Sometimes good performance seems to be in direct conflict with programmer-understandable types. This post is about ways to get fast parity checking on natural number types, while also maintaining as many invariants as possible automatically. The example here is definitely contrived—if you wanted fast numeric parity calculations you could just check the lowest bit. However, it’s a pretty good stand-in for more complicated encodings, and how we can play with them to make them both performant and easy to understand. Along the way we’ll briefly touch on some cool Haskell language extensions: type families, GADTs, and data kinds.
Let’s start with a tiny subset of the Prelude, and we’ll also define a
Bool type as well as the
not operation on it which we can use later.
Hopefully this is straightforward. Here’s our stand-in for an existing type with a focus on understandability by human programmers. We’re encoding natural numbers as Peano numerals, where
Z means 0, and
S means successor. Let’s also throw in a helpful
succ function for finding successors.
Our version of
succ here only needs to provide the
S constructor, which means it’s simple to write and easy to understand.
For whatever reason, the other constraint on our type is that we want fast parity checking, i.e. a way to tell whether a number is even or odd. Let’s write a version of that now.
Here we’re deconstructing our
Nat and recursively finding out whether it’s even by reducing the number by two, and checking the parity of that. Intuitively, this works, because we always end at one of the base cases and are making the number smaller at each step. However, it’s rather slow. We’re taking around \(n/2\) steps to check the parity of a number \(n\), which means the runtime of this function grows linearly with the size of its input. Let’s see if we can do better.
Here’s another idea: we can encode odd and even numbers separately. This way, there’s an easy way to check a number’s parity from its encoding in a single step. Maybe our new data type looks like this:
ZZ is the equivalent of
Nat, but now we have both an
SO (“odd successor”) and an
SE (“even successor”). This makes the
succ function a bit more complicated now:
We want the
SOs to alternate, which we can do as shown. However, note that now there’s a higher burden on people writing functions dealing with
ParityNat as opposed to just plain
Nats. Along with additional cases to handle, there are more places for implementations to contain errors. We’re also relying on people to use
succ now, rather than just the plain data constructors. At least our new parity checking functions are speedy:
There’s another encoding option, which puts more burden on the author of the encoding, but hopefully automatically enforces more constraints for later users of the library. We can push parity checking into the types. First, let’s make a new type for the parity itself.
If this seems weirdly familiar, that’s because it’s isomorphic to
Bool which we defined earlier. We can also define a “
not” operation on
Paritys, but first let’s take one step back. If we want to enforce parity in types, having
Odd terms isn’t that helpful. We need to promote
Parity to a kind, and promote
Odd to types. While we’re at it, let’s also add the type families extension so we can define the equivalent to our
not function, but at the type level instead of the term level. All in all, we’ll want the following extensions enabled:
Now we can define the type-level equivalent of
not, but for
Parity instead of
Bool. Let’s call it “
Opp”. Here’s its definition:
This may look bizarre, but it’s more or less just a different syntax for defining a function.
Opp takes a
n and returns a
Parity. The opposite of even is odd, and the opposite of odd is even. The ticks in front of
'Odd remind us that these have been promoted from terms to types.
We can now define our constructors analogous to
S. This stuff here is the reason we needed GADTs.
Breaking this down a bit further, a
Natural is a type which takes something of kind
Parity and gives us back a normal type (something of kind
*). Zero is even, and the
Succ of any
Natural has the opposite parity as that
The successor function is again trivial to write.
Weirdly, it now almost doesn’t make sense to have
odd functions. Because this information is encoded in the types, it’s already sort of carried along with every
Natural. However, just for the sake of completeness we can write something like this:
This was a brief tour of a few possible encodings for a small bit of data with additional outside constraints on it. The first encoding was very simple, but runtime for operations we care about a lot (
odd) was too slow. We switched to the more performant
ParityNat, but this came at the cost of ease of use when writing functions using that type. Finally, we sort of pushed the problem up to the type level so that anything of type
Natural 'Even has even parity, and likewise any
Natural 'Odd has odd parity. This did away with the need for parity checking as functions, but comes at the cost of a more complicated type system encoding of our desired result.
Like so many places in engineering, this provides an interesting example of multiple tradeoffs that have to be balanced. Normally I lean towards “clarity at most costs”, sacrificing performance to make programmers’ jobs more manageable. However, occasionally understandability has to be traded away for enhancements of the details of how our programs actually run, and this tradeoff is more common as systems become more heavily relied upon by others.