Related
Consider the following simple snippet of PureScript code
a :: Int
a = 5
b :: Int
b = 7
c = a + b
main ∷ Effect Unit
main = do
logShow c
The program successfully infers the type of C to be Int, and outputs the expected result:
12
However, it also produces this warning:
No type declaration was provided for the top-level declaration of c.
It is good practice to provide type declarations as a form of documentation.
The inferred type of c was:
Int
in value declaration c
I find this confusing, since I would expect the Int type for C to be safely inferred. Like it often says in the docs, "why derive types when the compiler can do it for you?" This seems like a textbook example of the simplest and most basic type inference.
Is this warning expected? Is there a standard configuration that would suppress it?
Does this warning indicate that every variable should in fact be explicitly typed?
In most cases, and certainly in the simplest cases, the types can be inferred unambiguously, and indeed, in those cases type signatures are not necessary at all. This is why simpler languages, such as F#, Ocaml, or Elm, do not require type signatures.
But PureScript (and Haskell) has much more complicated cases too. Constrained types are one. Higher-rank types are another. It's a whole mess. Don't get me wrong, I love me some high-power type system, but the sad truth is, type inference works ambiguously with all of that stuff a lot of the time, and sometimes doesn't work at all.
In practice, even when type inference does work, it turns out that its results may be wildly different from what the developer intuitively expects, leading to very hard to debug issues. I mean, type errors in PureScript can be super vexing as it is, but imagine that happening across multiple top-level definitions, across multiple modules, even perhaps across multiple libraries. A nightmare!
So over the years a consensus has formed that overall it's better to have all the top-level definitions explicitly typed, even when it's super obvious. It makes the program much more understandable and puts constraints on the typechecker, providing it with "anchor points" of sorts, so it doesn't go wild.
But since it's not a hard requirement (most of the time), it's just a warning, not an error. You can ignore it if you wish, but do that at your own peril.
Now, another part of your question is whether every variable should be explicitly typed, - and the answer is "no".
As a rule, every top-level binding should be explicitly typed (and that's where you get a warning), but local bindings (i.e. let and where) don't have to, unless you need to clarify something that the compiler can't infer.
Moreover, in PureScript (and modern Haskell), local bindings are actually "monomorphised" - that's a fancy term basically meaning they can't be generic unless explicitly specified. This solves the problem of all the ambiguous type inference, while still working intuitively most of the time.
You can notice the difference with the following example:
f :: forall a b. Show a => Show b => a -> b -> String
f a b = s a <> s b
where
s x = show x
On the second line s a <> s b you get an error saying "Could not match type b with type a"
This happens because the where-bound function s has been monomorphised, - meaning it's not generic, - and its type has been inferred to be a -> String based on the s a usage. And this means that s b usage is ill-typed.
This can be fixed by giving s an explicit type signature:
f :: forall a b. Show a => Show b => a -> b -> String
f a b = s a <> s b
where
s :: forall x. Show x => x -> String
s x = show x
Now it's explicitly specified as generic, so it can be used with both a and b parameters.
I am approaching the Haskell programming language, and I have a background of Scala and Java developer.
I was reading the theory behind type constructors, but I cannot understand if they can be considered types. I mean, in Scala, you use the keywords class or trait to define type constructors. Think about List[T], or Option[T]. Also in Haskell, you use the same keyword data, that is used for defining new types.
So, are type constructors also types?
Let's look at an analogy: functions. In some branches of mathematics, functions are called value constructors, because that's what they do: you put one or more values in, and they construct a new value out of those.
Type constructors are exactly the same thing, except on the type level: you put one or more types in, and they construct a new type out of those. They are, in some sense, functions on the type level.
Now, to our analogy: what is the analog of the question you are asking? Well, it is this: "Can value constructors (i.e. functions) be considered as values in functional programming languages?"
And the answer is: it depends on the programming language. Now, for functional programming languages, the answer is "Yes" for almost all (if not all) of them. It depends on your definition of what a "functional programming language" is. Some people define a functional programming language as a programming language which has functions as values, so the answer will be trivially "Yes" by definition. But, some people define a functional programming language as a programming language which does not allow side-effects, and in such a language, it is not necessarily true that functions are values.
