### Functors and Monads in Java - Don't fear Algebra.

## Functors

Before we explain what a monad is, let's explore simpler construct called a

*functor*. A functor is a typed data structure that encapsulates some value(s). From a syntactic perspective a functor is a container with the following API:
But mere syntax is not enough to understand what a functor is. The only operation that functor provides is map() that takes a function f. This function receives whatever is inside a box, transforms it and wraps the result as-is into a second functor. Please read that carefully. Functor<T> is always an immutable container, thus map never mutates the original object it was executed on. Instead, it returns the result (or results - be patient) wrapped in a brand new functor, possibly of different type R. Additionally functors should not perform any actions when identity function is applied, that is map(x -> x). Such a pattern should always return either the same functor or an equal instance.

Often Functor<T> is compared to a box holding instance of T where the only way of interacting with this value is by transforming it. However, there is no idiomatic way of unwrapping or escaping from the functor. The value(s) always stay within the context of a functor. Why are functors useful? They generalize multiple common idioms like collections, promises, optionals, etc. with a single, uniform API that works across all of them. Let me introduce a couple of functors to make you more fluent with this API:

An extra F type parameter was required to make Identity compile. What you saw in the preceding example was the simplest functor just holding a value. All you can do with that value is transforming it inside map method, but there is no way to extract it. This is considered beyond the scope of a pure functor. The only way to interact with functor is by applying sequences of type-safe transformations:

Or fluently, just like you compose functions:

From this perspective mapping over a functor is not much different than just invoking chained functions:

Why would you even bother with such verbose wrapping that not only does not provide any added value, but also is not capable of extracting the contents back? Well, it turns out you can model several other concepts using this raw functor abstraction. For example starting from Java 8 Optional is a functor with the map() method. Let us implement it from scratch:

Now it becomes interesting. An

`FOptional<T>`

functor *may*hold a value, but just as well it might be empty. It's a type-safe way of encoding`null`

. There are two ways of constructing `FOptional`

- by supplying a value or creating an `empty()`

instance. In both cases, just like with `Identity`

,`FOptional`

is immutable and we can only interact with the value from inside. What differs`FOptional`

is that the transformation function `f`

may not be applied to any value if it is empty. This means functor may not necessarily encapsulate exactly one value of type `T`

. It can just as well wrap an arbitrary number of values, just like `List`

... functor:
The API remains the same: you take a functor in a transformation - but the behavior is much different. Now we apply a transformation on each and every item in the FList, declaratively transforming the whole list. So if you have a list of customers and you want a list of their streets, it's as simple as:

It's no longer as simple as saying

Now you should see the first benefits of functors - they abstract away the internal representation and provide consistent, easy to use API over various data structures. As the last example let me introduce the

`customers.getAddress().street()`

, you can't invoke`getAddress()`

on a collection of customers, you must invoke `getAddress()`

on each individual customer and then place it back in a collection. By the way, Groovy found this pattern so common that it actually has a syntax sugar for that: `customer*.getAddress()*.street()`

. This operator, known as spread-dot, is actually a `map`

in disguise. Maybe you are wondering why I iterate over `list`

manually inside `map`

rather than using `Stream`

s from Java 8:`list.stream().map(f).collect(toList())`

? Does this ring a bell? What if I told you`java.util.stream.Stream<T>`

in Java is a functor as well? And by the way, also a monad?Now you should see the first benefits of functors - they abstract away the internal representation and provide consistent, easy to use API over various data structures. As the last example let me introduce the

*promise*functor, similar to`Future`

. `Promise`

"promises" that a value will become available one day. It is not yet there, maybe because some background computation was spawned or we are waiting for an external event. But it will appear sometime in the future. The mechanics of completing a`Promise<T>`

are not interesting, but the functor nature is:
Looks familiar? That is the point! The implementation of the functor is beyond the scope of this article and not even important. Enough to say that we are very close to implementing CompletableFuture from Java 8 and we almost discovered Observable from RxJava. But back to functors. Promise<Customer> does not hold a value of Customer just yet. It promises to have such value in the future. But we can still map over such functor, just like we did with FOptional and FList - the syntax and semantics are exactly the same. The behavior follows what the functor represents. Invoking customer.map(Customer::getAddress) yields Promise<Address>, which means map is non-blocking. customer.map() will customer promise to complete. Instead, it returns another promise, of a different type. When upstream promise completes, downstream promise applies a function passed to map() and passes the result downstream. Suddenly our functor allows us to pipeline asynchronous computations in a non-blocking manner. But you do not have to understand or learn that - because Promise is a functor, it must follow syntax and laws.

