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One thing I do not fully understand about Haskell is declaring functions and their types: is it something you have to do or is it just something you should do for clarity? Or are there certain scenarios where you need to do it, just not all?
You don’t need to declare the type of any function that uses only standard Haskell type system features. Haskell 98 is specified with global type inference, meaning that the types of all top-level bindings are guaranteed to be inferable.
However, it’s good practice to include type annotations for top-level definitions, for a few reasons:
Verifying that the inferred type matches your expectations
Helping the compiler producing better diagnostic messages when there are type mismatches
Most importantly, documenting your intent and making the code more readable for humans!
As for definitions in where clauses, it’s a matter of style. The conventional style is to omit them, partly because in some cases, their types could not be written explicitly before the ScopedTypeVariables extension. I consider the omission of scoped type variables a bit of a bug in the 1998 and 2010 standards, and GHC is the de facto standard compiler today, but it’s still a nonstandard extension. Regardless, it’s good practice to include annotations where possible for nontrivial code, and helpful for you as a programmer.
In practice, it’s common to use some language extensions that complicate type inference or make it “undecidable”, meaning that, at least for arbitrary programs, it’s impossible to always infer a type, or at least a unique “best” type. But for the sake of usability, extensions are usually designed very carefully to only require annotations at the point where you actually use them.
For example, GHC (and standard Haskell) will only infer polymorphic types with top-level foralls, which are normally left entirely implicit. (They can be written explicitly using ExplicitForAll.) If you need to pass a polymorphic function as an argument to another function like (forall t. …) -> … using RankNTypes, this requires an annotation to override the compiler’s assumption that you meant something like forall t. (… -> …), or that you mistakenly applied the function on different types.
If an extension requires annotations, the rules for when and where you must include them are typically documented in places like the GHC User’s Guide, and formally specified in the papers specifying the feature.
Short answer: Functions are defined in "bindings" and have their types declared in "type signatures". Type signatures for bindings are always syntactically optional, as the language doesn't require their use in any particular case. (There are some places type signatures are required for things other than bindings, like in class definitions or in declarations of data types, but I don't think there's any case where a binding requires an accompanying type signature according to the syntax of the language, though I might be forgetting some weird situation.) The reason they aren't required is that the compiler can usually, though not always, figure out the types of functions itself as part of its type-checking operation.
However, some programs may not compile unless a type signature is added to a binding, and some programs may not compile unless a type signature is removed, so sometimes you need to use them, and sometimes you can't use them (at least not without a language extension and some changes to the syntax of other, nearby type signatures to use the extension).
It is considered best practice to include type signatures for every top-level binding, and the GHC -Wall flag will warn you if any top-level bindings lack an associated type signature. The rationale for this is that top-level signatures (1) provide documentation for the "interface" of your code, (2) aren't so numerous that they overburden the programmer, and (3) usually provide sufficient guidance to the compiler that you get better error messages than if you omit type signatures entirely.
If you look at almost any real-world Haskell source code (e.g., browse the source of any decent library on Hackage), you'll see this convention being used -- all top-level bindings have type signatures, and type signatures are used sparingly in other contexts (in expressions or where or let clauses). I'd encourage any beginner to use this same convention in the code they write as they're learning Haskell. It's a good habit and will avoid many frustrating error messages.
Long answer:
In Haskell, a binding assigns a name to a chunk of code, like the following binding for the function hypo:
hypo a b = sqrt (a*a + b*b)
When compiling a binding (or collection of related bindings), the compiler performs a type-checking operation on the expressions and subexpressions that are involved.
It is this type-checking operation that allows the compiler to determine that the variable a in the above expression must be of some type t that has a Num t constraint (in order to support the * operation), that the result of a*a will be the same type t, and that this implies that b*b and so b are also of this same type t (since only two values of the same type can be added together with +), and that a*a + b*b is therefore of type t, and so the result of the sqrt must also be of this same type t which must incidentally have a Floating t constraint to support the sqrt operation. The information collected and type relationships deduced during this type checking allow the compiler to infer a general type signature for the hypo function automatically, namely:
hypo :: (Floating t) => t -> t -> t
(The Num t constraint doesn't appear because it's implied by Floating t).