The most famous example may be John Backus' FP, from his seminal paper Can Programming Be Liberated from the von Neumann Style? – a functional style and its algebra of programs. In FP, there is a hierarchy of "function-like" things. Functions can only deal with values, and functions themselves are not values. However, there is a concept of "functionals" which are "function constructors", i.e. they can take functions (and also values) as input and/or produce functions as output, but they cannot take functionals as input and/or produce them as output.
So, FP is arguably a functional programming language, but it does not have functions as values.
Note: functions as values is also called "first-class functions" and functions that take functions as input or return them as output are called "higher-order functions".
If we look at some types:
1 :: Int
[1] :: List Int
add :: Int → Int
map :: (a → b, List a) → b
You can see that we can easily say: any value whose type has an arrow in it, is a function. Any value whose type has more than one arrow in it, is a higher-order function.
Again, the same applies to type constructors, since they are really the same thing except on the type level. In some languages, type constructors can be types, in some they can't. For example, in Java and C♯, type constructors are not types. You cannot have a List<List> in C♯, for example. You can write down the type List<List> in Java, but that is misleading, since the two Lists mean different things: the first List is the type constructor, the second List is the raw type, so this is in fact not using a type constructor as a type.
What is the equivalent to our types example above?
Int :: Type
List :: Type ⇒ Type
→ :: (Type, Type) ⇒ Type
Functor :: (Type ⇒ Type) ⇒ Type
(Note, how we always have Type? Indeed, we are only dealing with types, so we normally don't write Type but instead simply write *, pronounced "Type"):
Int :: *
List :: * ⇒ *
→ :: (*, *) ⇒ *
Functor :: (* ⇒ *) ⇒ *
So, Int is a proper type, List is a type constructor that takes one type and produces a type, → (the function type constructor) takes two types and returns a type (assuming only unary functions, e.g. using currying or passing tuples), and Functor is a type constructor, which itself takes a type constructor and returns a type.
Theses "type-types" are called kinds. Like with functions, anything with an arrow is a type constructor, and anything with more than one arrow is a higher-kinded type constructor.
And like with functions, some languages allow higher-kinded type constructors and some don't. The two languages you mention in your question, Scala and Haskell do, but as mentioned above, Java and C♯ don't.
However, there is a complication when we look at your question:
So, are type constructors also types?
Not really, no. At least not in any language I know about. See, while you can have higher-kinded type constructors that take type constructors as input and/or return them as output, you cannot have an expression or a value or a variable or a parameter which has a type constructor as its type. You cannot have a function that takes a List or returns a List. You cannot have a variable of type Monad. But, you can have a variable of type Int.
So, clearly, there is a difference between types and type constructors.
Well, types and type constructors have a calculus of their own and they each have kinds. If you use :k (Maybe Int) in ghci for example, you'll get *, now this is a proper type and it (usually) has inhabitants. In this case Nothing, Just 42, etc. * now has a more descriptive alias Type.
Now you can look at the kind of the constructor that is Maybe, and :k Maybe will give you * -> *. With the alias, this is Type -> Type which is what you expect. It takes a Type and constructs a Type. Now if you see types as set of values, one good question is what set of values do Maybe has? Well, none because it is not really a type. You might attempt something like Just but that has type a -> Maybe a with kind Type, rather than Maybe with kind Type -> Type.
At least in Haskell, there is a hierarchy that can roughly be described as follows.
Terms are things that exist at run-time, values like 1, 'a', and (+), for example.
Each term has a type, like Int or Char or Int -> Int -> Int.
Each type has a kind, and all types have the same kind, namely *.
A type constructor like [], though, has kind * -> *, so it is not a type. Instead, it is a mapping from a type to a type.
There are other kinds as well, including (in addition to * and * -> *, with an example of each):
* -> * -> * (Either)
(* -> *) -> * -> * (ReaderT, a monad transformer)
Constraint (Num Int)
* -> Constraint (Num; this is the kind of a type class)
I'm new to this, and I may be missing something important. I've read part one of Category Theory for Programmers, but the most abstract math I did in university was Group Theory so I'm having to read very slowly.
I would ultimately like to understand the theory to ground my use of the techniques, so whenever I feel like I've made some progress I return to the fantasy-land spec to test myself. This time I felt like I knew where to start: I started with Functor.
