# Laziness

Most mainstream functional languages employ an eager (a.k.a. strict) evaluation strategy, where an expression is evaluated entirely even if its resulting value is not needed or only parts of it are needed. As we will see, there are sometimes advantages in lazily evaluating certain expressions. There are two important aspects of lazy evaluation:

1. suspending (a.k.a delaying) a computation until its result is actually needed (a.k.a demanded or forced); and

2. memoizing (a.k.a caching) the result of a suspended computation in case its value is demanded again.

Many eager languages, including Elm, offer additional language constructs for selectively introducing laziness. We will work through two example encodings — natural numbers and streams — that motivate and illustrate lazy evaluation.

## Natural Numbers

We will work through a few encodings of natural numbers and define some simple operations on them.

### First Version — Nat.elm

We start by inductively defining Natural numbers to be either Zero or the Successor of some other Natural number.

type Nat = Z | S Nat

Next, let’s define functions toInt and fromInt to convert Natural numbers to and from Integers.

fromInt : Int -> Nat
fromInt n =
if n <= 0
then Z
else S (fromInt (n-1))

If we take fromInt for a spin, we see that it busts the stack rather quickly.

> fromInt 0
Z : Nat.Nat

> fromInt 1
S Z : Nat.Nat

> fromInt 10
S (S (S (S (S (S (S (S (S (S Z))))))))) : Nat.Nat

> fromInt 10000
RangeError: Maximum call stack size exceeded

Ah, right, we should define it tail-recursively so that it runs in constant stack space.

fromInt n =
let foo acc n =
if n <= 0
then acc
else foo (S acc) (n-1)
in foo Z n

That should do the trick…

> fromInt 10000
RangeError: Maximum call stack size exceeded

Or not. Okay, it’s time for a little more investigative journalism. We could fire up Elm Reactor to start debugging. Or we can be lazy (pun intended) and continue to poke around at the REPL.

> let _ = fromInt 10000 in ()
() : ()

That’s interesting. The call to fromInt was not the problem. So the act of printing the resulting Nat causes the stack overflow? Let’s write our own tail-recursive printing function to test this hypothesis.

strNat : Nat -> String
strNat n =
let foo acc n = case n of
Z    -> acc
S n_ -> foo ("S" ++ acc) n_
in foo "Z" n

Sure enough, that does the trick…

> fromInt 10000 |> strNat
" ... SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSZ" : String

> fromInt 10000000 |> strNat
" ... SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSZ" : String

… until we run out of heap space, that is.

> fromInt 100000000 |> strNat
FATAL ERROR: JS Allocation failed - process out of memory

Okay, now that we have sorted out our call stack and heap space concerns, let’s return to the task of programming with Nats. First, a bit of refactoring:

foldl : (b -> b) -> b -> Nat -> b
foldl f acc n =
case n of
Z    -> acc
S n_ -> foldl f (f acc) n_

strNat = foldl ((++) "S") "Z"

And a function to convert Nats to Ints:

toInt : Nat -> Int
toInt = foldl ((+) 1) 0

We can add two Nats together by peeling Successor labels off of y one at a time and wrapping them around x.

plus : Nat -> Nat -> Nat
plus x y =
let foo acc n = case n of
Z    -> acc
S n_ -> foo (S acc) n_
in foo x y

Or:

plus x y = foldl S y x

Or:

plus x y = foldl S x y

Or, better yet:

plus = foldl S

This plus function encodes the usual notion of addition for our Nat type.

> plus (fromInt 0) (fromInt 0) |> strNat
"Z" : String

> plus (fromInt 0) (fromInt 2) |> strNat
"SSZ" : String

> plus (fromInt 10) (fromInt 2) |> strNat
"SSSSSSSSSSSSZ" : String

> plus (fromInt 10) (fromInt 2) |> toInt
12 : Int

We can define equality for Nats by peeling off one data constructor at a time.

eqNat : Nat -> Nat -> Bool
eqNat x y =
let foo x y = case (x, y) of
(Z, Z)       -> True
(S x_, S y_) -> foo x_ y_
_            -> False
in foo x y

This seems to work just fine…

> eqNat (fromInt 0) (fromInt 0)
True : Bool

> eqNat (fromInt 0) (fromInt 2)
False : Bool

> eqNat (fromInt 10) (fromInt 2)
False : Bool

… but it is really slow for some comparisons that seem like they should be easy to decide quickly.