Because the compiler can learn the type signatures of (most) bound names, like hypo, automatically as a side-effect of the type-checking operation, there's no fundamental need for the programmer to explicitly supply this information, and that's the motivation for the language making type signatures optional. The only requirements the language places on type signatures is that if they are supplied, they must appear in the same declaration list as the associated binding (e.g., both must appear in the same module, or in the same where clause or whatever, and you can't have a type signature without a binding), there must be at most one type signature for a binding (no duplicate type signatures, even if they are identical, unlike in C, say), and the type supplied in the type signature must not be in conflict with the results of type checking.
The language allows the type signature and binding to appear anywhere in the same declaration list, in any order, and with no requirement they be next to each other, so the following is valid Haskell code:
double :: (Num a) => a -> a
half x = x / 2
double x = x + x
half :: (Fractional a) => a -> a
Such silliness is not recommended, however, and the convention is to place the type signature immediately before the corresponding binding, though one exception is to have a type signature shared across multiple bindings of the same type, whose definitions follow:
ex1, ex2, ex3 :: Tree Int
ex1 = Leaf 1
ex2 = Node (Leaf 2) (Leaf 3)
ex3 = Node (Node (Leaf 4) (Leaf 5)) (Leaf 5)
In some situations, the compiler cannot automatically infer the correct type for a binding, and a type signature may be required. The following binding requires a type signature and won't compile without it. (The technical problem is that toList is written using polymorphic recursion.)
data Binary a = Leaf a | Pair (Binary (a,a)) deriving (Show)
-- following type signature is required...
toList :: Binary a -> [a]
toList (Leaf x) = [x]
toList (Pair b) = concatMap (\(x,y) -> [x,y]) (toList b)
In other situations, the compiler can automatically infer the correct type for a binding, but the type can't be expressed in a type signature (at least, not without some GHC extensions to the standard language). This happens most often in where clauses. (The technical problem is that type variables aren't scoped, and go's type involves the type variable a from the type signature of myLookup.)
myLookup :: Eq a => a -> [(a,b)] -> Maybe b
myLookup k = go
where -- go :: [(a,b)] -> Maybe b
go ((k',v):rest) | k == k' = Just v
| otherwise = go rest
go [] = Nothing
There's no type signature in standard Haskell for go that would work here. However, if you enable an extension, you can write one if you also modify the type signature for myLookup itself to scope the type variables.
myLookup :: forall a b. Eq a => a -> [(a,b)] -> Maybe b
myLookup k = go
where go :: [(a,b)] -> Maybe b
go ((k',v):rest) | k == k' = Just v
| otherwise = go rest
go [] = Nothing
It's considered best practice to put type signatures on all top-level bindings and use them sparingly elsewhere. The -Wall compiler flag turns on the -Wmissing-signatures warning which warns about any missing top-level signatures.
The main motivation, I think, is that top-level bindings are the ones that are most likely to be used in multiple places throughout the code at some distance from where they are defined, and the type signature usually provides concise documentation for what a function does and how it's intended to be used. Consider the following type signatures from a Sudoku solver I wrote many years ago. Is there much doubt what these functions do?
possibleSymbols :: Index -> Board -> [Symbol]
possibleBoards :: Index -> Board -> [Board]
setSymbol :: Index -> Board -> Symbol -> Board
While the type signatures auto-generated by the compiler also serve as decent documentation and can be inspected in GHCi, it's convenient to have the type signatures in the source code, as a form of compiler-checked comment documenting the binding's purpose.
Any Haskell programmer who's spent a moment trying to use an unfamiliar library, read someone else's code, or read their own past code knows how helpful top-level signatures are as documentation. (Admittedly, a frequently levelled criticism of Haskell is that sometimes the type signatures are the only documentation for a library.)
A secondary motivation is that in developing and refactoring code, type signatures make it easier to "control" the types and localize errors. Without any signatures, the compiler can infer some really crazy types for code, and the error messages that get generated can be baffling, often identifying parts of the code that have nothing to do with the underlying error.
For example, consider this program:
data Tree a = Leaf a | Node (Tree a) (Tree a)
leaves (Leaf x) = x
leaves (Node l r) = leaves l ++ leaves r
hasLeaf x t = elem x (leaves t)
main = do
-- some tests
print $ hasLeaf 1 (Leaf 1)
print $ hasLeaf 1 (Node (Leaf 2) (Leaf 3))
The functions leaves and hasLeaf compile fine, but main barfs out the following cascade of errors (abbreviated for this posting):
Leaves.hs:12:11-28: error:
• Ambiguous type variable ‘a0’ arising from a use of ‘hasLeaf’
prevents the constraint ‘(Eq a0)’ from being solved.
