# Posts tagged "haskell":

# donald stewart on haskell in the large

Last October Don Stewart gave a very interesting talk at *CodeNode* on
his experience using Haskell at a large scale. And by large, he means
millions of lines of code. Although he wasn't allowed to talk about
the very specifics of the code, his talk is full of interesting
remarks that i found resonate with my experience. They actually
convinced me that the next language i should try in production should
be OCaml.

The first thing that drew my attention in this talk was the sheer amount of code they've written: 3.4 millions of lines of code. Taking into account Haskell's conciseness, that's quite a lot. I've got experience with very substantial projects written in functional languages such as Clojure with two orders of magnitude less code! Althouth it's true that running on top of the JVM lets you reuse hundreds of libraries, and that might be closing the gap a bit.

A second interesting point was the fact that Don's team is using interpreted code (via their own Haskell compiler): virtual machines have many advantages, yes.

And i wasn't surprised at all to learn that their code is strict by default: my experience with the limited laziness of Clojure has been bad enough to reinforce my exasperation with space leaks in my hobbyist Haskell. Laziness is pretty cool, but also the source of the most difficult to track down bugs i've encountered in the latest years. More difficult in fact than any memory leak i can remember in those dreadful C++ years: there at least i had a very clear, fool-proof methodology to avoid those problems (unlike the case of mutable state: the peace of mind bought by immutability is priceless).

I also found myself nodding enthusiastically at Don's plea for sticking to basic, simple constructs and to avoid as much as possible the line noise of excessive infix operators. One can write Perl in any language. And, for an old lisper, his advocacy of a programming style based on writing interpreters for little languages, sounded right on the money.

The thing he mentions over and over that i haven't experienced first-hand in a real-life, sizeable project is the power of a real type system (no, Java's or C++'s aren't), and that's something i'd really like to try.

Then at some point he mentions they have their own module system, not as good as ML's, but… and i realised that there's a language out there that has all the good things the talk mentions and needs none of the workarounds they use. It's called OCaml, and i've been meaning to use it in earnest for some years now. Maybe it's finally time i dust that part of my bookshelf, if only to compare real-world development in a properly typed language with my experience with dynamic languages such as Clojure (which has not been by any means as bad as static typists seem to fear).

At any rate, there's lots more in this talk that is surely more interesting than my ramblings: recommended!

# enjoying haskell

I've been reading about Haskell quite a bit during the last months, writing some actual code, and liking the language more and more. After many years favouring dynamically typed languages, i'm beginning to really appreciate Haskell's type system and the benefits it brings to the table.

A common argument from the static typing camp is that the compiler is catching a whole class of bugs for you, to which dynamic types answer that a good test suite (which you need anyway for any serious development) will catch those relatively trivial bugs for you. I tend to agree with the dynamic faction on this issue, but then i think that the strength of static typing (coupled with good type inference) is not at all about the compiler catching typing bugs but, rather, as enforcing useful constraints. When you write Haskell, you have to think hard about your data types and the functions using them; and the compiler will keep complaining and, most importantly, the code will feel awkward and somehow ad hoc until you find a good set of types to solve your problem.

The limits to your freedom imposed by the type system entail, in my experience, a boost in the amount of thought and imagination that i put in my design and implementation, in much the same way as the constraints imposed by metric and rhythm to poetic writing boost creativity and actually help producing a beautiful text. Or, in fact, in the same way as any kind of constraint in any creative endeavour helps (paradoxically, at first sight) in attaining beauty, or, at least, in having fun during the process.

