## The fixed point combinator and memoization in Haskell

The Fibonacci numbers $F_n$ for $n \in \mathbb{N}_0$ are defined by
$F_n = n$ if $n \le 1$ and $F_n = F_{n-1} + F_{n-2}$ otherwise.
In Haskell this becomes

fib n | n <= 1    = n
| otherwise = fib (n-1) + fib (n-2)


The translation is almost verbatim. Unfortunately, as a show-case for functional programming, the definition of fib is pretty terrible. The problem can be seen from the evaluation tree for fib 5 below.

The tree has $F_6 = 8$ leaves. More generally, one can show by induction that the tree for fib n has $F_{n+1}$ leaves. Clearly computing $F_n$ by adding up $F_{n+1}$ numbers, of which $F_n$ are $1$ and $F_{n-1}$ are $0$, is not time efficient.

A more minor objection is that the domain of fib is not $\mathbb{N}_0$. In fact, thanks to Haskell’s polymorphism, fib is defined on values of any numeric type. For example fib 1.5 evaluates to fib 0.5 + fib (-0.5), which, since both arguments are $\le 1$, evaluates to $0.5-0.5 = 0$. It seems remarkable that, without any thought (in fact, with the deliberate absence of thought), we have extended the domain of the Fibonacci function to all numbers understood by Haskell. Anyway, this objection is easily addressed:

fib' :: Integer -> Integer
fib' n | n <  0    = undefined
| n <= 1    = n
| otherwise = fib' (n-1) + fib' (n-2)


Here undefined is the error value, also known as ‘bottom’, that silently inhabits every type.

#### Memoization

To address the serious problem, we need to tell Haskell to remember, or memoize the values of fib, rather than recompute them each time. The following idiom works well and is amenable to generalization.

fibM = \n -> values !! n
where values = [fibAux m | m <- [0..]]
fibAux n | n <= 1    = n
| otherwise = fibM (n-2) + fibM (n-1)


To understand why this works, one has to have some understanding of Haskell’s evaluation model. Haskell is lazy. Roughly put: no value is evaluated unless its necessary for the computation to proceed. Thus if fibM is evaluated at 30 (say), the apparently infinite list [fibAux m | m <- [0..]] is only evaluated as far as its 31st member. Moreover, since values appears at the top level of the ‘where’ in the definition of fibM, it is shared between all invocations of fibM. (To use the correct terminology, it is a constant applicative form’.) Sharing a list means sharing all its members, so each list member, and so each value of fibM is computed at most once.

This point is maybe made more clearly by considering a simpler definition, that fails to memoize correctly.

fibM' n | n <= 1    = n
| otherwise = values !! (n-1) + values !! (n-2)
where values = [fibM' m | m <- [0..]]


The problem is that the list values could potentially depend on $n$ (considering replacing [0..] with [0..n]). and so cannot be shared between all invocations of fibM'. The difference with fibM is that the closure for fibM includes values; in contrast, each evaluation of fibM' n creates a new closure, with a new values list. (So fibM' uses quadratic storage, as well as exponential time—what a triumph!)

Writing fibM needs some care. We have to introduce an auxillary part-memoized’ function, and in turn we have to be careful in fibAux to use memoized values, rather than recursive calls to fibAux. Maybe we could make all this a one-time effort, by writing a function memoize :: (a -> a) -> (a -> a), such that memoize fib evaluates to a memoized version of fib?

A bit of thought suggests this is too ambitious. For example, what if the type a does not support testing for equality? Then we cannot even know if we are evaluating the memoized function on an argument already seen. But surely, we can write a memoizer for functions with domain and range the natural numbers less than $2^{64}$, with type signature memoizeInt :: (Int -> Int) -> (Int -> Int)?

I don’t understand Haskell or functional programming well enough to be completely confident here, but my belief is that no such memoizer can be defined. Consider the following failed attempt.

memoizeInt :: (Int -> Int) -> (Int -> Int)
memoizeInt f = \m -> values !! m
where values = [f m | m <- [0..]]


This is a useful memoizer for a function such as fac n = product [1..n] that may require a lot of work for each evaluation, but do not call themselves recursively. For instance, in ghci 7.8.4 on my laptop, evaluating fac (10^7) takes 4.5s; after defining facM = memoizeInt fac, the first evaluation of facM (10^7) takes 11s, and subsequent evaluations of facM (10^7) are instantaneous. (The result is in every case 0, because of the limited range of the Int type.)

