07 March 2021

Review: Why FP Matters (Hughes)

"Why Why FP Matters Matters"

A few years ago, I came across the blog post Why Why Functional Programming Matters Matters. It was a major factor in my decision to invest heavily into learning functional programming, so it's safe to say it's had a huge impact on my development as a software engineer. In fact, I highly recommend reading everything Reginald Braithwaite has ever written; I can't claim to have read it all, but I can say that I've never regretted any minute spent doing it.

Today, however, I want to talk about the paper that blog post talks about. It is titled "Why Functional Programming Matters", and if you have never read it, you should go read it right now. Don't worry, I'm not going anywhere. As a piece of static text, I have all the patience in the world.

As a brief summary, the paper argues that functional programming yields better programs because it adds constraints on programs, which in turn lets the programmer make assumptions about program segments (i.e. functions), which increases reasoning ability, and thus composability, and therefore makes it possible to use new types of "glue" between code segments, thereby improving code reuse. I'm not going to repeat all of the arguments, you just read them. (Seriously, go read the paper if you haven't.)

A couple weeks ago I decided to reread the paper, to check if everything I've learned in the past dozen years since I first read it would allow me to gather any new insight. I decided that a superficial reading would not be enough, and that to force me to really carefully read everything I would reimplement the code samples. I also thought this would be a nice test of how Clojure holds up to Hughes' definition of functional programming.

This has led me to a few realizations that I had not made while simply reading the paper, which is the subject of this blog post. If you're interested in more details than are presented here, you can take a look at the entire Clojure code here.

Types

In modern days, Haskell has taken the mantle of the Avatar of Functional Programming, to the point that many programmers do not meaningfully distinguish "functional programming" from "strong static typing". This is obviously a problem for Clojure as it is, as the scholars say in a disdain-dripping tone, untyped.

Interestingly, Hughes' paper does not discuss static typing at all, and does not seem to include it in his definition of functional programming. Instead, he defines functional programming as the absence of an assignment operator.1

Furthermore, while it may look superifically like Haskell, the language used in the paper (Miranda) does not seem to respect our modern notions of static typing either. For example, we can extract the following code snippet from the paper:

listof * ::= Nil | Cons * (listof *)
treeof * ::= Node * (listof (treeof *))

which would roughly correspond to the Haskell type definitions:

data List a = Nil | Cons a (List a)
data Tree a = Node a (List (Tree a))

So far, so good. But then Hughes writes this function to fold over a tree:

foldtree f g a (Node label subtrees) = f label (foldtree f g a subtrees)
foldtree f g a (Cons subtree rest) = g (foldtree f g a subtree)
                                       (foldtree f g a rest)
foldtree f g a Nil = a

This really threw me off and it took me a while to realize that the reason I had trouble translating that function to a meaningful implementation in my Clojure code was that is it not properly typed: the fourth argument is assumed to be either a Tree or a Node. Knowing that, it becomes trivial to implement, but I was certainly not expecting my source material to have type issues.

Higher-order functions

According to Hughes, the first new type of glue that is enabled by the absence of assignments — or, more generally, side-effects — is higher-order functions. I do not fully agree with that, as it seems perfectly possible to use higher-order functions in an imperative language, with some of the functions performing side-effects.

Most of the functions defined in Section 3 of the paper translate fairly directly to Clojure. For example:

(foldr f x) Nil = x
(foldr f x) (Cons a l) = f a ((foldr f x) l)

can be written in "pure" Clojure as:

(defn foldr
  [f init]
  (fn [ls]
    (if (empty? ls)
      init
      (f (first ls) ((foldr f init) (rest ls))))))

or, if we want to get closer to the pattern matching syntax used in the paper, using core.match:

(defn foldr
  [f init]
  (fn [ls]
    (match ls
      ([] :seq) init
      ([hd & tl] :seq) (f hd ((foldr f init) tl)))))

The main sticking point throughout this section is currying: Clojure has multi-argument functions, and thus no automatic currying. This is why we have to explicitly return a function here.