From the fantasy-land spec:
Functor
u.map(a => a) is equivalent to u (identity)
u.map(x => f(g(x))) is equivalent to u.map(g).map(f) (composition)
map method
map :: Functor f => f a ~> (a -> b) -> f b
A value which has a Functor must provide a map method. The map method takes one argument:
u.map(f)
f must be a function,
i. If f is not a function, the behaviour of map is unspecified.
ii.f can return any value.
iii. No parts of f's return value should be checked.
map must return a value of the same Functor
I don't understand the phrase "a value which has a functor".
What does it mean for a value to "have a functor"? I don't think "is a functor" or "belongs to a functor" would make any more sense here.
What is the documentation trying to say about u?
My understanding is that a functor is a mapping between categories (or in this case from a category to itself). We are working in the category of types, in which objects are types and morphisms are families of functions that take a type to another type, or a type to itself.
As far as I can tell, a functor maps a to Functor a. It has some kind of constructor like,
Functor :: a -> F a
and a map function like,
map :: Functor f => (a -> b) -> f a -> f b
The a refers to any type, and the family of functions (a -> b) refers to all morphisms pointing from any particular type a to any other particular type b. So it doesn't even really make sense to me to distinguish between values which do or do not "have a functor" because if at least one functor exists, then it exists "for" every type... so every value can be mapped by a function that belongs to a morphism between two types that has been lifted by a functor. What I mean is: please show me an example of a value which "does not have a functor".
The functor is the mapping; it isn't the type F a, and it isn't values of the type F a. So in the docs, the value u is not a functor, and I don't think it "has a" functor... it is a value of type F a.
As you say, a functor in programming is a mapping between types, but JavaScript has not types, so you have to imagine them. Here, imagine that the function f goes from type a to type b. The value u is the result of some functor F acting on a. The result of u.map(f) is of the type F b, so you have a mapping from F a (that's the type of u) to F b. Here, map is treated as a method of u.
That's as close as you can get to a functor in JavaScript. (It doesn't help that the same letter f is alternatively used for the function and the functor in the documentation.)
I've been doing dev in F# for a while and I like it. However one buzzword I know doesn't exist in F# is higher-kinded types. I've read material on higher-kinded types, and I think I understand their definition. I'm just not sure why they're useful. Can someone provide some examples of what higher-kinded types make easy in Scala or Haskell, that require workarounds in F#? Also for these examples, what would the workarounds be without higher-kinded types (or vice-versa in F#)? Maybe I'm just so used to working around it that I don't notice the absence of that feature.
(I think) I get that instead of myList |> List.map f or myList |> Seq.map f |> Seq.toList higher kinded types allow you to simply write myList |> map f and it'll return a List. That's great (assuming it's correct), but seems kind of petty? (And couldn't it be done simply by allowing function overloading?) I usually convert to Seq anyway and then I can convert to whatever I want afterwards. Again, maybe I'm just too used to working around it. But is there any example where higher-kinded types really saves you either in keystrokes or in type safety?
So the kind of a type is its simple type. For instance Int has kind * which means it's a base type and can be instantiated by values. By some loose definition of higher-kinded type (and I'm not sure where F# draws the line, so let's just include it) polymorphic containers are a great example of a higher-kinded type.
data List a = Cons a (List a) | Nil
The type constructor List has kind * -> * which means that it must be passed a concrete type in order to result in a concrete type: List Int can have inhabitants like [1,2,3] but List itself cannot.
I'm going to assume that the benefits of polymorphic containers are obvious, but more useful kind * -> * types exist than just the containers. For instance, the relations
data Rel a = Rel (a -> a -> Bool)
or parsers
data Parser a = Parser (String -> [(a, String)])
both also have kind * -> *.