The problem in fibM'' = memoizeInt fib is that when Haskell evaluates fibM'' n, the recursive evaluations fib (n-2) and fib (n-1) call the original fib function, not the memoized function fibM''. We would like to somehow ‘reach into’ the definition of fib so that the recursive calls are redirected, but the immutability of Haskell values makes this impossible.

#### Recursive functions as fixed points

The difficulty seems to be with functions that call themselves. Ignoring memoization, how do compilers cope such functions? We digress to consider a strategy sheds some light on the memoization problem. Consider the function fibBuilder below.

fibBuilder :: (Int -> Int) -> (Int -> Int)
fibBuilder f = g
where g n | n <= 1    = f n
| otherwise = f (n-2) + f (n-1)


The input of fibBuilder is a function. The output is another function which we shall see is a ‘better approximation’ to a correct Fibonacci function. Consider for example

[id m | m <- [0,1,2,3]
[g m | m <- [0,1,2,3]] where g = fibBuilder id
[h m | m <- [0,1,2,3]] where h = (fibBuilder (fibBuilder id))


The output for the identity function id is clearly [0,1,2,3]. Chasing through the evaluations of g we get g 2 $\rightarrow$ id 0 + id 1  $\rightarrow$ 0 + 1 $\rightarrow$ 1 and g 3  $\rightarrow$ id 1 + id 2 $\rightarrow$ 1 + 2 $\rightarrow$ 3. Hence h 3 $\rightarrow$ g 2 + g 1 where g = fibBuilder id evaluates to  1 + 1, and so to 2. Thus g agrees with fib on 0, 1, 2 and h agrees with fib on 0, 1, 2, 3.

An easy inductive argument shows that a function defined, like g and h, but with $r$ calls to fibBuilder agrees with fib on $\{0,1,\ldots, r+1\}$. We want to somehow ‘pass to the limit’, and get the function fibInf defined by calling fibBuilder infinitely many times. So fibInf should be the same as fibBuilder fibInf. By analogy with the infinite list ones = 1 : ones, we boldly define

fibInf = fibBuilder fibInf


The result is disappointing: no evaluation of fibInf ever terminates. For example, fibInf 3 $\rightarrow$ (fibBuilder fibInf) 3 $\rightarrow$ fibInf 2 + fibInf 1 $\rightarrow$ (fibBuilder fibInf) 2 + fibInf 1 $\rightarrow$ fibInf 1 + fibInf 0 + fibInf 1. Now the evaluation gets stuck, since each summand evaluates to itself. But there is a simple solution: extend the builder function so that it knows the initial conditions for the Fibonacci numbers, as well as the pattern of recursion. (This amounts to deleting one f in the original definition.)

fibBuilder' :: (Int -> Int) -> (Int -> Int)
fibBuilder' f = g
where g n | n <= 1    = n
| otherwise = f (n-2) + f (n-1)

fibInf' = fibBuilder' fibInf'


This works. While non-obvious to define, neither of the functions above calls itself recursively. So, at least in this case, we’ve seen that a compiler that can cope with functions that return other functions can cope with a truly recursive function.

An alternative definition, emphasising that fibInf and fibBuilder fibInf are the same is

fibInf' = fix fixBuilder' where fix k = f (fix k)

Thus the defining property of fibInf' also serves as a working Haskell definition. For more theory on the fixed point combinator see the Wikipedia page. If there was a prize for ‘function that seems the most useless, and is in fact the most useful’, fix would be a strong contender.

#### Memoizing builder functions

In the same spirit of ‘defining property as working definition’, we might try to define a memoized version of fib by

fibMInf = fibBuilder' (memoizeInt fibMInf)


Remarkably enough, this works. For example, chasing through the evaluation of fibMInf 3 we get fibMInf 3 $\rightarrow$ fibBuilder' (memoizeInt fibMInf) 3 $\rightarrow$ f 1 + f 2 where f = memoizeInt fibMInf. The closure for f has the list values, defined by values = [f m | m <- [0..]]. So, as if by magic, we now call a memoized function in the recursive evaluations.

So although we haven’t been able to define a good memoizer with type (Int -> Int) -> (Int -> Int), we can use the naive attempt above to write a good memoizer for builder functions.

memoizeBuilder :: ((Int -> Int) -> (Int -> Int)) -> (Int -> Int)
memoizeBuilder builder = fix (builder . memoizeInt)

fibMInf' = memoizeBuilder fibBuilder'