There are many other great examples of functors, for example representing value or error in a compositional manner. But it is high time to look at monads.

##

## From Functors to Monads

I assume you understand how functors work and why are they a useful abstraction. But functors are not that universal as one might expect. What happens if your transformation function (the one passed as an argument to map()) returns functor instance rather than simple value? Well, a functor is just a value as well, so nothing bad happens. Whatever was returned is placed back in a functor so all behaves consistently. However imagine you have this handy method for parsing Strings:

Exceptions are side-effects that undermine type system and functional purity. In pure functional languages, there is no place for exceptions. After all, we never heard about throwing exceptions during math classes, right? Errors and illegal conditions are represented explicitly using values and wrappers. For example tryParse() takes a String but does not simply return an int or silently throw an exception at runtime. We explicitly tell, through the type system, that tryParse() can fail, there is nothing exceptional or erroneous in having a malformed string. This semi-failure is represented by an optional result. Interestingly Java has checked exceptions, the ones that must be declared and handled, so in some sense, Java is purer in that regard, it does not hide side-effects. But for better or worse checked exceptions are often discouraged in Java, so let's get back to tryParse(). It seems useful to compose tryParse with String already wrapped in FOptional:

That should not come as a surprise. If

`tryParse()`

would return an `int`

you would get`FOptional<Integer> num`

, but because `map()`

function returns `FOptional<Integer>`

itself, it gets wrapped twice into awkward `FOptional<FOptional<Integer>>`

. Please look carefully at the types, you must understand why we got this double wrapper here. Apart from looking horrible, having a functor in functor ruins composition and fluent chaining:
Here we try to map over the contents of

`FOptional`

by turning `int`

into +Date+. Having a function of `int -> Date`

we can easily transform from `Functor<Integer>`

to `Functor<Date>`

, we know how it works. But in case of `num2`

the situation becomes complicated. What `num2.map()`

receives as input is no longer an `int`

but an `FOoption<Integer>`

and obviously`java.util.Date`

does not have such a constructor. We broke our functor by double wrapping it. However having a function that returns a functor rather than simple value is so common (like`tryParse()`

) that we can not simply ignore such requirement. One approach is to introduce a special parameterless `join()`

method that "flattens" nested functors:
It works but because this pattern is so common, special method named

`flatMap()`

was introduced. `flatMap()`

is very similar to `map`

but expects the function received as an argument to return a functor - or *monad*to be precise:
We simply concluded that

`flatMap`

is just a syntactic sugar to allow better composition. But`flatMap`

method (often called `bind`

or `>>=`

from Haskell) makes all the difference since it allows complex transformations to be composed in a pure, functional style. If `FOptional`

was an instance of monad, parsing suddenly works as expected:
Monads do not need to implement

`map`

, it can be implemented on top of `flatMap()`

easily. As a matter of fact `flatMap`

is the essential operator that enables a whole new universe of transformations. Obviously just like with functors, syntactic compliance is not enough to call some class a monad, the `flatMap()`

operator has to follow monad laws, but they are fairly intuitive like associativity of `flatMap()`

and identity. The latter requires that `m(x).flatMap(f)`

is the same as`f(x)`

for any monad holding a value `x`

and any function `f`

. We are not going to dive too deep into monad theory, instead let's focus on practical implications. Monads shine when their internal structure is not trivial, for example `Promise`

monad that will hold a value in the future. Can you guess from the type system how `Promise`

will behave in the following program? First, all methods that can potentially take some time to complete return a `Promise`

:
We can now compose these functions as if they were all blocking using monadic operators:

This becomes interesting.

`flatMap()`

must preserve monadic type therefore all intermediate objects are `Promise`

s. It is not just about keeping the types in order - preceding program is suddenly fully asynchronous! `loadCustomer()`

returns a `Promise`

so it does not block. `readBasket()`

takes whatever the `Promise`

has (will have) and applies a function returning another `Promise`

and so on and so forth. Basically, we built an asynchronous pipeline of computation where the completion of one step in the background automatically triggers next step.
Reference:

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