The previously-mentioned foldtree function is interesting:

foldtree f g a (Node label subtrees) = f label (foldtree f g a subtrees)
foldtree f g a (Cons subtree rest) = g (foldtree f g a subtree)
                                       (foldtree f g a rest)
foldtree f g a Nil = a

A direct implementation of this in Clojure would read:

(defn foldtree
  [node-fn list-fn zero [node children]]
  (node-fn node
           (reduce list-fn
                   zero
                   (map #(foldtree node-fn list-fn zero %) children))))

assuming a representation of lists as Clojure vectors and nodes as two-element vectors.

This lets us write the last function of Section 3, maptree:

maptree f = foldtree (Node . f) Cons Nil

as follows:

(defn maptree
  [f]
  #(foldtree
     (fn [node children] [(f node) children])
     conj [] %))

Again, currying is the main difference here: (Node . f) is a succint way to express a function that takes one argument, applies f to it, then waits for a second argument to construct a Node. With our chosen representation of nodes as tuples, the Clojure equivalent of the Node function would be vector called with two arguments, and while Clojure does have a direct equivalent for . (comp), there is no direct syntax for delaying evaluation until another argument is provided. The most faithful transcription of (Node . f) would be to explicitly curry vector:

(comp (fn [x] (fn [y] (vector x y))) f)

Overall, if we're willing to wrap all of our functions in this way (including redefining the standard library functions), all of the examples of Section 3 translate in a very direct way.

One interesting point that does not really apper until we try to reuse maptree in Section 5 is that there is no good way to write a lazy foldr in Clojure. Clojure does have some support for laziness in the form of lazy lists2, but the language overall is strict.

To be fully lazy, foldr needs the function f itself to be lazy in its second argument, which cannot be generally expected of Clojure functions. If we were ready to rewrite all our functions, as we could do for currying, we could have foldr looking something like:

(defn foldr
  [f init]
  (fn [ls]
    (match ls
      ([] :seq) init
      ([hd & tl] :seq) (f hd (delay ((foldr f init) tl))))))

but f would need to know to force its second argument, and that gets messy very quickly.

This limited support for laziness may not be as problematic as one might at first think, though. Consider, for example, the function sum described in the paper:

sum = foldr (+) 0

which readily translates to

(def sum (foldr + 0))

Even in Miranda (or Haskell), this is not lazy in any meaningful sense: the + function is not able to return a partial result and will always need to fully evaluate both of its arguments. Most use-cases of foldr where we want laziness are actually cases where the expected result is a list. In a lazy language, such lazy list operations can be built atop foldr using the laziness of cons, but realistically they are provided by the standard library (map, filter, etc.) and what they are built upon is only of limited relevance. These operations can be built lazily even in the absence of a lazy fold.

For example, the paper builds a function maptree defined by:

maptree f = foldtree (Node . f) Cons Nil

Just like foldr, it is pretty hard to build a lazy foldtree in Clojure. However, the operations for which that matters are rare, and examples like maptree can easily be built to be lazy by not building them on top of fold:

(defn maptree
  [f [value children]]
  [(f value) (map #(maptree f %) children)])

This would translate in Miranda to:

maptree f (Node value children) = Node (f value) (map (maptree f) children)

which, even in Miranda, I like better than the version using foldtree.

Lazy lists

Section 4 of the paper is ostensibly about numerical computation, but is really about lazy lists. As I would expect, Clojure stands up very well here, as it has good support for lazy lists to start with and the code in this section is not too reliant on currying. For example, the square root computation is given in the paper as:

next n x = (x + n/x) / 2
repeat f a = Cons a (repeat f (f a))
relative eps (Cons a (Cons b rest))
  = b, if abs (a/b - 1) <= eps
  = relative eps (Cons b rest), otherwise
relativesqrt a0 eps n = relative eps (repeat (next n) a0)

and translates fairly directly to Clojure as:

(defn next
  [n x]
  (/ (+ x
        (/ n 1.0 x))
     2.0))

(defn repeat
  [f a]
  (cons a (lazy-seq (repeat f (f a)))))

(defn relative
  [eps [a b & rs]]
  (if (>= eps (Math/abs (dec (/ a 1.0 b))))
    b
    (relative eps (cons b rs))))

(defn relativesqrt
  [a0 eps n]
  (relative eps (repeat #(next n %) a0)))

where (cons hd (lazy-seq tl)) is how you build a lazy list in Clojure.