> eqNat (fromInt 10000) (fromInt 10000000)
False : Bool

> eqNat (fromInt 0) (fromInt 10000000)
False : Bool

The problem is that, under eager evaluation, both Natural numbers are evaluated completely before calling eqNat, which then very quickly decides the last two disequalities.

### Delaying Evaluation with Thunks — ThunkNat.elm

A common approach to delaying the evaluation of an expression e of type a in an eager language is to define a function \() -> e, called a thunk, that waits for a dummy argument before evaluating the expression.

We will port the implementations above in order to delay computing the representations of natural numbers. In our new representation of Nats, a Successor value stores the delayed computation of the Nat that it succeeds. The force function is used to evaluate a suspended computation.

type Nat = Z | S (Thunk Nat)

type alias Thunk a = () -> a

force : Thunk a -> a
force thunk = thunk ()

Note that implementing a function like the following is not a good idea, because a call to delay will force its argument to be evaluated!

delay : a -> Thunk a
delay e = \() -> e

To implement fromInt, we no longer need to implement a (tail-recursive) helper function because there are no direct recursive calls; instead, the latter case immediately returns a Successor value (which may some time later lead to a call to fromInt).

fromInt n =
if n <= 0
then Z
else S (\_ -> fromInt (n-1))

Notice that our new representation of non-Zero numbers is quite different from before:

> import Nat as N
> import ThunkNat exposing (..)

> N.fromInt 10
S (S (S (S (S (S (S (S (S (S Z))))))))) : Nat.Nat

> fromInt 10
S <function> : ThunkNat.Nat

Unlike fromInt, toInt does need to make recursive calls immediately, because the resulting type (Int) does not have the notion of delayed computation built in to its representation. Therefore, we will want to employ the tail-recursive helper strategy.

toInt n =
let foo acc n = case n of
Z    -> acc
S n_ -> foo (1 + acc) (force n_)
in foo 0 n

As before, fromInt and toInt are inverses:

> fromInt 100000000 |> toInt
100000000 : Int

Notice how toInt uses force to evaluate all of the nested suspensions that are stored within a Nat. A function like this is called monolithic, whereas a function like fromInt is called incremental because it does not trigger the evaluation of all nested suspensions.

Another example of a monolithic function is strNat.

strNat n =
let foo acc n = case n of
Z    -> acc
S n_ -> foo ("S" ++ acc) (force n_)
in foo "Z" n

However, this function is no longer needed for its original purpose above, because printing the representation of a Successor value is now very quick.

> fromInt 10000
S <function> : ThunkNat.Nat

> fromInt 10000 |> strNat
" ... SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSZ" : String

We can now return to our motivation for delaying the evaluation of Nats.

eqNat x y =
let foo x y = case (x, y) of
(Z, Z)       -> True
(S x_, S y_) -> foo (force x_) (force y_)
(_, _)       -> False
in foo x y

When x and y represent the same number, eqNat evaluates all of the nested suspensions in both x and y. Otherwise, it evaluates only enough of their delayed representations in order to demonstrate a difference in corresponding data constructors. As a result, all of the following comparisons run quickly, unlike with our original Nat.elm implementation.

> eqNat (fromInt 0) (fromInt 0)
True : Bool

> eqNat (fromInt 0) (fromInt 2)
False : Bool

> eqNat (fromInt 10) (fromInt 2)
False : Bool

> eqNat (fromInt 10000) (fromInt 10000000)
False : Bool

> eqNat (fromInt 0) (fromInt 10000000)
False : Bool

We finish porting the original module with the following incremental implementation of plus.

plus x y = case y of
Z    -> x
S y_ -> S (\_ -> plus x (force y_))

As a result, the following comparison evaluates quickly.