Probable fix: use a type annotation to specify what ‘a0’ should be.
Leaves.hs:12:19: error:
• Ambiguous type variable ‘a0’ arising from the literal ‘1’
prevents the constraint ‘(Num a0)’ from being solved.
Probable fix: use a type annotation to specify what ‘a0’ should be.
Leaves.hs:12:27: error:
• No instance for (Num [a0]) arising from the literal ‘1’
Leaves.hs:13:11-44: error:
• Ambiguous type variable ‘a1’ arising from a use of ‘hasLeaf’
prevents the constraint ‘(Eq a1)’ from being solved.
Probable fix: use a type annotation to specify what ‘a1’ should be.
Leaves.hs:13:19: error:
• Ambiguous type variable ‘a1’ arising from the literal ‘1’
prevents the constraint ‘(Num a1)’ from being solved.
Probable fix: use a type annotation to specify what ‘a1’ should be.
Leaves.hs:13:33: error:
• No instance for (Num [a1]) arising from the literal ‘2’
With programmer-supplied top-level type signatures:
leaves :: Tree a -> [a]
leaves (Leaf x) = x
leaves (Node l r) = leaves l ++ leaves r
hasLeaf :: (Eq a) => a -> Tree a -> Bool
hasLeaf x t = elem x (leaves t)
a single error is immediately localized to the offending line:
leaves (Leaf x) = x
^
Leaves.hs:4:19: error:
• Occurs check: cannot construct the infinite type: a ~ [a]
Beginners might not understand the "occurs check" but are at least looking at the right place to make the simple fix:
leaves (Leaf x) = [x]
So, why not add type signatures everywhere, not just at top-level? Well, if you literally tried to add type signatures everywhere they were syntactically valid, you'd be writing code like:
{-# LANGUAGE ScopedTypeVariables #-}
hypo :: forall t. (Floating t) => t -> t -> t
hypo (a :: t) (b :: t) = sqrt (((a :: t) * (a :: t) :: t) + ((b :: t) * (b :: t) :: t) :: t) :: t
so you want to draw the line somewhere. The main argument against adding them for all bindings in let and where clauses is that those bindings are often short bindings easily understood at a glance, and they're localized to the code that you're trying to understand "all at once" anyway. The signatures are also potentially less useful as documentation because bindings in these clauses are more likely to refer to and use other nearby bindings of arguments or intermediate results, so they aren't "self-contained" like a top-level binding. The signature only documents a small portion of what the binding is doing. For example, in:
qsort :: (Ord a) => [a] -> [a]
qsort (x:xs) = qsort l ++ [x] ++ qsort r
where -- l, r :: [a]
l = filter (<=x) xs
r = filter (>x) xs
qsort [] = []
having type signatures l, r :: [a] in the where clause wouldn't add very much. There's also the additional complication that you'd need the ScopedTypeVariables extension to write it, as above, so that's maybe another reason to omit it.
As I say, I think any Haskell beginner should be encouraged to adopt a similar convention of writing top-level type signatures, ideally writing the top-level signature before starting to write the accompanying bindings. It's one of the easiest ways to leverage the type system to guide the design process and write good Haskell code.
I've been playing with Haskell, trying to create a very simple website using Servant and Lucid. At the moment I reached the stage "My code works, I have no idea why".
I tried creating Bootstrap button. According the the doc, it should be defined as:
<button type="button" class="btn btn-primary">Primary</button>
So I found Lucid.Html5 doc: https://hackage.haskell.org/package/lucid-2.9.11/docs/Lucid-Html5.html and worked out the function that creates a button:
button_ :: Term arg result => arg -> result
After spending some time trying to work out the correct syntax, I came up with this:
-- correctly replicates the html pasted above
button_ [type_ "button", class_ "btn btn-primary"] "Primary"
Typically I would have called it a victory and focused on other tasks, but this one looks like a true piece of magic to me.
The doc says "button_" is a function that takes an argument "arg" and returns a value of a generic type "result". However, in my application "button_" clearly takes two arguments and returns "Html ()".
-- f arg arg again ??
button_ [type_ "button", class_ "btn btn-primary"] "Primary"
It must do something with the "Term" typeclass, but I'm not sure how to understand it. Can someone help me with this ? I tried loading the module into ghci and inspecting types with ":t", but that didn't help me too much.