The main surprise in Section 4 did not come from any language issue, but rather from the example computation chosen by Hughes. At the end of the section, he presents:

super (integrate sin 0 4)

as an example of a quickly converging integral. The definition of super is given as:

super s = map second (repeat improve s)
improve s = elimerror (order s) s
elimerror n (Cons a (Cons b rest)) =
  Cons ((b * (2^n) - a)/(2^n-1)) (elimerror n (Cons b rest))
order (Cons a (Cons b (Cons c rest))) = round (log2 ((a - c)/(b - c) - 1))
round x = x rounded to the nearest integer
log2 x = the logarithm of x to the base 2

and integrate is given as:

integrate f a b = integ f a b (f a) (f b)
integ f a b fa fb = Cons ((fa + fb) * (b - a) / 2)
                         map addpair (zip2 (integ f a m fa fm)
                                           (integ f m b fm fb))
where m = (a + b) / 2
      fm = f m

While translating that into Clojure poses no particular issue, the resulting computation actually diverges almost immediately:

whyfp.core=> (take 10 (super (integrate2 #(Math/sin %) 0 4)))
(1.0617923583434352
 1.5780150025674877
 1.690241935320747
 ##-Inf
 ##NaN
 ##NaN
 ##NaN
 ##NaN
 ##NaN
 ##NaN)

I do not have a good explanation for why this happens. The other final example:

improve (integrate f 0 1)
where f x = 1/(1 + x * x)

does converge to \(\pi/4\) as claimed.

AI & tree walking

The last (pre-conclusion) section of the paper is about extending the lazy list approach from Section 4 to trees, with a typical "AI" game-tree-walking running example. Provided we use a lazy implementation of maptree as previously discussed, there is no new information in this section about how Clojure holds up as a language supporting functional programming, nor indeed any real new information about the advantages of functional programming.

This section was a lot more effort to write code for, because it builds upon undefined ("left as an exercise") functions for generating the game tree being walked.

Before we conclude...

So far, Clojure seems to be holding up pretty well. The main issue I encountered while trying to convert the code in this paper was around currying; I personally do not believe that currying is a good idea to begin with, so I am not bothered by that.

I am, however, a little bit bothered by the fact that all of the use-cases for laziness in this paper boiled down to lazy lists. Lazy lists are an important use-case, but they make a poor argument for making your entire language lazy, as it is quite easy to integrate lazy lists in an otherwise eager language (as Clojure demonstrates).

So before I conclude this post, I did want to show an example of laziness that, as far as I can tell, is not easily expressed as a lazy list, and which I have no idea how to express in Clojure. This example comes from the advent of code 2019 set of problems. Without going too far into the details, by day 6 we have built a small virtual machine interpreter (dubbed "IntMachine") that one could model, as I did, as a function that looks like3:

execIntCode :: [Int] -> [Int] -> [Int]

Pretty straightforward, right? It's a function that takes two lists of integers, one that represents whatever the machine is going to read and one that represents the internal code of the machine, and it writes out some output, also as a list of ints.

