I understand that abstraction is about taking something more concrete and making it more abstract. That something may be either a data structure or a procedure. For example:
I'd like to offer an answer for the greatest possible audience, hence I use the Lingua Franca of the web, Javascript.
Let's start with an ordinary piece of imperative code:
// some data
const xs = [1,2,3];
// ugly global state
const acc = [];
// apply the algorithm to the data
for (let i = 0; i < xs.length; i++) {
acc[i] = xs[i] * xs[i];
}
console.log(acc); // yields [1, 4, 9]
In the next step I introduce the most important abstraction in programming - functions. Functions abstract over expressions:
// API
const foldr = f => acc => xs => xs.reduceRight((acc, x) => f(x) (acc), acc);
const concat = xs => ys => xs.concat(ys);
const sqr_ = x => [x * x]; // weird square function to keep the example simple
// some data
const xs = [1,2,3];
// applying
console.log(
foldr(x => acc => concat(sqr_(x)) (acc)) ([]) (xs) // [1, 4, 9]
)
As you can see a lot of implementation details are abstracted away. Abstraction means the suppression of details.
Another abstraction step...
// API
const comp = (f, g) => x => f(g(x));
const foldr = f => acc => xs => xs.reduceRight((acc, x) => f(x) (acc), acc);
const concat = xs => ys => xs.concat(ys);
const sqr_ = x => [x * x];
// some data
const xs = [1,2,3];
// applying
console.log(
foldr(comp(concat, sqr_)) ([]) (xs) // [1, 4, 9]
);
And another one:
// API
const concatMap = f => foldr(comp(concat, f)) ([]);
const comp = (f, g) => x => f(g(x));
const foldr = f => acc => xs => xs.reduceRight((acc, x) => f(x) (acc), acc);
const concat = xs => ys => xs.concat(ys);
const sqr_ = x => [x * x];
// some data
const xs = [1,2,3];
// applying
console.log(
concatMap(sqr_) (xs) // [1, 4, 9]
);
The underlying principle should now be clear. I'm still dissatisfied with concatMap
though, because it only works with Array
s. I want it to work with every data type that is foldable:
// API
const concatMap = foldr => f => foldr(comp(concat, f)) ([]);
const concat = xs => ys => xs.concat(ys);
const sqr_ = x => [x * x];
const comp = (f, g) => x => f(g(x));
// Array
const xs = [1, 2, 3];
const foldr = f => acc => xs => xs.reduceRight((acc, x) => f(x) (acc), acc);
// Option (another foldable data type)
const None = r => f => r;
const Some = x => r => f => f(x);
const foldOption = f => acc => tx => tx(acc) (x => f(x) (acc));
// applying
console.log(
concatMap(foldr) (sqr_) (xs), // [1, 4, 9]
concatMap(foldOption) (sqr_) (Some(3)), // [9]
concatMap(foldOption) (sqr_) (None) // []
);
I broadened the application of concatMap
to encompass a larger domain of data types, nameley all foldable datatypes. Generalization emphasizes the commonalities between different types, (or rather objects, entities).
I achieved this by means of dictionary passing (concatMap
's additional argument in my example). Now it is somewhat annoying to pass these type dicts around throughout your code. Hence the Haskell folks introduced type classes to, ...um, abstract over type dicts:
concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
concatMap (\x -> [x * x]) ([1,2,3]) -- yields [1, 4, 9]
concatMap (\x -> [x * x]) (Just 3) -- yields [9]
concatMap (\x -> [x * x]) (Nothing) -- yields []
So Haskell's generic concatMap
benefits from both, abstraction and generalization.