In my experience, the process of writing a program or library in any
language is always a struggle to find *the* right way of expressing the
solution to a problem, as the culmination of a series of
approximations. The code feels better, more expressive and
easy-flowing with each rewrite, until something just clicks and i get
this feeling that i've finally captured the essence of the problem (a
litmus test being that then it's natural to extend the solution to
cases i hadn't thought of when writing the solution, as if, somehow,
the new solutions were already there, waiting for you to *discover*
them). And i'm finding that the powerful type system offered by
Haskell (think not only of vanilla Hindley-Milner, but also of
extensions such as GADTs or type families) is helping me reaching the
(local) optimum quicker, that satisfying constraints means i'm closer
to the final solution when my code compiles for the first time. You
often hear Haskell programmers saying something similar ("once my code
compiles, it works"), and i think it's mostly true, except that the
real reason is not that the compiler is catching trivial typing bugs,
but, rather, that the constraints imposed by the type system are
making you think harder and find the right solution. Same thing with
monads, and the clean separation they provide for stateful
computations: again, you must think carefully about the right
combination of monads and pure code to solve the problem, and most of
the time your code will simply not type check if you don't get the
solution right.

There are two more ways that Haskell's type system is helping me writing better programs. Two ways that are especially poignant when the code becomes sizeable enough. The first one is self-documentation: seeing the type of my functions (or asking the interpreter for them) instantly informs me of almost everything i need to know to use them; in fact, when writing in dynamic languages i keep annotating function signatures with this same information, only that there i'm all by myself to ensure that this information is right. PLT contract system is but a recognition of the usefulness of typing in this regard, although i much prefer the terseness and notational elegance of Haskell's type signatures over the much more verbose and, to my eyes, somewhat clunky notation used by PLT (which is not really PLT's fault, being as it is a very schemish notation). Let me stress here that having a REPL such as ghci is a god-send (and, to me, a necessity for really enjoying the language): it will tell me the type of an expression in much the same way as decent Lisp or Scheme environments will report a function's signature.

The second way Haskell's lending a helping hand with non-trivial code base is refactoring. As i mentioned above, i rewrite my programs several times as a rule, and rewrites almost always involve modifying data structures or adding new ones. As i grow older, i find it more and more difficult to keep in my head all the places and ways a given data structure is used in my programs, and with dynamic languages i'm often falling back to grepping the source code to find them. And again, their plasticity often works against me, in that they let me use those data structures in crooked ways, or forget to take into account new fields or constructors for a modified data type. Haskell's compiler has proved an invaluable ally to my refactorings and, by comparison, modifying and maintaining my bigger dynamic programs is not as fun as it used to be.

As an aside, types are not the only thing i'm finding enjoyable about Haskell. Its astonishing capabilities to express very abstract problems with a remarkable economy of expression (due, in part, to its highly tuned syntax) are extremely useful. To my mind, they mimic the process by which in math we solve harder and harder problems by abstracting more and more, cramming together more relevant information in less space (some cognitive science writers will tell you that thought and even consciousness consists on our ability to compress information). That means that i can express my solutions by capturing them in very high level description: initially, that makes them harder to understand, but once i feel comfortable with the basic concepts and operations, they scale up much, much better than more verbose, less sophisticated ones. Using these new hard-earned concepts, i can solve much harder problems without adding to the complexity of the code in a significant way (one could say, using a loose analogy, that the solutions grow logarithmically with complexity instead of polynomically or exponentially). A direct consequence of this expressiveness is that some well-written Haskell programs are, hands down, the most beautiful pieces of code i've ever seen (just pick a random post at, say, a Neighbohood of Infinity and you'll see what i mean; or read Richard Bird's Sodoku solver and compare his solution with one written in your favourite programming language).

Finally, let me say that i find programming in Haskell more difficult than programming in any other language i've used, with perhaps the only exception of Prolog. Sometimes, considerably so. And that's perfectly fine with me. For one thing, it makes it more interesting and rewarding. In addition, i'm convinced that that's the price to pay for being able to solve harder problems. I take issue with the frequent pleas to the effect that programming should be effortless or trivial: writing good programs is hard, and mastering the tools for doing it well takes, as with any other engineering or scientific discipline, hard work (why, i don't heard anyone complaining that building bridges or computing the effects of gravitational lensing is too difficult). There's no silver bullet.