The Term typeclass is very convenient—we don't need different term functions for creating elements with or without attributes—but can be a bit difficult to understand.
The definition of button_ is
-- | #button# element
button_ :: Term arg result => arg -> result
button_ = term "button"
term is a method of the Term typeclass, and has type:
term :: Text -> arg -> result
That is: you give it the name of the element, some argument whose type depends on the particular instance, and it returns some result whose type depends on the particular instance. But what instances are available? There are three:
Term Text Attribute
-- here, term :: Text -> Text -> Attribute
This one is for creating attributes, not elements.
Applicative m => Term (HtmlT m a) (HtmlT m a)
-- here, term :: Text -> HtmlT m a -> HtmlT m a
This one is for creating elements without attributes. The arg we pass as argument to term is some piece of html representing the children, and we get another piece of html in return.
(Applicative m, f ~ HtmlT m a) => Term [Attribute] (f -> HtmlT m a)
-- here, term :: Text -> [Attribute] -> HtmlT m a -> HtmlT m a
This one is the most confusing, and the one that is being used in your code. Here, arg is a list of Attribute values. That much is clear. But result is the type of a function! After passing the attributes, we are left with another function HtmlT m a -> HtmlT m a which allows us to supply the contents of the button ("Primary" in your case).
The f ~ HtmlT m a is another wrinkle that isn't really relevant for this answer. It simply says that f is equal to HtmlT m a. Why not put it directly then? Well, in certain cases it can help drive type inference in desirable ways.
I've started a course on introduction to F#, and I've been having some trouble with two assignments. The first one had me creating two functions, where the first function takes an input and adds it with four, and the second one calculates sqrt(x^2+y^2). Then I should write function expressions for them both, but for some reason it gives me the error "Unexpected symbol'|' in implementation file".
let g = fun n -> n + 4;;
let h = fun (x,y) -> System.Math.Sqrt((x*x)+(y*y));;
let f = fun (x,n) -> float
|(n,0) -> g(n)
|(x,n) -> h(x,n);;
The second assignment asks me to create a function, which finds the sequence of Fibonaccis numbers. I've written the following code, but it seems to forget about the 0 in the beginning since the output always is n+1 and not n.
let rec fib = function
|0 -> 0
|1 -> 1
|n -> fib(n-1) + fib(n-2)
;;
Keep in mind that this is the first week, so I should be able to create these with those methods.
Your first snippet mostly suffers from two issues:
In F#, there is a difference between float and int. You write integer values as 4 or 0 and you write float values as 4.0 or 0.0. F# does not automatically convert integers to floats, so you need to be consistent.
Your syntax in the f function is a bit odd - I'm not sure what float is supposed to mean there and the fun and function constructs behave differently.
So, starting with your original code:
let g = fun n -> n + 4;;
This works, but I would not write it as an explicit function using fun - you can use let to define functions too and it is simpler. Also, you only need ;; in F# Interactive, but if you're using any decent editor with command for sending code to F# interactive (via Alt+Enter) you do not need that.
However, in your f function, you want to return float so you need to modify g to return float too. This means replacing 4 with 4.0:
let g n = n + 4.0
The h function is good, but you can again write it using let:
let h (x,y) = System.Math.Sqrt((x*x)+(y*y));;
In your f function, you can either use function to write a function using pattern matching, or you can use more verbose syntax using match (function is just a shorthand for writing a function and then pattern matching on the input):
let f = function
| (n,0.0) -> g(n)
| (x,n) -> h(x,n)
let f (x, y) =
match (x, y) with
| (n,0.0) -> g(n)
| (x,n) -> h(x,n)
Also note that the indentation matters - you need spaces before |.
I'm going to address your first block of code, and leave the Fibonacci function for later. First I'll repost your code, then I'll talk about it.
let g = fun n -> n + 4;;
let h = fun (x,y) -> System.Math.Sqrt((x*x)+(y*y));;
let f = fun (x,n) -> float
|(n,0) -> g(n)
|(x,n) -> h(x,n);;
First comment: If you're defining a function and assigning it immediately to a name, like in all these examples, you don't need the fun keyword. The usual way to define functions is to write them as let (name) (parameters) = (function body). So your code above would become:
let g n = n + 4;;
let h (x,y) = System.Math.Sqrt((x*x)+(y*y));;
let f (x,n) = float
|(n,0) -> g(n)
|(x,n) -> h(x,n);;
I haven't made any other changes, so your f function still has an error in it. Let's address that error next.