We can take this further in Haskell, however, by having types with even higher-order kinds. For instance we could look for a type with kind (* -> *) -> *. A simple example of this might be Shape which tries to fill a container of kind * -> *.
data Shape f = Shape (f ())
Shape [(), (), ()] :: Shape []
This is useful for characterizing Traversables in Haskell, for instance, as they can always be divided into their shape and contents.
split :: Traversable t => t a -> (Shape t, [a])
As another example, let's consider a tree that's parameterized on the kind of branch it has. For instance, a normal tree might be
data Tree a = Branch (Tree a) a (Tree a) | Leaf
But we can see that the branch type contains a Pair of Tree as and so we can extract that piece out of the type parametrically
data TreeG f a = Branch a (f (TreeG f a)) | Leaf
data Pair a = Pair a a
type Tree a = TreeG Pair a
This TreeG type constructor has kind (* -> *) -> * -> *. We can use it to make interesting other variations like a RoseTree
type RoseTree a = TreeG [] a
rose :: RoseTree Int
rose = Branch 3 [Branch 2 [Leaf, Leaf], Leaf, Branch 4 [Branch 4 []]]
Or pathological ones like a MaybeTree
data Empty a = Empty
type MaybeTree a = TreeG Empty a
nothing :: MaybeTree a
nothing = Leaf
just :: a -> MaybeTree a
just a = Branch a Empty
Or a TreeTree
type TreeTree a = TreeG Tree a
treetree :: TreeTree Int
treetree = Branch 3 (Branch Leaf (Pair Leaf Leaf))
Another place this shows up is in "algebras of functors". If we drop a few layers of abstractness this might be better considered as a fold, such as sum :: [Int] -> Int. Algebras are parameterized over the functor and the carrier. The functor has kind * -> * and the carrier kind * so altogether
data Alg f a = Alg (f a -> a)
has kind (* -> *) -> * -> *. Alg useful because of its relation to datatypes and recursion schemes built atop them.
-- | The "single-layer of an expression" functor has kind `(* -> *)`
data ExpF x = Lit Int
| Add x x
| Sub x x
| Mult x x
-- | The fixed point of a functor has kind `(* -> *) -> *`
data Fix f = Fix (f (Fix f))
type Exp = Fix ExpF
exp :: Exp
exp = Fix (Add (Fix (Lit 3)) (Fix (Lit 4))) -- 3 + 4
fold :: Functor f => Alg f a -> Fix f -> a
fold (Alg phi) (Fix f) = phi (fmap (fold (Alg phi)) f)
Finally, though they're theoretically possible, I've never see an even higher-kinded type constructor. We sometimes see functions of that type such as mask :: ((forall a. IO a -> IO a) -> IO b) -> IO b, but I think you'll have to dig into type prolog or dependently typed literature to see that level of complexity in types.
Consider the Functor type class in Haskell, where f is a higher-kinded type variable:
class Functor f where
fmap :: (a -> b) -> f a -> f b
What this type signature says is that fmap changes the type parameter of an f from a to b, but leaves f as it was. So if you use fmap over a list you get a list, if you use it over a parser you get a parser, and so on. And these are static, compile-time guarantees.
I don't know F#, but let's consider what happens if we try to express the Functor abstraction in a language like Java or C#, with inheritance and generics, but no higher-kinded generics. First try:
interface Functor<A> {
Functor<B> map(Function<A, B> f);
}
The problem with this first try is that an implementation of the interface is allowed to return any class that implements Functor. Somebody could write a FunnyList<A> implements Functor<A> whose map method returns a different kind of collection, or even something else that's not a collection at all but is still a Functor. Also, when you use the map method you can't invoke any subtype-specific methods on the result unless you downcast it to the type that you're actually expecting. So we have two problems:
The type system doesn't allow us to express the invariant that the map method always returns the same Functor subclass as the receiver.
Therefore, there's no statically type-safe manner to invoke a non-Functor method on the result of map.
There are other, more complicated ways you can try, but none of them really works. For example, you could try augment the first try by defining subtypes of Functor that restrict the result type:
interface Collection<A> extends Functor<A> {
Collection<B> map(Function<A, B> f);
}
interface List<A> extends Collection<A> {
List<B> map(Function<A, B> f);
}
interface Set<A> extends Collection<A> {
Set<B> map(Function<A, B> f);
}
interface Parser<A> extends Functor<A> {
Parser<B> map(Function<A, B> f);
}
// …
This helps to forbid implementers of those narrower interfaces from returning the wrong type of Functor from the map method, but since there is no limit to how many Functor implementations you can have, there is no limit to how many narrower interfaces you'll need.
