Arrows first encounter

UPDATE: My post appeared on Reddit's programming feed. Thank you! Also, my post's title should've been "Point-free programming style first encounter", but I'm too lazy to change it ^_^

I was really busy lately so no post around for a while. Surprisingly, I wasn't busy wrinting code but actually thinking about the problem: this is feasible in Haskell as the code is more concise and the typing system keeps you from doing stupid design mistakes (or at least make them appear earlier than in other programming language).

Now, done with the ritual Haskell-awe prologue, here my post for today.

Lately, I've been working to a program manipulating objects which turned out to be monomials, which I represented as a pair of (Double,[Int]), i.e.:

(4.0,[1,2,-1,-2]) -> 4.0 x(1,1) x(2,2) x(3,-1) x(4,-2)

where x(i,j) are unknowns indexed by a pair. Since I assume that the first component of the unknowns' pairs will be the sequence 1,2,3,... I only need to specify the second component. I know, it's kinda weird, but it has to do with my domain problem which is related to finite difference operators for PDE.

Now I want to define a "product" operator which works like this:

(k,[b,c,d]) * (h,[e,f,g]) = (k*h,[a+e, c+f, d+g])

I can define this like follows:

prod (k,ll) (h,rl) = (k*h, zipWith (+) ll rl)

This is perfectly correct and good as it's very expressive.

Now, I want to apply the same function monadically to list arguments, i.e.:

prod :: [(Double,[Int])] -> [(Double,[(Double,[Int])] -> [(Double,[Int])]

To do this, I'll use liftM2 which take a function of two arguments and applies it to every pair formed taking an element from the first list and an element from the second. So:

prodM = liftM2 prod

But I'll never use prod by itself, so let's put this in a lambda abstraction and rename prodM to prod:

prod = liftM2 $ \(k,ll) (h,rl) -> (k*h, zipWith (+) ll rl)

Again, this is correct but my Haskell soul is becoming more strict and such a formulation sounds bad to me (I could've felt perfectly right with some month ago). I don't like lambdas when used this way, I see as an intrusive element in the perfect world of function combinators/operators (and liftM2 is such a combinator). More, I had to specify that prod requires two tuples while it'd be neat if the compiler could infere this automatically: I see this as code replication issue.

The solution to this dilemma is using point-free style programming. I'll abuse arrow combinators a bit. Specifically, there's the (***) operator which takes two function f and g and applies them to a tuple:

(f *** g) (a,b) = (f a, g b)

What if I need to combine two tuples using two separate binary operators? I.e.:

(f ***' g) (a,b) (c,d) = (f a c, g b d)

Arrows module doesn't provides such a combinator but we could build one. First, if we combine f and g using (***), we get a pair of two functions as f and g are partially evaluated. What we need now is to interpose (***) between these two functions. How to do this? Simple, we just need to uncurry (***). So, now:

(f ***' g) = uncurry (***) . (f *** g)

and we're done as f ***' g returns a function taking two tuples and apply f and g separately to each component. This is good as we can rewrite prod as:

prod = liftM2 $ uncurry (***) . ((*) *** (zipWith (+)))

which is essentially a smart combination of functions. Notice that no arguments of prod has been specified: this is where point-free programming style really lies. Changing the way prod is written will change the type of its arguments, in single place, without requiring you to rewrite them in two places (i.e. the argument list).

Random numbers (where it all started)

At first, Haskell seemed a reasonable language... until Monads appeared! Until then, purity was a clear plus of Haskell but... what about a random number generator (RNG)?

A pure function is deterministic: it depends only on its inputs and, as computers are deterministic as well, its output must be the same given the same input data.

Conversely, an RNG function (at least as it's known in the imperative world) must be non-deterministic and return a different outcome whenever it's called (without arguments).

So, how to code an RNG function in Haskell? Enter Monads.

Haskell can't give up on purity, so the solution is simple: instead of returning a value, an RNG function returns a computation.

A computation is like a machine with some pipes going in. By itself, the machine doesn't do anything. But, if you run it, providing fuel through the pipes, it will eventually produce something.

The machine is pure, while the fuel is impurity (or side-effects). The fuel comes from the environment under which the machine is run.

So, an RNG function is actually a machine which, when run under the special enviroment of the IO Monad, provides a number. More, you can build a special RNG machine (i.e. write an RNG computation) producing an infinite list of random numbers: just fill in sufficient fuel until more random numbers are needed!

In this way, Haskell let's you stay on the pure side when building the computation, letting you do all sort of things a computer theorist would do to a program (applying transformations, proving properties...), and getting you dirty when running your computation.

Sometimes it's hard to catch up with the meaning of Monads at first because the first Monad you encounter is the IO Monad, which is implicitly run by the Haskell run-time environment. So, you lose the concept that a Monad computation must be run to produce something useful.