> eqNat (fromInt 0) (plus (fromInt 10000000) (fromInt 10000000))
False : Bool

#### Digression

What happens if we ask Elm to compare Nats using built-in equality?

> fromInt 0 == fromInt 0
True : Bool

> fromInt 0 == fromInt 1
False : Bool

> fromInt 1 == fromInt 2
Error: Trying to use `(==)` on functions. There is no way to know if
functions are "the same" in the Elm sense.

Elm throws a run-time error when trying to compare two different function values. Fair enough. Notice that “physical equality” between function values is supported, however.

> fromInt 1 == fromInt 1
Error: Trying to use `(==)` on functions. There is no way to know if
functions are "the same" in the Elm sense.

> let foo = fromInt 1 in foo == foo
True : Bool

### Memoizing Thunks — LazyNat.elm

Defining suspensions and incremental functions can be really valuable techniques, but there’s no free lunch. The representation of a thunk is a closure, which is a function to evaluate along with bindings for all free variables referred to by the function. Delaying computations willy nilly, then, can lead to a huge number of these closures building up. So one should restrict the use of thunks to situations where the benefits of being able to define incremental functions outweighs the overheads associated with delayed computations.

Another concern is that the same delayed computation may be demanded more than once. If the computation takes significant resources to evaluate, then redoing the work every time is undesirable. In a pure language with only strict evaluation, there is no recourse: every time a thunk is forced, it must be re-evaluated. As a result, many strict languages offer special-purpose constructs for manipulating delayed computations with the guarantee that the result of forcing a delayed computation is cached in case it is forced again in the future. The term lazy evaluation is often used to describe support for delayed computations with the guarantee of evaluating any such computation at most once.

In Elm, the Lazy library provides support for lazy evaluation. (Lazy is a community library, so an elm-package.json file is required to declare this dependency.) The lazy function turns a Thunk a into a Lazy a value, which force evaluates, reusing the result of any previous call to force.

lazy  : (() -> a) -> Lazy a
force : Lazy a -> a

The native JavaScript implementation of lazy uses a mutable variable (called isForced) to track whether the particular thunk has been evaluated and a mutable variable (called value) to store this result.

It is simple to port ThunkNat.elm to use the Lazy library in order to obtain the benefits of memoization. First, we redefine the type of Nat as follows.

type Nat = Z | S (Lazy Nat)

Then, we sprinkle a call to lazy in front of every thunked value. The resulting implementation can be found in LazyNat.elm. (Use diff or vimdiff to see how similar the two files are.)

Using this implementation, we now expect that force-ing an expensive suspension for the second time should be practically instantaneous. As we discussed above, the worst case for eqNat is when both its arguments are equal. So let’s use a call to eqNat as an example of a slow computation.

> import LazyNat exposing (..)

> foo i = eqNat (fromInt i) (fromInt i)
<function> : Int -> Bool

> foo 100000
True : Bool

> foo 1000000
True : Bool

The last operation above is quite slow. So, we should be able to delay its evaluation, force and memoize its result, and reevaluate it a second time nearly instantaneously. But the second force is just as slow as the first!

> import Lazy exposing (..)

> slow = lazy (\_ -> foo 1000000)
Lazy <function> : Lazy.Lazy Bool

> force slow
True : Bool

> force slow   -- still slow... :-(
True : Bool

Good thing we still have our investigative journalist hats on.

> force slow
True : Bool

> (force slow, force slow, force slow, force slow, force slow)
(True,True,True,True,True) : ( Bool, Bool, Bool, Bool, Bool )

These two expressions require about the same amount of time to evaluate, which suggests that caching is kicking in for the latter case. So it appears that the native memo tables do not persist across REPL operations.

Optional Exercise — Write an Elm program that measures the time it takes to evaluate the previous two expressions (for example, using Time).