I'm learning haskell and have been writing various fairly trivial functions to get a handle on the language. I wrote this function:
doubleOneOrTail :: [t]->[t]
doubleOneOrTail [x] = [x,x]
doubleOneOrTail (x:xs) = xs
Which does exactly what is says. Doubles the list of 1 element, or returns the tail of a list of multiple elements. This generic syntax will work for a list of single elements or list of lists, etc. I can rewrite this function to the following:
doubleOneOrTail :: [[t]]->[[t]]
doubleOneOrTail [[x]] = [[x,x]]
doubleOneOrTail (x:xs) = xs
This will throw an error if I type the following:
doubleOneOrTail [1,2,3]
but it does accept this:
doubleOneOrTail [[[[1,2,3],[2]]]]
Treats it as a list with a single element (that element being a list of lists) and doubles it.
Clearly the pattern [a]->[a] isn't matching a list of single elements but in some way matching any order of lists. Whlie [[a]]->[[a]] is also matching multiple orders of lists (though not lists of just single elements). If anyone could explain how this is working it would be greatly appreciated.
A secondary question is it possible (or even desirable) to have a function declaration that specifically takes a particular order of a list. Say only lists of lists?
It sounds like you have a pretty good understanding of what's surprising you here: [a] is a list of any type of thing at all, including a list of lists. You can work out by hand what the type inferencer does automatically. Suppose that, using your first definition, we write:
let xs = [[[1,2], [3,4]], [[5]]]
in doubleOneOrTail xs
At this point GHC has to make sure the types match up for us.
xs :: [[[Integer]]] -- really could be any Num type, but assume Integer
Now since we're calling doubleOneOrTail with an [[[Integer]]] as argument, GHC must unify the types [a] and [[[Integer]]], which means finding some concrete type to substitute for a that makes the two types match up. There is only one correct answer:
[a] ~ [[[Integer]]]
a ~ [[Integer]]
Therefore, we are doubling or tailing a list of things, where each thing is a list of list of numbers. And since the types do indeed unify, GHC gives the all-clear, the function call compiles, and as a result we get [[[5]]].
As to your second question, whether you could or should restrict to a particular depth of lists: usually you should not. This kind of function is called parametrically polymorphic, meaning it works for any kind of a at all, no matter what it is; this is a useful property that it's good to preserve when possible. If your function doesn't need to look at values of its a type in order to perform correctly, it should allow them to be of any type.
Suppose that you still wanted to restrict its type? I don't know of a way to restrict it to depth-one lists without also adding some other incidental restriction. For example, you could say that it must be a list of numbers (and hope that nobody defines a Num instance for lists!):
doubleOneOrTail :: Num a => [a] -> [a]
Or you could limit it to a very specific type, such as [Int]. This would guarantee it can only ever be called with that type.
doubleOneOrTail :: [Int] -> [Int]
But as discussed above, all of these approaches unnecessarily restrict the type of your function. Better to define it as generally as possible, and find some other way to satisfy whatever other concerns make you want to restrict its type.
Little addition to #amalloy answer. Most probably you'll be fine with Enum class constraint:
doupbleOneOrTrail :: (Enum t) => [t] -> [t]
So your function accepts list of chars, numbers, booleans, etc
λ: doubleOneOrTail "foo"
"oo"
λ: doubleOneOrTail [False]
[False,False]
Related
Is there a way to check how many elements (,) has? I know I can access first and second element of a tuple with fst and snd but and thought I could somehow sum elements and then compare it with fst tuple snd tuple and check like this:
tuple = (1,2)
sum tuple == fst tuple + snd tuple
and then I get True for this case and get False for triple = (1,2,3). Eitherway I cant ask fst (1,2,3) nor I can do sum tuple
Is there a way to check if I have a tuple or not?
Something like this:
is_tuple :: (a,b) -> a
is_tuple (a,_) = a
but to get True when I input tuple and False when I give (1,2,3) or (1,2,3,4) and so on... as a input.
i.e:
is_tuple :: Tuple -> Bool
is_tuple x = if x is Tuple
then True
else False
?
Is there a way to check how many elements (,) has?
No, because the answer is always 2.
The type (,) is the type constructor for a tuple of two values, or a 2-tuple. It is a distinct type from (,,), the constructor for 3-tuples. Likewise, both those types are distinct from (,,,), the constructor for 4-tuples, and so on.