(EDIT: And note that this only works because Functor<B> appears as the result type, and so the child interfaces can narrow it. So AFAIK we can't narrow both uses of Monad<B> in the following interface:
interface Monad<A> {
<B> Monad<B> flatMap(Function<? super A, ? extends Monad<? extends B>> f);
}
In Haskell, with higher-rank type variables, this is (>>=) :: Monad m => m a -> (a -> m b) -> m b.)
Yet another try is to use recursive generics to try and have the interface restrict the result type of the subtype to the subtype itself. Toy example:
/**
* A semigroup is a type with a binary associative operation. Law:
*
* > x.append(y).append(z) = x.append(y.append(z))
*/
interface Semigroup<T extends Semigroup<T>> {
T append(T arg);
}
class Foo implements Semigroup<Foo> {
// Since this implements Semigroup<Foo>, now this method must accept
// a Foo argument and return a Foo result.
Foo append(Foo arg);
}
class Bar implements Semigroup<Bar> {
// Any of these is a compilation error:
Semigroup<Bar> append(Semigroup<Bar> arg);
Semigroup<Foo> append(Bar arg);
Semigroup append(Bar arg);
Foo append(Bar arg);
}
But this sort of technique (which is rather arcane to your run-of-the-mill OOP developer, heck to your run-of-the-mill functional developer as well) still can't express the desired Functor constraint either:
interface Functor<FA extends Functor<FA, A>, A> {
<FB extends Functor<FB, B>, B> FB map(Function<A, B> f);
}
The problem here is this doesn't restrict FB to have the same F as FA—so that when you declare a type List<A> implements Functor<List<A>, A>, the map method can still return a NotAList<B> implements Functor<NotAList<B>, B>.
Final try, in Java, using raw types (unparametrized containers):
interface FunctorStrategy<F> {
F map(Function f, F arg);
}
Here F will get instantiated to unparametrized types like just List or Map. This guarantees that a FunctorStrategy<List> can only return a List—but you've abandoned the use of type variables to track the element types of the lists.
The heart of the problem here is that languages like Java and C# don't allow type parameters to have parameters. In Java, if T is a type variable, you can write T and List<T>, but not T<String>. Higher-kinded types remove this restriction, so that you could have something like this (not fully thought out):
interface Functor<F, A> {
<B> F<B> map(Function<A, B> f);
}
class List<A> implements Functor<List, A> {
// Since F := List, F<B> := List<B>
<B> List<B> map(Function<A, B> f) {
// ...
}
}
And addressing this bit in particular:
(I think) I get that instead of myList |> List.map f or myList |> Seq.map f |> Seq.toList higher kinded types allow you to simply write myList |> map f and it'll return a List. That's great (assuming it's correct), but seems kind of petty? (And couldn't it be done simply by allowing function overloading?) I usually convert to Seq anyway and then I can convert to whatever I want afterwards.
There are many languages that generalize the idea of the map function this way, by modeling it as if, at heart, mapping is about sequences. This remark of yours is in that spirit: if you have a type that supports conversion to and from Seq, you get the map operation "for free" by reusing Seq.map.
In Haskell, however, the Functor class is more general than that; it isn't tied to the notion of sequences. You can implement fmap for types that have no good mapping to sequences, like IO actions, parser combinators, functions, etc.:
instance Functor IO where
fmap f action =
do x <- action
return (f x)
-- This declaration is just to make things easier to read for non-Haskellers
newtype Function a b = Function (a -> b)
instance Functor (Function a) where
fmap f (Function g) = Function (f . g) -- `.` is function composition
The concept of "mapping" really isn't tied to sequences. It's best to understand the functor laws:
(1) fmap id xs == xs
(2) fmap f (fmap g xs) = fmap (f . g) xs
Very informally:
The first law says that mapping with an identity/noop function is the same as doing nothing.
The second law says that any result that you can produce by mapping twice, you can also produce by mapping once.
This is why you want fmap to preserve the type—because as soon as you get map operations that produce a different result type, it becomes much, much harder to make guarantees like this.
I don't want to repeat information in some excellent answers already here, but there's a key point I'd like to add.
You usually don't need higher-kinded types to implement any one particular monad, or functor (or applicative functor, or arrow, or ...). But doing so is mostly missing the point.
