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Last active Aug 29, 2015
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WIP: Quick, informal introduction to parametricity

Parametricity, Part 1

The basic idea of parametricity is that from the parametric type signatures of "generic" pure functions we can formulate theorems about the the way it behaves. And much more so than functions of specific types.

Assumptions

  1. We assume pure functions only. The notion doesn't hold if a function is dependent on anything outside of it's explicit inputs. It can't. Thankfully all business logic can actually be implemented using pure functions which even a compiler can verify (in my experience) if you write in a decent programming language and don't have insane engineers on your team. Obviously your mileage may vary (YMMV).
  2. The definition language supports parametric polymorphism. Parametric polymorphism is when a function or a data type can be written generically so that it can handle values identically without depending on their type.

Motivations

We want to be able to "reason" about code. How it behaves, or, at least, how we can expect it to behave. This is because, as engineers, we hate not knowing the state of the world, or, more specifically, our programs. By understanding some basic guarantees of how we can expect a function to behave, we can build functionality on top of it with less surprises. What software engineer likes surprises, especially in production? Not me. At least not when I am on call. ;)

Examples

Let's look at some examples of what I mean by theorems about functions from it's generic type signature.

Below is a type signature of a function. Assuming it is a pure function and that a can be any type, what can the only behavior of this function be?

f1 :: a -> a
f1 a = ???

If it isn't obvious to you yet, then let's put a few different types for a and see:

f1str :: String -> String
f1str s = ???

Assuming the only thing the result of function f1str can depend on is the inputs (in this case just one input), then it could be any "transformation" of the input s. Now in this case the type is String -> String, there are many kinds of transformations that we can apply to the input to derive the result in a pure way. For example, we could apply a hash function on the input value to produce the result value. We could reverse the string or just take up to the first three characters of the input string and provide that as the resultant value. There are infinite (when you factor in all combinations of these transformations from String -> String) possibilities.

However, let's try Int -> Int, there are not that many coinciding "transforms" that are available for both Strings and Ints. Now extending this to any type from input to output we realize there is only one possible definition, which is the identity function:

identity :: a -> a
identity a = a

Yes, ok, that example was trivial, but you see how when we made the types more and more generic we could make more and more assertions about the definition of the function?

That's a critical piece of the idea behind parametricity.

Cool, now let's take a look at a less trivial example.

f2 :: (a -> b) -> [a] -> [b]
f2 f as = ???

We are given a function from type a to type b. Let's say this is something like String -> Int where we take a string and give its length as the result value. So this means after taking that function, len, we would be taking a list of strings and returning a list of ints.

So what can we say about this function, even the concrete one defined above? Well, the most likely definition of that is:

f2 :: (String -> Int) -> [String] -> [Int]
f2 f ss = map f ss

Well that just means that f2 is map, right? The more generic we make the types the more sure we are that we can't have another definition for this.

Is the power of this stuff starting to strike you yet?

Ok, let's continue on with elaborating on how reasoning about code using parametricity might be a good thing. Perhaps even kind of powerful.

Now let's look at the following:

f3 :: (a -> b) -> Maybe a -> Maybe b
f3 f ma = ???

First we need to understand (maybe?) what the Maybe type is about. Imagine you live in a world where null, nil, None, NULL or other such construct do not exist. It's a blissful land where if you are told there will be a value of type PandaBear somewhere, there will be an actual PandaBear value. Same for GrizzlyBear. No ifs, buts, or nulls.

Sometimes you really aren't sure if there will be a value of that type there. Sometimes it might be nothing or "blank". So we have to be able to tell the calling code that somehow. We do that with the type Maybe a.

This is what the sum type definition of Maybe a looks like in Haskell:

data Maybe a = Just a | Nothing

Now we need to look at how we would use this:

sqrt :: Int -> Maybe Double
sqrt i
  | i > 0 = Just $ Math.sqrt i
  | otherwise = Nothing

-- We would use this like:
--   sqrt 1   => Just 1
--   sqrt 0   => Nothing
--   sqrt -1  => Nothing
--   sqrt 9   => Just 3

defaulting :: a -> Maybe a -> a
defaulting a1 (Just a2) = a2
defaulting a1 Nothing = a1

-- We would use the above like:
--   g1 :: Maybe Int -> Int
--   g1 = defaulting 6
--   g1 $ Just 3   => 3
--   g1 $ Nothing  => 6

There are two functions here using Maybe a of some kind. The sqrt function takes an Int value and if it is not zero of greater it will return a "nothing" value since our return type is a Double (rather than Complex). If we had made the type signature of sqrt to be Int -> Double and we know that no such value of null, nil, None, etc can exist for types then we can infer a logical error in the type signature exists and that the definition of sqrt is problematic due to it's ambiguity when given zero or less as input value. This is mightily helpful to help you design better APIs.

I the second function we have a defaulting function which takes a default value as a first argument and a "maybe" value of the same underlying type, a in this case, and in the case that the Maybe a value is "something" it will return that. Otherwise it returns the default.

Hopefully you can see the value of the Maybe a type now. Let's get back to the original function we were going to investigate: (a -> b) -> Maybe a -> Maybe b. Again, assuming it is a pure function, what can we assert from the type signature?

It seems obvious that if we can assume nothing about type a or b because they can be ANYTHING AT ALL, then there is only one definition of the function that can possibly exist. Let's see what this might look like with the definition expanded for f3 :: (a -> b) -> Maybe a -> Maybe b:

-- Here we are using pattern matching which if you aren't used to Haskell, Erlang, 
-- Scala or other languages that have this might be alien to you. Basically we are 
-- "branching" the code path between the cases when a `Just x` value is provided
-- versus when a `Nothing` value is provided as input. It yields (in my opinion) 
-- far more readable code than `if/else` branching.
-- There is also the benefit that the compiler can warn you when you haven't covered
-- all value constructors for a type you are matching on. :) I like types. Sue me.
f3 :: (a -> b) -> Maybe a -> Maybe b
f3 f (Just x) = Just $ f x
f3 f Nothing  = Nothing

This reduces to the following definition where you can find fmap in Haskell's Prelude (read: automatically imported standard libraries, it's a good approximation anyway).

f3 :: (a -> b) -> Maybe a -> Maybe b
f3 f ma = fmap f ma

The basic idea here is that we apply the function from a -> b on the value of Maybe a in the case that it is a Just a value. If it is a Nothing value it will result in a Nothing in the the type of Maybe b for the result of the function.

Conclusion

Hopefully I have been able to illustrate the value of using the basic idea of parametricity (even in a very informal way without real math(s)) as both a way to reason about other people's code and also offering an example of how inspecting types signatures using basic understanding of parametricity can show logical design level errors in your own API design.

Next time, we will conquer space and domesticate alien species...or something...with our new found super powers. Cheers.

Parametricity, Part 2

TODO: Write a compelling application of parametricity for working developers to see the value.

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