This is just driving home the point: programs work well when they’re built out of functions, and you want your language to encourage you to build things out of functions. In this case, the sort function is built out of two smaller ones: it’s the essence of function composition.

# Laziness

So I fully admit that laziness is one of the features of Haskell that does have downsides. I don’t think every language should be lazy, but I did want to say a little about it in regards to the sorting example here.

I tend to think that people overstate how hard it makes reasoning about space: it actually follows pretty straightforward rules, which you can generally step through in yourself (compared to, for instance, rewrite rules, which are often black magic!)

In modern programming, people will tend to use laziness it anyway. Python is a great example: the itertools library is almost entirely lazy. Actually making use of the laziness, though, is difficult and error-prone. Above, for instance, the heapToList function is lazy in Haskell, but strict in Python. Converting it to a lazy version is not the most difficult thing in the world:

def heapToList(tree):
try:
while True:
yield popMin(tree)
except IndexError:
pass

But now, suddenly, the entire list API won’t work. What’s more, if we try and access the first element of the returned value, we mutate the whole thing: anyone else looking at the output of the generator will have it mutated out from under them!

Laziness fundamentally makes this more reusable. Take our popMin function: if we just want to view the smallest element, without reconstructing the rest of the tree, we can actually use popMin as-is. If we don’t use the second element of the tuple we don’t pay for it. In Python, we need to write a second function.

# Testing

Testing the sort function in Haskell is ridiculously easy. Say we have an example sorting function that we trust, maybe a slow but obvious insertion sort, and we want to make sure that our fast heap sort here does the same thing. This is the test:

quickCheck (\xs -> sort (xs :: [Int]) === insertionSort xs)

In that single line, the QuickCheck library will automatically generate random input, run each sort function on it, and compare the two outputs, giving a rich diff if they don’t match.

# Conclusion

This post was meant to show a few features like pattern-matching, algebraic data types, and function-based APIs in a good light. These ideas aren’t revolutionary any more, and plenty of languages have them, but unfortunately several languages don’t. Hopefully the example here illustrates a little why these features are good, and pushes back against the idea that algebraic data types are too complex for mainstream languages.

# Update 5/10/2019

This got posted to /r/haskell and hackernews. You can find me arguing in the comments there a little bit: I’m oisdk on hackernews and u/foBrowsing on reddit.

There are two topics that came up a bunch that I’d like to add to this post. First I’ll just quote one of the comments from Beltiras:

Friend of mine is always trying to convert me. Asked me to read this yesterday evening. This is my take on the article:

Most of my daily job goes into gluing services (API endpoints to databases or other services, some business logic in the middle). I don’t need to see yet another exposition of how to do algorithmic tasks. Haven’t seen one of those since doing my BSc. Show me the tools available to write a daemon, an http server, API endpoints, ORM-type things and you will have provided me with tools to tackle what I do. I’ll never write a binary tree or search or a linked list at work.

If you want to convince me, show me what I need to know to do what I do.

and my response:

I wasn’t really trying to convince anyone to use Haskell at their day job: I am just a college student, after all, so I would have no idea what I was talking about!

I wrote the article a while ago after being frustrated using a bunch of Go and Python at an internship. Often I really wanted simple algebraic data types and pattern-matching, but when I looked up why Go didn’t have them I saw a lot of justifications that amounted to “functional features are too complex and we’re making a simple language. Haskell is notoriously complex”. In my opinion, the res, err := fun(); if err != nil (for example) pattern was much more complex than the alternative with pattern-matching. So I wanted to write an article demonstrating that, while Haskell has a lot of out-there stuff in it, there’s a bunch of simple ideas which really shouldn’t be missing from any modern general-purpose language.

As to why I used a binary tree as the example, I thought it was pretty self-contained, and I find skew heaps quite interesting.

The second topic was basically people having a go at my ugly Python; to which I say: fair enough! It is not my best. I wasn’t trying necessarily to write the best Python I could here, though, rather I was trying to write the “normal” implementation of a binary tree. If I was to implement a binary tree of some sort myself, though, I would certainly write it in an immutable style rather than the style here. Bear in mind as well that much of what I’m arguing for is stylistic: I think (for instance) that it would be better to use reduce in Python more, and I think the move away from it is a bad thing. So of course I’m not going to use reduce when I’m showing the Python version: I’m doing a compare and contrast!

1. Yes, I know about the new dataclasses feature. However, it’s wrapped up with the (also new) type hints module, and as such is much more complicated to use. As the purpose of the Python code here is to provide something of a lingua franca for non-Haskellers, I decided against using it. That said, the problems outlined are not solved by dataclasses.