I think the mistake you're making here is to think that fun and function are interchangeable. They're not. fun is standard function definition, but function is something else. It's a very common pattern in F# to write functions like the following:
let someFunc parameter =
match parameter with
| "case 1" -> printfn "Do something"
| "case 2" -> printfn "Do something else"
| _ -> printfn "Default behavior"
The function keyword is shorthand for one parameter plus a match expression. In other words, this:
let someFunc = function
| "case 1" -> printfn "Do something"
| "case 2" -> printfn "Do something else"
| _ -> printfn "Default behavior"
is exactly the same code as this:
let someFunc parameter =
match parameter with
| "case 1" -> printfn "Do something"
| "case 2" -> printfn "Do something else"
| _ -> printfn "Default behavior"
with just one difference. In the version with the function keyword, you don't get to pick the name of the parameter. It gets automatically created by the F# compiler, and since you can't know in advance what the name of the parameter will be, you can't refer to it in your code. (Well, there are ways, but I don't want to make you learn to run before you have learned to walk, so to speak). And one more thing: while you're still learning F#, I strongly recommend that you do NOT use the function keyword. It's really useful once you know what you're doing, but in your early learning stages you should use the more explicit match (parameter) with expressions. That way you'll get used to seeing what it's doing. Once you've been doing F# for a few months, then you can start replacing those let f param = match param with (...) expressions with the shorter let f = function (...). But until match param with (...) has really sunk in and you've understood it, you should continue to type it out explicitly.
So your f function should have looked like:
let f (x,n) =
match (x,n) with
|(n,0) -> g(n)
|(x,n) -> h(x,n);;
I see that while I was typing this, Tomas Petricek posted a response, and it addresses the incorrect usage of float, so I won't duplicate his explanation of why you're going to get an error on the word float in your f function. And he also explained about ;;, so I won't duplicate that either. I'll just say that when he mentions "any decent editor with command for sending code to F# interactive (via Alt+Enter)", there are a lot of editor choices -- but as a beginner, you might just want someone to recommend one to you, so I'll recommend one. First off, though: if you're on Windows, you might be using Visual Studio already, in which case you should stick to Visual Studio since you know it. It's a good editor for F#. But if you don't use Visual Studio yet, I don't recommend downloading it just to play around with F#. It's a beast of a program, designed for professional software developers to do all sorts of things they need to do in their jobs, and so it can feel a bit overwhelming if you're just getting started. So I would actually recommend something more lightweight: the editor called Visual Studio Code. It's cross-platform, and will run perfectly well on Linux, OS X or on Windows. Once you've downloaded and installed VS Code, you'll then want to install the Ionide extension. Ionide is a plugin for VS Code (and also for Atom, though the Atom version of Ionide is updated less often since all the Ionide developers use VS Code now) that makes F# editing a real pleasure. There are actually three extensions you'll find: Ionide-fsharp, Ionide-FAKE, and Ionide-Paket. Download and install all three: FAKE and Paket are two tools for F# programming that you might not need yet, but once you do need them, you'll already have them installed.
Okay, that's enough to get you started, I think.
I am beginner in Haskell .
The convention used in function definition as per my school material is actually as follows
function_name arguments_separated_by_spaces = code_to_do
ex :
f a b c = a * b +c
As a mathematics student I am habituated to use the functions like as follows
function_name(arguments_separated_by_commas) = code_to_do
ex :
f(a,b,c) = a * b + c
Its working in Haskell .
My doubt is whether it works in all cases ?
I mean can i use traditional mathematical convention in Haskell function definition also ?
If wrong , in which specific cases the convention goes wrong ?
Thanks in advance :)
Let's say you want to define a function that computes the square of the hypoteneuse of a right-triangle. Either of the following definitions are valid
hyp1 a b = a * a + b * b
hyp2(a,b) = a * a + b * b
However, they are not the same function! You can tell by looking at their types in GHCI
>> :type hyp1
hyp1 :: Num a => a -> a -> a
>> :type hyp2
hyp2 :: Num a => (a, a) -> a
Taking hyp2 first (and ignoring the Num a => part for now) the type tells you that the function takes a pair (a, a) and returns another a (e.g it might take a pair of integers and return another integer, or a pair of real numbers and return another real number). You use it like this
>> hyp2 (3,4)
25
Notice that the parentheses aren't optional here! They ensure that the argument is of the correct type, a pair of as. If you don't include them, you will get an error (which will probably look really confusing to you now, but rest assured that it will make sense when you've learned about type classes).
