Functional Programming without Lambda - Part 2 Lifting, Functor, Monad

本是古典何须时尚 2021-11-16 10:22 157阅读 0赞

## Lifting ##

Now, let's review `map` from another perspective. `map :: (T -> R) -> [T] -> [R]` accepts 2 parameters, a function `f :: T -> R` and a list `list :: [T]`. `[T]` is a generic type paramterized by `T`, it's not the same as `T`, but definitely shares some properties of `T`. So, an interesting interpretation of `map(f) :: [T] -> [R]` is that `map` turns a function of type `T -> R` into a function of `[T] -> [R]`, this is called **lifting**.

Take the square function `x -> x * x` as an example, `map(x -> x * x)` turns the function on `Int` into a function on `[Int]`. Therefore, it makes sense to name `map(x -> x * x)` as `squareForList`. You can even simply name it as `square`, which is the overloaded version of `square` for `[Int]`.

The concept of lifting is the key to understand the advanced abstractions in functional programming. Lifting allows you to reuse a function of type `T -> R` (or `T1 -> T2 -> R` ...) in the context of List, [Maybe][], [Lazy][], [Promise][], etc. That saves you the work to implement similar functions from scratch just for the context.

Let me explain why lifting matters by changing the string conversion problem in the previous chapter a bit. In the original problem, we got a function `convert :: String -> String`, what if the input string is not directly available, but asynchronously fetched from a web service? Do you want to chain the `convert` to the callback for the asynchronous HTTP response? You can use callback, but that makes you lose functional composition.

Just like `map` lifts a function on `T` into a function on `[T]`, we just wanted to lift it to `Promise<T>`. Here `Promise<T>` stands for an asynchronously available value of type `T`. So, we'll introduce a function `fmap :: (T -> R) -> Promise<T> -> Promise<R>`, meaning `fmap` turns a function of type `T -> R` into a function of type `Promise<T> -> Promise<R>`. See the following example:

// Java 6
    F1<String, String> convert = _(split(" "), reverse, map(toUpperCase), join("_"));
    // fmap turns a function of type "T -> R" into a function of type "Promise<T> -> Promise<R>"
    F1<Promise<String>, Promise<String>> convertForPromise = fmap(convert);
    // await() blocks until the async result available
    String result = convertForPromise.apply(promise(URL)).await();

More details [here][].

`promise(URL) :: Promise<String>` stands for a string value which will be available in the future. Calling `await` on the promise object will block until the string is available. `fmap` turns `convert :: String -> String` into `convertForPromise :: Promise<String> -> Promise<String>` which can work on a promise. By the way, if you like we can omit the `convert` function by inlining it as:

`fmap(_(split(" "), reverse, map(toUpperCase), join("_")))`

## Functor ##

As I mentioned in the previous section, Promise, Maybe, List, Lazy, and so on are all contexts. The idea behind is the functional abstraction named [Functor][]. In Java, a functor can be defined as follows:

interface class Functor<T> {
      <R> Functor<R> fmap(F1<T, R> f);
    }

then, `Promise<T>` will implement the `fmap`:

class Promise<T> implements Functor<T> {
      <R> Promise<R> fmap(F1<T, R> f) {
        ...
      }
    }

But as I have said before, we are not in favor of the OO-style API design. A better way to define functor in Java is as follows:

public class Promises {
      public static <T, R> F1<Promise<T>, Promise<R>> fmap(F1<T, R> f) {
        return Promises.<T, R>fmap().apply(f);
      }
    }

It essentially means if we can define a function `fmap` to lift a function of type `T -> R` into a function of type `Functor<T> -> Functor<R>`, then `Functor<T>` is a functor. In addition, there're 2 properties named Functor Laws as the semantics constraints to ensure the type makes sense:

fmap id      = id
    fmap (p . q) = (fmap p) . (fmap q)

Don't be scared, it's actually very simple. Just like we put the FILO constraint on the `push` and `pop` of the `Stack` type to make sure it behaves as what we want.

If you feel too abstract, take a look at the example of `List<T>` or `Promise<T>`. More often than not, your functor class satisfies the laws automatically. However, keep in mind that you may always want to test the functor laws for your functor class, just like you want to test FILO for a `Stack` implementation. See unit tests of `Promise<T>` for the functor laws [here][].

## Monad ##

Lifting a function of type `T -> R` into a function of type `Functor<T> -> Functor<R>` allows us to reuse the existing functions in a different context, but sometimes the basic function we have is not as plain as `toUpperCase :: String -> String`. Let's look at the following problem:

> Given 1) a function `Promise<String> asyncGet(String url)` which accepts an URL and returns a promise of the web page; 2) `n` hyperlinked web pages, the contents of one page is the URL of the next page, url1 -> page1 (url2) -> page2 (url3) -> page3 (url4) ... page\_n (url1), please write a function `Promise<String> asyncGetK(String url, int k)` which starts from the `url`, goes forward by `k` steps, returns the page.

If what we have is a sync function `String get(String url)`, that would be a simple loop like:

// Java 6
    String getK(String url, int k) {
      String page = url;
      for (int i = 0; i < k; i++) {
        page = get(page);
      }
      return page;
    }

The point here is that the result of the previous `get` can be directly passed to the next `get`, because the type matches. In other words, we can compose multiple `get` functions together.

But since we only have `asyncGet` of type `String -> Promise<String>`, the result type `Promise<String>` of a previous `asyncGet` doesn't match the parameter type `url :: String` of the next `asyncGet`, we are unable to compose them together directly. So, we'd really like to lift `asyncGet :: String -> Promise<String>` into `asyncGetPromise :: Promise<String> -> Promise<String>` then it's composable.

The idea is great, but what would happen if we apply `fmap` to `asyncGet`. Since the type of `fmap` is `(T -> R) -> Promise<T> -> Promise<R>`, then the type of `fmap(asyncGet)` would be `Promise<String> -> Promise<Promise<String>>`. Ooops, that's too much! But if we have a `join :: Promise<Promise<T>> -> Promise<T>` to flatten a nested promise, then we will get `_(fmap(asyncGet), join) :: Promise<String> -> Promise<String>`. Combining `fmap` and `join` together, we get a function `flatMap :: (T -> Promise<R>) -> Promise<T> -> Promise<R>`, which is exactly what we want.

Being able to define a function `fmap` makes a type a Functor, likewise being able to define a function `flatMap` makes a type a Monad. Then the code would be like:

// Java 6
    String getK(String url, int k) {
      F1<Promise<String>, Promise<String>> asyncGetPromise = flatMap(asyncGet);
      Promise<String> page = unit(url);
      for (int i = 0; i < k; i++) {
        page = asyncGetPromise(page);
      }
      return page.await();
    }

It really shares the same structure as the sync code. That is isomorphic!

转载于:https://www.cnblogs.com/weidagang2046/p/fp-without-lambda-2.html

[Maybe]: http://en.wikibooks.org/wiki/Haskell/Understanding_monads/Maybe
[Lazy]: https://sankarsan.wordpress.com/2009/10/04/laziness-in-c-4-0-lazyt/
[Promise]: http://en.wikipedia.org/wiki/Futures_and_promises
[here]: https://github.com/weidagang/fava/blob/master/tests/fava/PromiseTest.java
[Functor]: http://en.wikibooks.org/wiki/Haskell/The_Functor_class