When you write a function with type (Foo, Bar) -> Baz, the typechecker will reject any attempts to call the function with a tuple of a different number of values (or something that isn’t a tuple at all). Therefore, your isTuple function only has one logical implementation,
isTuple :: (a, b) -> Bool
isTuple _ = True
…since it is impossible to ever actually call isTuple with a value that is not a 2-tuple.
Without using typeclasses, it is impossible in Haskell to write a function that accepts a tuple of arbitrary size; that is, you cannot be polymorphic over the size of a tuple. This is because, unlike lists, tuples are heterogenous—they can contain values of different types. A function that accepts a tuple of varying length would have no way to predict which elements of the tuple are of which type, and therefore it wouldn’t be able to actually do anything useful.
Very rarely, when doing advanced, type-level trickery, it can be useful to have a type that represents a tuple of varying length, which in Haskell is frequently known as an HList (for heterogenous list). These can be implemented as a library using fancy typeclass machinery and type-level programming. However, if you are a beginner, this is definitely not what you want.
It is difficult to actually give advice on what you should do because, as a commenter points out, your question reads like an XY problem. Consider asking a different question that gives a little more context about the problem you were actually trying to solve that made you want to find the list of a tuple in the first place, and you’ll most likely get more helpful answers.
I have a list of strings in Haskell and I need to get those elements with odd length in another list. How can this be done using higher order functions like foldr, foldl, foldr1, foldl1, filter, map, and so on? I will very much appreciate your help. Can list comprehension be used in this case?
It seems that you are aware that filter exists (since you've mentioned), but perhaps are uncertain how it works. If you're trying to extract a specific subset of a list, this seems to be the right path. If you look at its type-signature, you'll find it's pretty straight-forward:
(a -> Bool) -> [a] -> [a]
That is, it takes a function that returns True or False (i.e. true to contain in the new set, false otherwise) and produces a new list. Similarly, Haskell provides a function called odd in Prelude. It's signature looks as follows:
Integral a => a -> Bool
That is, it can take any Integral type and returns True if it is odd, false otherwise.
Now, let's consider a solution:
filter odd [1..10]
This will extract all the odd numbers between [1,10].
I noticed you mentioned list comprehensions. You probably do not want to use this if you are already given a list and you are simply filtering it. However, a list comprehension would be a perfectly acceptable solution:
[x | x <- [1..10], odd x]
In general, list comprehensions are used to express the generation of lists with more complicated constraints.
Now, to actually answer your question. Since we know we can filter numbers, and if we're using Hoogle searching for the following type (notice that String is simply [Char]):
[a] -> Int
You will see a length function. With some function composition, we can quickly see how to create a function which filters odd length. In summary, we have odd which is type Int -> Bool (in this case) and we have length which is [a] -> Int or-- specifically-- String -> Int. Our solution now looks like this:
filter (odd . length) ["abc","def","eh","123","hm","even"]
Here ya go.
getOddOnes = filter . flip (foldr (const (. not)) id) $ False
Note: if you turn this in for your homework, you'd best be prepared to explain it!
I'm really new to Haskell and have been going through the 99 problems translation. This is my solution to number 9:
pack :: (Eq a) => [a] -> [[a]]
pack (xs)
| null xs = []
| otherwise =
let (matched, unmatched) = span (== head xs) xs
in [matched] ++ pack unmatched
I don't get how I'm allowed to do | null xs = [] when the type signature says the function returns a [[]]. I've seen other solutions to the same problem do the same thing.
I mean, I'm not complaining, but is this something specifically allowed? Are there any caveats I have to look out for?
I'm using the GHCi on a default Windows 7 Haskell Platform 2013.2.0.0 installation, if that helps.
[] is an empty list. It has the following type:
[] :: [b] -- Note: I'm using b instead of a because your function already uses a
That b can be everything. Can you choose a b such that [b] ~ [[a]]? (~ is equality for types)? Yes, just use b ~ [a] and the type of [] becomes:
[] :: [b] :: [[a]] -- When b ~ [a]
So [] is also a value of type [[a]]. [] is a valid value for any type of list, be it a list of a's or a list of lists of a's.
[Integer] is the type meaning "a list of integer values". Can there be a value in this type that doesn't actually contain any integers? Yes, the empty list [] contains zero integers.
[[Integer]] is the type meaning "a list of lists of integer values". Can there be a value in this type that doesn't actually contain any lists? Yes, the empty list [] contains zero lists.
Note that [] with type [[Integer]] is quite different from [[]] with the same type. The first represents the empty list of lists. The second is a non-empty list; it contains exactly one element, which is itself the empty list. A box containing one empty box is not the same thing as a box containing nothing at all! We could of course have [[], [], [], []] as well, where the outer non-empty list contains several elements, each of which is a empty list.