All that said, please don't read the above as an apostasy letter announcing the embracement of a new religion. There's still much to be said in favour of dynamic languages, specially those in the Lisp family, whose malleability (fostered by their macro systems) is also a strength, in that they allow you to replicate some of the virtues i've been extolling in this post. Haskell lacks the power of homoiconicity, its template mechanisms feeling all but cranky, and that's a serious drawback in some contexts (i have yet to decide how serious, as i have yet to decide how much i'm missing in reflection capabilities). As always, it is a matter of trade-offs and, fortunately, nobody will charge you for high treason for using the language better fit to the problem at hand, or so i hope.

# programmers go bananas

## Introduction: lists galore

I learned programming backwards, plunging right on into C and, shortly after, C++ and Java from the very beginning. I was knee deep in complex data structures, pointers and abstruse template syntax in no time. And the more complex it all felt, the more i thought i was learning. Of course, i was clueless.

Reading SICP and learning about functional programming changed it
all. There were many occasions for revelation and awe, but one of my
most vivid recollections of that time is my gradual discovery of the
power of simplicity. At about half way into SICP i realised in wonder
that that beautiful and powerful world was apparently being
constructed out of extremely simple pieces. Basically, everything was
a list. Of course there were other important ingredients, like
procedures as first-class objects, but the humble list was about the
only data structure to be seen. After mulling on it for a little bit,
i saw where lists draw their power from: recursion. As you know, lists
are data types recursively defined: a list is either the empty list or
an element (its *head*) followed by another list (its *tail*):

list = []

list = a : list

where i'm borrowing Haskell's notation for the empty list ([]) and the list constructor (:), also known by lispers as () and cons. So that was the trick, i thought: lists have recursion built-in, so to speak, and once you've read a little bit about functional programming you don't need to be sold on the power and beauty of recursive programs.

It is often the case that powerful and beautiful yet simple constructs have a solid mathematical foundation, and only when you grasp it do you really realize how powerful, beautiful and amazingly simple that innocent-looking construct is. Lists, and recursive operations on them, are an excellent case in point. But the path connecting them to their mathematical underpinnings is a long and winding one, which lays in the realm of Category Theory.

I first became acquainted of the relationship between categories and recursive programming reading Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, by Erik Meijer, Maarten Fokkinga and Ross Paterson. Albeit very enjoyable, this paper presupposes a high degree of mathematical sophistication on the reader's side. I will try in this article to give you a simplified overview of the concepts involved, including Category Theory, its application to programming languages and what funny names like catamorphism, anamorphism or lambda-lifting have to do with your everyday list manipulations. Of course, i'll be only scratching the surface: interspersed links and the Further reading section provide pointers to more in-depth explorations of this wonderland.

## Categorical interlude

Categories are (relatively) simple constructs. A category consists of
a set **O** of objects, and a set **A** of arrows between elements of
**O**. Arrows are composable: if there's an arrow from *a* to *b*, and another
one from *b* to *c*, there must be an arrow from *a* to *c* (where *a*, *b* and *c*
are elements of **O**). Besides, they are *associative*: if you have arrows
from *a* to *b*, *b* to *c*, and *c* to *d*, you can go from *a* to *d* via two
different paths, namely, first from *a* to *c* and then from *c* to *d*, or
first from *a* to *b* and then from *b* to *d*. Finally, for each element *a*
in **O** there's an *identity* arrow which goes from *a* to itself (called an
*identity*), such that following this arrow changes nothing. These
properties are better visualized with a diagram (or a bit of
mathematical notation), as shown in the image on the right.

