If we can do 2 we can do 4, and 8, and 16.
By this point we’re encoding the operation of Sum by deconstructing our Nat into its binary form.
If we’re operating on _27 then we would deconstruct it as 16+8+2+1 in our implicit searching whilst summing.
But this idea is not satisfying, because there’s no way in this approach to encode an arbitrary binary solution.
We have to encode the 8 and 16 and etc cases manually.
Also, it turns out, it’s not that efficient.
Here is an implementation with the first extra layer, handling two Succ layers at once:Again with the entirely unscientific benchmarks, but it sped up my local calculation compile time by about 40%.
Not as fast as we would expect.
We should expect a halving of the compile time based purely on the algorithm.
What’s the problem?The problem is the encoding of the number itself: Succ.
It’s too many nested types.
Each time you add one you increase the nesting and the entire thing becomes more cumbersome for the compiler to handle and compare.
Imagine we strip four layers of Succ from _11: the compiler has to examine _11, count four layers of type nesting, unwrap it, then apply four layers of type nesting to the other number.
While we decreased the complexity of the numerical algorithm, we didn’t do much to the actual job the compiler has to do.
Wouldn’t it be great if we had just one operation to do at every stage, instead of four or more like in the above example?Binary?Succ is the problem.
It’s elegant, but not fit for purpose.
We know a binary form of Sum makes Sum, by some measure, more efficient.
We should try and move this binary encoding into the very definition of Nat.
This becomes extraordinarily simple if you realise the the Succ encoding is just the unary encoding of integers.
It’s just the repetition of a symbol to represent the magnitude of the number, like a tally chart.
Usually in unary that symbol is 1 and here it happens to be Succ.
So we just have to replace Succ with two wrappers, one representing 0 and one representing 1 for the binary expansion of integers.
I chose Zero and One respectively.
We have a choice to make though — which direction our encoding runs in.
For example, 4 could be One[Zero[Zero[b0]] or it could be Zero[Zero[One[b0]]].
We could encode the binary expansion 100 forwards or backwards.
(Here b0 is the ‘initial’ binary number representing 0, like Nat.
_0 represents the initial unary number (also 0)).
If you imagine you were doing a long sum or multiplication by hand, like one of these from primary school: 1783 + 434——– .
7You’ll remember that we operate from the right inwards, from the lowest denomination.
For that reason, we should choose a reverse binary expansion for our new Nat, so all our encodings start at the lowest denomination on the outer layer.
So One[X] becomes a simple arithmetic operation, and represents the number 2 * X + 1.
Zero[X] similarly represents the number 2 * X.
Let’s call our new version BNat to avoid any confusion.
Here’s what a basic setup might look like:But before we go any further, we have to ask ourselves something.
Why stop at Binary?Binary is the natural thing software developers reach for, for obvious reasons.
But is it the best choice?We saw above that we’re trying to minimise the length of the expansion of the number N when encoded in some flavour of Nat, some base, unary being base 1 and binary being base 2.
The ‘most efficient’ encoding in these terms is when the base used to represent our N is infinite — give each number its own unique symbol and suddenly every type is 1 deep and there’s no recursion at all.
Solved it!Obviously that’s absurd.
We’re also limiting the number of symbols themselves.
That’s what Succ is optimised for: there is only one symbol and the typeclasses as a result are very simple and easy to reason about.
We’d like to be more balanced.
We’re trying to minimise the number of symbols used — our base — while also minimising the length of the number N in our base.
So we’re trying to minimise 1 / L*B where L = Length of expansion of N in base B and B = Chosen base, with respect to B.
We know L = log_b(N) so we have to find the minimum of 1 / (B * log_b(N)).
This is the same as finding the minimum of B / log(B), where log is the natural logarithm.
Do the maths yourself, plug it into wolfram alpha, whatever you like.
Turns out the function is minimised when B = e, the natural logarithm base, roughly 2.
Which is sadly no good to us, since we definitely can’t operate with a non-integer base in the scala compiler.
Handily though, B=3 gives a lower value in the above function than B=2.