For instance, the State Monad has runState function. But, as I said in one of my previous post, IO has its own run function: unsafePerformIO, from Foreign module.

So, let's see how to build an RNG computation:

randomList r = do g <- getStdGen
return $ randomRs r g

randomList takes a pair representing a range of numbers and return a computation that, when run, provides an infinite list of numbers. Indeed, randomList's type is:

Num a => (a,a) -> IO [a]

Specifically, when you state:

g <- getStdGen

you are saying something like: "g is a pipe. When you run this computation, please provide a StdGen value here". randomRs is a plain pure function taking a StdGen (which is basically a seed with a next function iterated to provide an infinite list of random number from it) and the range to which random numbers must be belong to.

How do we run this machine? Simple: put unsafePerformIO in front of it.

randomList r = unsafePerformIO $
do g <- getStdGen
return $ randomRs r g

Now randomList produces an infinite list of random numbers because it'll run the computation inside it.

But, beware! randomList is now and impure function masked as a pure function. Haskell could tricked by this masking and make the wrong thing misled by the purity assumption.

So, I promised myself not to write another Monad tutorial, but I did, at last!

Again a post about Haskell performances

I've found an interesting article describing how to make an Haskell program run as fast as the OCaml version, which is often regarded as the workhorse among FP languages. Three points grabbed my attention the most:

1. ghc 6.8 seems to boost the performances of Haskell programs: in some cases the benchmarks show programs run twice as faster as those compiled with ghc 6.6.
2. Faster Haskell programs are written similarly to OCaml version.
3. OCamls versions are no longer than their Haskell versions.

Something to think about!

"Making Haskell programs faster and smaller"

I've just found this article through a nice blog about programming languages. Haven't read it yet, but surely something to put apart!

The typical Haskell programmer's customs

When learning a new programming language, it's very important to learn the best practices arising from the community. For example, in Java you learn pretty soon to name your stuff in CamelCase: you'll make your code more readable by others and, at the same time, you're likely to avoid the mistakes others did before.

Haskell is no different and I found a good article by Bryan O’Sullivan, somewhat related to this.

Is Haskell really expressive?

My idea of an expressive program is one whose meaning (or at least the meaning of its functions when taken apart) is kind of intuitive.

Haskell is often remarkably pointed out as an expressive language. Nevertheless, I'd say that a language is simply a weapon in the hand of the developer and that expressiveness mostly depend solely on him.

Indeed, consider the following Haskell piece of code:

cartesian = (foldl (liftM2 $ flip (:)) [[]]).(map (\n -> [0..n-1]))

Apart from its name, would you tell that this function evaluates a multiple cartesian product? Actually you need to have an intermediate knowledge of Haskell and its libraries to catch the meaning of this function:

• You should know that foldl is like a generic "summation"
• You should know that liftM2 will lift a function taking two arguments into a Monad (the List Monad actually).
• You should know that semantics of the List Monad has to do with non-determinism, i.e. passing two lists to a lifted function will result in applying that function to each and every pair built from them.

Writing programs this way is really concise but I sometimes feel like it's not really intuitive. My understanding of Haskell programs improves everyday and lots of Haskell snippets I've read in the past are now less obscure. But I think my most importat breakthrough in understanding the language will happen when I'm able to make the above snippet's meaning apparent even for the almost occasional (and non-Haskell) programmer.

Haskell concurrency

Here's an article introducing Haskell concurrency constructs for the two paradigms out there: message passing (à-la Erlang) and shared-memory (the rest of us).

HAppS

http://bluebones.net/2007/09/simple-haskell-web-programming-with-happs/

Control.Monad.Fix: recursive lambda abstractions

http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Fix.html

CleverCSS

http://sandbox.pocoo.org/clevercss-hs/

Tracing your code... the dirty way!

Problem: you want to trace your Haskell program but you're scared that touching anything might thrash your type hackery.

Solution: Use Debug.Trace.trace wherever you want to print something on the console:

Before: map f (x:xs) = (f x):(map f xs)
After: map f (x:xs) = trace "Here I am" $ (f x):(map f xs)

Explanation: I was trying to understand how HAppS worked and I really missed the possibility to put a harmless print statemente here and there in the code just to get the feeling of what was happening in the behinds, as I'd do in an imperative language like python.

The problem is that we have to deal with Haskell purity so if you want to print something in a Haskell program, you have to do it inside the IO Monad.

So far so good, but IO is not available everytime you need it... at least not the "official" one. In fact Haskell allows you to enter the IO Monad anytime you need it and perform an IO action straight away using the unsafePerformIO function.

"Unsafe" means that you use it at your own risk, since you're doing something impure in a pure environment. Of course, trace has nothing to do with the real meaning of the traced funcion, so it's perfectly safe to use it that way.

I spent almost an hour to figure out how to do this, so I hope this trick will save you time.