A category captures a *mathematical world* of objects and their
relationships. The canonical example of a category is **Set**, which
contains, as objects, (finite) sets and, as arrows, (total)
functions between them. But categories go far beyond modeling
sets. For instance, one can define a category whose objects are
natural numbers, and the 'arrows' are provided by the relation
"less or equal" (that is, we say that there is an arrow joining
two numbers *a* and *b* if *a* is less or equal than *b*)/./ What we are
trying to do with such a definition is to somehow capture the
essence of ordered sets: not only integers are ordered but also
dates, lemmings on a row, a rock's trajectory or the types of the
Smalltalk class hierarchy. In order to abstract what all those
categories have in common we need a way to go from one category to
another preserving the shared structure in the process. We need
what the mathematicians call an *isomorphism*, which is the
technically precise manner of stating that two systems are, in a
deep sense, analogous; this searching for commonality amounts to
looking for concepts or abstractions, which is what mathematics
and (good) programming is all about (and, arguably, intelligence
itself, if you are to believe, for instance, Douglas Hofstadter's
ideas).

To boot, our definition of a category already contains the concept
of isomorphic objects. Think of an arrow from *a* to *b* as an
operation that transforms *a* in *b*. An arrow from *b* to *a* will make
the inverse transformation. If composing both transformations
gives you the identity, you are back to the very same object *a*,
and we say that *a* and *b* are isomorphic: you can transform one into
the other and back at will. In a deep sense, this concept captures
a generic way of expressing equality that pervades all maths: if
you're not afraid of a little bit of maths, Barry Mazur's essay
When is a thing equal to some other thing? is an excellent
introduction to Category Theory with an emphasis in the concept of
equality. Among many other things, you will learn how the familiar
natural numbers can be understood as a category, or how an object
is completely defined by the set of its transformations (and,
therefore, how to actually get rid of objects and talk only of
transformations; i know this is stretching and mixing metaphors
(if not plain silly), but this stress in arrows, as opposed to
objects, reminded me of Alan Kay's insistence on focusing on
messages rather than objects). Another introductory article with
emphasis on categories as a means to capture sameness is R. Brown
and T. Porter's Category Theory: an abstract setting for analogy
and comparison.

Not only objects inside a category can be transformed into each other.
We reveal the common structure of two disjoint categories by means of
a functor mapping across two categories. A functor consists of two
functions: one that maps each object of the first category to an
object in the second, and another one putting in correspondence arrows
in one category with arrows in the second. Besides, these functions
must preserve arrow composition. Let me spell this
mathematically. Consider to categories, C and C' with object sets **O**
and **O'** and arrow sets **A** and **A'**. A functor F mapping C to C' will
consist then of two functions (Fo, Fa); the first one taking elements
of **O** to elements of **O'**:

Fo: **O** -> **O'**

Fo(a) in **O'** for every a in **O**

and the second one taking arrows from **A** to arrows in **A'**:

Fa: **A** -> **A'**

Fa(**f**) in **A'** for every **f** in **A**

and such that, if **f** is an arrow from a to b in C, Fa(**f**) is an arrow
from Fo(a) to Fo(b) in C'. Moreover, we want that following arrows in C
is 'analogous' to following them in C', i.e., we demand that

Fa(**fg**) = Fa(**f**)Fa(**g**)

In the left hand side above, we are composing two arrows in C and then
going to C', while in the right hand side we first take each arrow to
C' and, afterwards, compose them in there. If C and C' have the same
structure, these two operations must be equivalent. Finally, F must
preserve identities: if **i** is the identity arrow for an element a in **O**,
Fa(**i**)must be the identity arrow for Fo(a) in **O'**. The diagram on the
left shows a partial graph (i'm not drawing the identity arrows and
their mappings) of a simple functor between two categories, and ways
of going from an object a in the first category to an object x in the
second one which are equivalent thanks to the functor's properties.

As you can see in this simple example, the functor gives us the ability of seeing the first category as a part of the second one. You get a category isomorphism in the same way as between objects, i.e., by demanding the existence of a second functor from C' to C (you can convince yourself that such a functor does not exist in our example, and, therefore, that the two categories in the diagram are not isomorphic).