In general I've found that when people don't see the usefulness of functors/monads/whatevers, it's often because they're thinking of these things one at a time. Functor/monad/etc operations really add nothing to any one instance (instead of calling bind, fmap, etc I could just call whatever operations I used to implement bind, fmap, etc). What you really want these abstractions for is so you can have code that works generically with any functor/monad/etc.
In a context where such generic code is widely used, this means any time you write a new monad instance your type immediately gains access to a large number of useful operations that have already been written for you. That's the point of seeing monads (and functors, and ...) everywhere; not so that I can use bind rather than concat and map to implement myFunkyListOperation (which gains me nothing in itself), but rather so that when I come to need myFunkyParserOperation and myFunkyIOOperation I can re-use the code I originally saw in terms of lists because it's actually monad-generic.
But to abstract across a parameterised type like a monad with type safety, you need higher-kinded types (as well explained in other answers here).
For a more .NET-specific perspective, I wrote a blog post about this a while back. The crux of it is, with higher-kinded types, you could potentially reuse the same LINQ blocks between IEnumerables and IObservables, but without higher-kinded types this is impossible.
The closest you could get (I figured out after posting the blog) is to make your own IEnumerable<T> and IObservable<T> and extended them both from an IMonad<T>. This would allow you to reuse your LINQ blocks if they're denoted IMonad<T>, but then it's no longer typesafe because it allows you to mix-and-match IObservables and IEnumerables within the same block, which while it may sound intriguing to enable this, you'd basically just get some undefined behavior.
I wrote a later post on how Haskell makes this easy. (A no-op, really--restricting a block to a certain kind of monad requires code; enabling reuse is the default).
The most-used example of higher-kinded type polymorphism in Haskell is the Monad interface. Functor and Applicative are higher-kinded in the same way, so I'll show Functor in order to show something concise.
class Functor f where
fmap :: (a -> b) -> f a -> f b
Now, examine that definition, looking at how the type variable f is used. You'll see that f can't mean a type that has value. You can identify values in that type signature because they're arguments to and results of a functions. So the type variables a and b are types that can have values. So are the type expressions f a and f b. But not f itself. f is an example of a higher-kinded type variable. Given that * is the kind of types that can have values, f must have the kind * -> *. That is, it takes a type that can have values, because we know from previous examination that a and b must have values. And we also know that f a and f b must have values, so it returns a type that must have values.
This makes the f used in the definition of Functor a higher-kinded type variable.
The Applicative and Monad interfaces add more, but they're compatible. This means that they work on type variables with kind * -> * as well.
Working on higher-kinded types introduces an additional level of abstraction - you aren't restricted to just creating abstractions over basic types. You can also create abstractions over types that modify other types.
Why might you care about Applicative? Because of traversals.
class (Functor t, Foldable t) => Traversable t where
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t
Once you've written a Traversable instance, or a Traversal for some type, you can use it for an arbitrary Applicative.
Why might you care about Monad? One reason is streaming systems like pipes, conduit, and streaming. These are entirely non-trivial systems for working with effectful streams. With the Monad class, we can reuse all that machinery for whatever we like, rather than having to rewrite it from scratch each time.
Why else might you care about Monad? Monad transformers. We can layer monad transformers however we like to express different ideas. The uniformity of Monad is what makes this all work.
What are some other interesting higher-kinded types? Let's say ... Coyoneda. Want to make repeated mapping fast? Use
data Coyoneda f a = forall x. Coyoneda (x -> a) (f x)
This works or any functor f passed to it. No higher-kinded types? You'll need a custom version of this for each functor. This is a pretty simple example, but there are much trickier ones you might not want to have to rewrite every time.
Recently stated learning a bit about higher kinded types. Although it's an interesting idea, to be able to have a generic that needs another generic but apart from library developers, i do not see any practical use in any real app. I use scala in business app, i have also seen and studied the code of some nicely designed sgstems and libraries like kafka, akka and some financial apps. Nowhere I found any higher kinded type in use.
It seems like they are nice for academia or similar but the market doesn't need it or hasn't reached to a point where HKT has any practical uses or proves to be better than other existing techniques. To me it's something that you can use to impress others or write blog posts but nothing more than that. It's like multiverse or string theory. Looks nice on paper, gives you hours to speak about but nothing real
(sorry if you don't have any interest in theoratical physiss). One prove is that all the answers above, they all brilliantly describe the mechanics fail to cite one true real case where we would need it despite being the fact it's been 6+ years since OP posted it.