Now looking at hyp1 one way to read the type a -> a -> a is it takes two things of type a and returns something else of type a. You use it like this
>> hyp1 3 4
25
Now you will get an error if you do include parentheses!
So the first thing to notice is that the way you use the function has to match the way you defined it. If you define the function with parens, you have to use parens every time you call it. If you don't use parens when you define the function, you can't use them when you call it.
So it seems like there's no reason to prefer one over the other - it's just a matter of taste. But actually I think there is a good reason to prefer one over the other, and you should prefer the style without parentheses. There are three good reasons:
It looks cleaner and makes your code easier to read if you don't have parens cluttering up the page.
You will take a performance hit if you use parens everywhere, because you need to construct and deconstruct a pair every time you use the function (although the compiler may optimize this away - I'm not sure).
You want to get the benefits of currying, aka partially applied functions*.
The last point is a little subtle. Recall that I said that one way to understand a function of type a -> a -> a is that it takes two things of type a, and returns another a. But there's another way to read that type, which is a -> (a -> a). That means exactly the same thing, since the -> operator is right-associative in Haskell. The interpretation is that the function takes a single a, and returns a function of type a -> a. This allows you to just provide the first argument to the function, and apply the second argument later, for example
>> let f = hyp1 3
>> f 4
25
This is practically useful in a wide variety of situations. For example, the map functions lets you apply some function to every element of a list -
>> :type map
map :: (a -> b) -> [a] -> [b]
Say you have the function (++ "!") which adds a bang to any String. But you have lists of Strings and you'd like them all to end with a bang. No problem! You just partially apply the map function
>> let bang = map (++ "!")
Now bang is a function of type**
>> :type bang
bang :: [String] -> [String]
and you can use it like this
>> bang ["Ready", "Set", "Go"]
["Ready!", "Set!", "Go!"]
Pretty useful!
I hope I've convinced you that the convention used in your school's educational material has some pretty solid reasons for being used. As someone with a math background myself, I can see the appeal of using the more 'traditional' syntax but I hope that as you advance in your programming journey, you'll be able to see the advantages in changing to something that's initially a bit unfamiliar to you.
* Note for pedants - I know that currying and partial application are not exactly the same thing.
** Actually GHCI will tell you the type is bang :: [[Char]] -> [[Char]] but since String is a synonym for [Char] these mean the same thing.
f(a,b,c) = a * b + c
The key difference to understand is that the above function takes a triple and gives the result. What you are actually doing is pattern matching on a triple. The type of the above function is something like this:
(a, a, a) -> a
If you write functions like this:
f a b c = a * b + c
You get automatic curry in the function.
You can write things like this let b = f 3 2 and it will typecheck but the same thing will not work with your initial version. Also, things like currying can help a lot while composing various functions using (.) which again cannot be achieved with the former style unless you are trying to compose triples.
Mathematical notation is not consistent. If all functions were given arguments using (,), you would have to write (+)((*)(a,b),c) to pass a*b and c to function + - of course, a*b is worked out by passing a and b to function *.
It is possible to write everything in tupled form, but it is much harder to define composition. Whereas now you can specify a type a->b to cover for functions of any arity (therefore, you can define composition as a function of type (b->c)->(a->b)->(a->c)), it is much trickier to define functions of arbitrary arity using tuples (now a->b would only mean a function of one argument; you can no longer compose a function of many arguments with a function of many arguments). So, technically possible, but it would need a language feature to make it simple and convenient.
I need a function with following attributes.
(c->d)->(a->b->c)->a->b->d
my function:
funktionD = (.) . (.)
but :t funktionD
funktionD :: (a -> b) -> (c -> d -> a) -> c -> d -> b
ist this equal?
It is. Type variables--written in lowercase--are just that, variables. You can rename them all you like as long as the pattern of which are the same variable stays the same.
Furthermore, for essentially the same reason, for the type signature you gave there is only one possible function of that type (excluding functions that crash or go into infinite loops, that is). Something to think about!
I think it is. Replace a with c (and vice versa), then replace b with d (and vice versa) and they are the same.