You have probably guessed by now one nifty property of functors: they let us going meta and define a category whose objects are categories and whose arrows are functors. Actually, Eilenberg and MacLane's seminal paper General theory of natural transformations used functors and categories of categories to introduce for the first time categories (natural transformations are structure-preserving maps between functors: this Wikipedia article gives an excellent overview on them).

But enough maths for now: it is high time to show you how this rather abstract concepts find their place in our main interest, programming languages.

## Categories and programming languages

About the only similarity between C and Haskell programming is that one spends a lot of time typing ASCII arrows. But of course, Haskell's are much more interesting: you use them to declare the type of a function, as in

floor:: Real -> Int

The above stanza declares a function that takes an argument of type real and returns an integer. In general, a function taking a single argument is declared in Haskell following the pattern

fun:: a -> b

where *a* and *b* are types. Does this ring a bell? Sure it does: if we
identify Haskell's arrows with categorical ones, the language types
could be the objects of a category. As we have seen, we need identities

id:: a -> a

id x = x

and arrow composition, which in Haskell is denoted by a dot

f:: b -> c

g:: a -> b

fg:: a -> b -> c

fg = f . g

Besides, associativity of arrow composition is ensured by Haskell's referential transparency (no side-effects: if you preserve referential transparency by writing side-effect free functions, it won't matter the order in which you call them): we've got our category. Of course, you don't need Haskell, or a statically typed language for that matter: any strongly typed programming language can be modelled as a category, using as objects its types and as arrows its functions of arity one. It just happens that Haskell's syntax is particularly convenient, but one can define function composition easily in any decent language; for instance in Scheme one would have

(define (compose f g) (lambda (x) (f (g x)))

Functions with more than one arguments can be taken into the picture by means of currying: instead of writing a function of, say, 2 arguments:

(define (add x y) (+ x y))

(add 3 4)

you define a function which takes one argument (x) and returns a function which, in turn, takes one argument (y) and returns the final result:

(define (add x) (lambda (y) (+ x y)))

((add 3) 4)

Again, Haskell offers a pretty convenient syntax. In Haskell, you can
define `add`

as:

add x y = x + y

which gets assigned, when applied to integers, the following type:

add:: Int -> (Int -> Int)

that is, `add`

is not a function from pairs of integers to integers, but
a function that takes an integer and returns a function of type
`Int -> Int`

.
Finally, we can also deal with functions taking no arguments and
constant values by introducing a special type, called unit or 1 (or
void in C-ish), which has a unique value (spelled `()`

in
Haskell). Constants of our language (as, e.g., `True`

or 43.23) are then
represented by arrows from 1 to the constant's type; for instance,
`True`

is an `1 -> Boolean`

arrow. The unit type is an example of what in
category theory is known as a terminal object.

Now that we have successfully modelled our (functional) programming
language as a category (call it C), we can use the tools of the theory
to explore and reason about the language constructs and
properties. For instance, functors will let me recover the original
motivation of this post and explore lists and functions on them from
the point of view of category theory. If our language provides the
ability to create lists, its category will contain objects (types) of
the 'list of' kind; e.g. `[Int]`

for lists of integers, `[Boolean]`

for
lists of Booleans and so on. In fact, we can construct a new
sub-category CL by considering list types as its objects and functions
taking and returning lists as its arrows. For each type a we have a
way of constructing a type, [a] in the sub-category, i.e., we have a
map from objects in C to objects in CL. That's already half a
functor: to complete it we need a map from functions in C to functions
in CL. In other words, we need a way to transform a function acting on
values of a given type to a function acting on lists of values of the
same type. Using the notation of the previous section:

Fo(a) = [a]

Fa(f: a -> b) = f': [a] -> [b]

`Fa`

is better known as `map`

in most programming languages. We call the
process of going from `f`

to `f'`

lifting (not to be confused with a
related, but not identical, process known as lambda lifting), and it's
usually pretty easy to write an operator that lifts a function to a
new one in CL: for instance in Scheme we would write:

(define (lift f) (lambda (lst) (map f lst)))

and for `lift`

to truly define a functor we need that it behaves well
respect to function composition:

(lift (compose f g)) = (compose (lift f) (lift g))

We can convince ourselves that this property actually holds by means of
a simple example. Consider the function `next`

which takes an integer to
its successor; its lifting `(lift next)`

will map a list of integers to
a list of their successors. We can also define `prev`

and `(lift prev)`

mapping (lists of) integers to (lists of) their predecessors.
`(compose next prev)`

is just the identity, and, therefore,
`(lift (compose next prev))`

is the identity too (with lifted
signature). But we obtain the same function if we compose `(lift next)`

and `(lift prev)`

in CL, right? As before, there's nothing specific to
Scheme in this discussion. Haskell even has a
Functor type class
capturing these ideas. The class defines a generic lift operation,
called `fmap`

that, actually, generalizes our list lifting to arbitrary
type constructors:

fmap :: (a -> b) -> (f a -> f b)

where `f a`

is the new type constructed from `a`

. In our previous
discussion, `f a = [a]`

, but if your language gives you a way of
constructing, say, tuples, you can lift functions on given types to
functions on tuples of those types, and repeat the process with any
other type constructor at your disposal. The only condition to name it a
functor, is that identities are mapped to identities and composition is
preserved:

fmap id = id

fmap (p . q) = (fmap p) . (fmap q)

I won't cover usage of type constructors (and their associated functors) other than lists, but just mention a couple of them: monads, another paradigmatic one beautifully (and categorically) discussed by Stefan Klinger in his Programmer's Guide to the IO Monad - Don't Panic (also discussed at LtU), and the creation of a dance and music library, for those of you looking for practical applications.

## To be continued…

Returning to lists, what the lifting and categorical description above buys us is a way to formalize our intuitions about list operations, and to transfer procedures and patterns on simple types to lists. In SICP's section on Sequences as conventional interfaces, you will find a hands-on, non-mathematical dissection of the basic building blocks into which any list operation can be decomposed: enumerations, accumulators, filters and maps. Our next step, following the bananas article i mentioned at the beginning, will be to use the language of category theory to provide a similar decomposition, but this time we will talk about catamorphisms, anamorphisms and similarly funny named mathematical beasts. What the new approach will buy us is the ability to generalize our findings beyond the list domain and onto the so-called algebraic types. But this will be the theme of a forthcoming post. Stay tunned.

## Further reading

The best introductory text on Category Theory i've read is Conceptual Mathematics : A First Introduction to Categories by F. William Lawvere (one of the fathers of Category Theory) and Stephen Hoel Schanuel. It assumes no sophisticated mathematical background, yet it covers lots of ground. If you feel at home with maths, the best option is to learn from the horse's mouth and get a copy of Categories for the Working Mathematician, by Mac Lane.

The books above do not deal with applications to Computer Science, though. For that, the canonical reference is Benjamin Pierce's Basic Category Theory for Computer Scientists, but i find it too short and boring: a far better choice is, in my opinion, Barr and Well's Category Theory and Computer Science. A reduced version of Barr and Well's book is available online in the site for their course Introduction to Category Theory. They are also the authors of the freely available Toposes, Triples and Theories, which will teach you everything about monads, and then more. Marteen Fokkinga is the author of this 80-pages Gentle Introduction to Category Theory, with a stress on the calculational and algorithmical aspects of the theory. Unless you have a good mathematical background, you should probably take gentle with a bit of salt.

Let me close by mentioning a couple of fun applications of Category Theory, for those of you that know a bit about it. Haskell programmers will like this implementation of fold and unfold (as a literate program) using F-(co)algebras and applied to automata creation, while those of you with a soft spot for Physics may be interested in John Baez's musings on Quantum Mechanics and Categories.