I have already read the Wikipedia article and searched for obvious places but I'm stuck. Can someone simple tell me what exactly is a Kind ? What is it used for ?
Scala examples are most appreciated
In short: a kind is to types what a type is to values.
What is a value? 1, 2, 3 are values. So are "Hello" and "World", true and false, and so forth.
Values belong to types. Types describe a set of values. 1, 2 and 3 belong to the type Nat, "Hello" and "World" to the type Text, true and false to the type Boolean.
Functions take one or more values as arguments and produce one or more values as results. In order to meaningfully do something with the arguments, the function needs to make some assumptions about them, this is done by constraining their types. So, function parameters and return values typically also have types.
Now, a function also has a type, which is described by the types of its inputs and outputs. For example, the abs function which computes the absolute value of a number has the type
Number -> NonNegativeNumber
The function add which adds two numbers has the type
(Number, Number) -> Number
The function divmod has the type
(Number, Number) -> (Number, Number)
Okay, but if functions take values as arguments and produce values as results, and functions are values, then functions can also take functions as arguments and return functions as results, right? What's the type of such a function?
Let's say we have a function findCrossing which finds the point where some other function (of numbers) crosses the y-axis. It takes as an argument the function and produces as a result a number:
(Number -> Number) -> Number
Or a function makeAdder which produces a function which takes a number and adds a specific number to it:
Number -> (Number -> Number)
And a function which computes the derivative of another function:
(Number -> Number) -> (Number -> Number)
Let's look at the level of abstraction here: a value is something very concrete. It only means one thing. If we were to order our levels of abstraction here, we could say that a value has the order 0.
A function, OTOH is more abstract: a single function can produce many different values. So, it has order 1.
A function which returns or accepts a function is even more abstract: it can produce many different functions which can produce many different values. It has order 2.
Generally, we call everything with an order > 1 "higher order".
Okay, but what about kinds? Well, we said above that 1, 2, "Hello", false etc. have types. But what is the type of Number? Or Text? Or Boolean?
Well, its type is Type, of course! This "type of a type" is called a kind.
And just like we can have functions which construct values out of values, we can have functions which construct types out of types. These functions are called type constructors.
And just like functions have types, type constructors have kinds. For example, the List type constructor, which takes an element type and produces a list type for that element has kind
Type -> Type
The Map type constructor, which takes a key type and a value type and produces a map type has kind
(Type, Type) -> Type
Now, continuing the analogy, if we can have functions which take functions as arguments, can we also have type constructors which take type constructors as arguments? Of course!
An example is the Functor type constructor. It takes a type constructor and produces a type:
(Type -> Type) -> Type
Notice how we always write Type here? Above, we had many different types like Number, Text, Boolean etc. Here, we always have only kind of type, namely Type. That gets tedious to (warning, bad pun ahead) type, so instead of writing Type everywhere, we just write *. I.e. Functor has the kind
(* -> *) -> *
and Number has the kind
*
Continuing the analogy, Number, Text and all others of kind * have order 0, List and all others of kind * -> * or more generally (*, …) -> (*, …) have order 1, Functor and all of kind (* -> *) -> * or * -> (* -> *) (and so forth) have order 2. Except in this case we sometimes also call it rank instead of order.
Everything above order/rank 1 is called higher-order, higher-rank or higher-kinded.
I hope the analogy is clear now: types describe sets of values; kinds describe sets of types.
Aside: I completely ignored currying. Basically, currying means that you can transform any function which takes two values into a function which takes one value and returns a function which takes the other value, similarly for three, four, five, … arguments. This means that you only ever need to deal with functions with exactly one parameter, which makes languages much simpler.
However, this also means that technically speaking, add is a higher-order function (because it returns a function) which is muddying the definitions.
The most eloquent explanation for kinds/higher-kinds I've seen so far and also in the context of Scala is Daniel Spiewak's High Wizardry in the Land of Scala. There are many versions, he gave this talk a few times, here I've chosen the longest one, but you can quickly google and find others.
The most important message from this talk is exactly the answer given by #Jörg W Mittag:
"kind is to types what a type is to values"
Another place for a more theoretical view on the subject is the paper Generics of a Higher Kind, also in the context of Scala.