So we’re abandoning binary, and ploughing full steam ahead on to the provably more efficient base ternary.
Ternary EncodingSame rules apply as binary.
We’re going to produce a reverse ternary encoding of the natural numbers using three symbols instead of two this time: Zero, One, and Two.
The arithmetic operations these represent are as follows:Zero[X] = 3XOne[X] = 3X + 1Two[X] = 3X + 2Here is the basic setup:We’re going to be defining Sum for TNat, but we need a couple of utilities first.
They serve as good examples to see what ternary algorithms look like:IncrAn easy way of increasing a TNat by 1 would be nice.
In unary’s case this operation was just to wrap a Succ round it.
Here’s an Incr implementation for TNat:Four cases, telling us how to increment each possible form of a TNat :incr0 says 0 + 1 = 1incrZero says (3x) + 1 = (3x + 1) which is One[x]incrOne says (3x + 1) + 1 = (3x + 2) which is Two[x]incrTwo says (3x + 2) + 1 = 3(x + 1) which is Zero[Incr[x]]Hopefully by now such typelevel functions are becoming easy to deconstruct.
TrebleNow we’re working with a more complex structure, there is one thing it makes no sense to express: Zero[t0].
That’s a different way of just saying t0 — since Zero represents multiplying by 3.
We can’t allow different representations of the same number as it would make any automatic calculation very difficult indeed.
So we need a safe way of wrapping a TNat in Zero, of trebling it:and now we can safely apply Zero to any number we like.
SumSum is… not nice to implement in ternary.
That’s one of the reasons we try to minimise the number of symbols used in the above efficiency calculation.
The Sum typeclass gets really big really fast.
There are 11 different cases, compared to Succ‘s 2 cases.
I prevaricated over whether to include it here or not and in the end decided not to, in case it scared you off.
This article is already too long.
But it is available for you linked at the end of the article.
It’s mostly grunt work anyway, there’s nothing ground-breaking in there.
If you understand the approach above in Incr and Treble then figuring out Sum is just elbow grease.
But, it does exist and the definition looks like this:Let’s take it for a spin.
BenchmarksLet’s dump some random numbers in to our new addition typeclass and see what falls out.
We will be using the following method to test:Recall the Succ encoding could not compile 22+22=44 for us.
Well:So far so good, we’ve broken our previous record!.Let’s kick things up to 11:Those are billions.
Compiling in negligible time.
Quintillions?.This is in the realm of long overflow:Negligible.
10⁴⁵?Negligible!.10¹⁵⁰?We’ve finally hit something that takes a noticeable amount of time to compile (it also breaks the github syntax highlighter).
Two random numbers in the order of 10¹⁵⁰.
Succ was hitting this time calculating 20+5.
Let’s see how far we can push it: 10³⁰⁰?So we’ve pretty much hit the limit here — 40s is not suitable for use in a project, and any longer and the compiler will start crashing.
10³⁰⁰ though is a 10²⁹⁹ times improvement over Succ, so it’s a pretty good improvement¹.
And it is definitely large enough to encode your string lengths or matrix dimensions, no matter how much data you’re processing.
The above TNat (including the promised Sum typeclass) is available in my experimental library Numerology.
Also included is BNat (the binary version we skipped over), many more typeclasses than were demonstrated here and various other experiments to do with compile-time arithmetic.
It’s not yet in stable release (experiments are still on-going), but the code as it is is more than ready to be used in production (and is used, in TypeChecked’s private repos) and the functionality currently available will not be regressed at any point.
DISCLAIMERThe above code may break your compiler.
In the Numerology repo there is a .
jvmopts file including the following line: -Xss50m.
This extravagant value provides plenty of stack space to compile every example in this post and in the repo.
 Fun fact: If we expressed a number in the order of 10³⁰⁰ in terms of unary Nat and Succ, and we did something wrong and the compiler gave us an error message about that number, the error message would be (approximately) 10³³ gigabytes in size.
That is many orders of magnitude more than all the information on the internet.
.. More details