On day 7, we need to plug these machines together. This is straightforward enough to do when you do it linearly. Here is a diagram taken from the puzzle explanation:

    O-------O  O-------O  O-------O  O-------O  O-------O
0 ->| Amp A |->| Amp B |->| Amp C |->| Amp D |->| Amp E |-> (to thrusters)
    O-------O  O-------O  O-------O  O-------O  O-------O

This translates directly to code that will look similar in pretty much any language (we define the function run as all five machines are running the same "Amp" code):

chain code [pa, pb, pc, pd, pe] =
  let run inputs = Lib.execIntcode inputs code
      a_out = run (pa:[0])
      b_out = run (pb:a_out)
      c_out = run (pc:b_out)
      d_out = run (pd:c_out)
      e_out = run (pe:d_out)
  in e_out

This is super-straightforward: run the first machine, collect its output, use that as the input for the next machine. Right? So far, so good.

In the second part of that same problem, we need to add a feedback loop: suddenly we aslo want the output of the last machine to serve as additional input to the first machine. Here is the diagram from the problem description:

      O-------O  O-------O  O-------O  O-------O  O-------O
0 -+->| Amp A |->| Amp B |->| Amp C |->| Amp D |->| Amp E |-.
   |  O-------O  O-------O  O-------O  O-------O  O-------O |
   |                                                        |
   '--------------------------------------------------------+
                                                            |
                                                            v
                                                     (to thrusters)

How do you handle that? None of my code thus far had been written with this kind of use-case in mind, but because I had written it in a lazy language, the solution was as simple as:

feedback code [pa, pb, pc, pd, pe] =
  let run inputs = Lib.execIntcode inputs code
      a_out = run (pa:0:e_out)
      b_out = run (pb:a_out)
      c_out = run (pc:b_out)
      d_out = run (pd:c_out)
      e_out = run (pe:d_out)
  in last e_out

That's right. I can just plug the output of the fifth machine directly to the input of the first machine. Just like that.

I find this use-case way more convincing than lazy list filtering, but maybe that's just because I have been used to lazy lists in Clojure for a long time, whereas this kind of full-language laziness is still fairly new to me.

Conclusion

The paper was first written in 1989, at which point I was not actively involved in software engineering. I imagine the argument for higher-order functions, garbage collection4 and immutable data was not an easy one to make. This is 2021, though, and hopefully by now everyone knows that higher-order functions are good and that sections of code that are free of side effects are much easier to work with, even in languages and programs where they do not comprise the entirety of the code base.

This leads me to think that the main open question left here is that of laziness. My personal opinion at this point is that laziness is just not worth it in general. The IntMachine example above is great, but the cost is a much degraded development experience on a daily basis, for the entire code base. And I have only ever had that one example of useful non-list laziness.

The price of laziness today is that it makes code harder to debug5, and it makes side effects harder to reason about. You may be thinking "hang on, isn't the whole point of this paper that you should not have side effects?", and you'd be right. But cute numerical computations aside, real code does have side effects (even when you hide them in a monad) because it does need to actually do something. Also, in this context I consider performance as a side effect, and while laziness does not strictly speaking make performance worse6, it does make it a lot harder to predict and reason about.

So, for now, I tend to prefer the Clojure approach, which yields most of the benefits of laziness with few of its downsides. However, better tooling for lazy languages could tip my opinion on this topic.


  1. I personally like this definition. I tend to think of it as imperative programming being ultimately based on the notion of a Turing machine, and functional programming being ultimately based on lambda calculus.

  2. By "good support" here I do not just mean that lazy lists exist: they are a very central abstraction in the standard library and most collection functions work lazily and return lazy lists, even when provided with non-lazy lists as inputs.

  3. This code is split between a library with the IntMachine code and the solution for day 7.

  4. While garbage collection is not explicitly addressed in the paper, higher-order functions with immutable data are fairly hard to implement without it, not to mention laziness. I believe garbage collection is as essential to functional programming as functions themselves.

  5. Stack traces don't make much sense in a lazy context.

  6. In fact, one could argue that lazy semantics at the language level give a compiler more freedom to implement optimizations, and therefore could lead to better performance overall. Haskell performance in general is quite good for such a high-level language. My issue here is not raw performance but predictability.

Tags: fp papers