If it helps, think of the type [[Integer]] as representing list of rows, where each row is a list of integers. For example, the following:
11, 12, 13;
21, 22, 23, 24;
31;
is one way of visualising the [[Integer]] value [[11, 12, 13], [21, 22, 23, 24], [31]], where I've used commas to separate elements of the inner lists, and also semicolons to terminate each row (also line breaks to make it easy to read).
In that scheme, [[]] is the list consisting of one empty row. so you'd write it as just a single line ending in a semicolon. Whereas [] is a list with no rows at all, not even empty ones. So you'd write it as a blank file with no semicolons.
If that helped, then it should be easy to see how that applies to more abstract types like [[a]]. In general tough, [] with some list type (regardless of what type is written between the brackets) is always the list consisting of zero of the element type; it doesn't matter whether the element type itself is a list (or anything else with a concept of "empty").
Because [] is of type [[a]] in this case: it is a list containing exactly 0 alpha lists.
In general, the empty list can match any list type because every element (all zero of them) is of the correct type.
You already have a lot of great examples, but this might be a useful way to think about it.
Consider a type isomorphic to Haskell's lists, but without the syntactic sugar:
data List a = Nil
| Cons a (List a)
Each type in Haskell is classified by its kind. This List type is of kind * -> *, which you can think of as a sort of type-level function that takes a basic type (of kind *) and returns another basic type.
You'll notice that the Cons constructor is also parameterized this way; it will be different for every different type a passed to the List type. But the Nil constructor is not parameterized that way; this means that the empty list constructor is the same for every type a that you may pass to List a. Nil remains polymorphic even when List a is constrained to a single type because its value does not depend on what a we choose!
So, the type of Nil is List a. This corresponds to [] in standard list notation. But how does [[]] translate to this desugared list type? The value is Cons Nil Nil and the type is List (List a)! In this syntax it is clearly not an empty list; it is a single-element list containing the empty list. It's still fully polymorphic, since nothing has yet constrained a to a single type, but it's definitely not empty.
The confusing thing about your example is that the name of the overall list type is the same as one of its constructors. If you see [[a]] in code you have to look at whether it's in a type context or a value context. In the former, it would mean (in our desugared notation) the type List (List a) while in the latter it would mean the value Cons (Cons a Nil) Nil.
This is not an answer, but may I suggest using pattern matches instead of head and null?
pack :: (Eq a) => [a] -> [[a]]
pack xs = case xs of
[] -> []
x:_ ->
let (matched, unmatched) = span (== x) xs
in [matched] ++ pack unmatched
This has the benefit the compiler will statically check that you don't access the first element of the list when the list is empty. Generally, use of head is considered non-idiomatic.
The following would be perhaps instructive: instead of
pack :: (Eq a) => [a] -> [[a]]
pack (xs)
| null xs = []
...
try to write it thus:
pack :: (Eq a) => [a] -> [[a]]
pack (xs)
| null xs = xs -- since xs is the empty list (WRONG!)
...
Based on the following (wrong) reasoning: We know that we must return the empty list when the argument is empty, hence, we can save typing [] (which would require pressing AltGr, for example on german keyboards) and return the argument right away -- it is the empty list, after all, as the null check confirmed.
The type checker will disagree, however, on this point.
As a matter of fact, for the typechecker there exist infinitely many different empty lists, each possible list element type has its own empty list. In most cases, however, the type checker will identify the correct one when you give him [] - after all, in our example it knows that it must be a list of lists of as from the type signature.
I'm doing some exercises from "Real World Haskell". One is to design a safe version of init :: [a] -> [a].
I'm supposed to start from safeInit :: [a] -> Maybe [a]
This is what I have at the moment.
safeInit :: [a] -> Maybe [a]
safeInit [] = Nothing
safeInit [a] = if length [a] <= 1
then Nothing
else Just (take (length [a] -1) [a])
In GCHi, when testing safeInit [1,2] I get the error message
* Exception: ch4exercise.hs:(21,1)-(24,44): Non-exhaustive patterns in function safeInit
I was under the impression that [a] simply stands for a list (of any size) of a's. What am I doing wrong?
As a type, [a] does stand for "a list of any size of as". As a pattern however, [a] stands for "a list containing exactly one element, which shall henceforth be known by the name a". Similarly [a,b] would mean "a list containing two elements, the first of which shall be known as a and the second of which shall be known as b" and so. [], as you already seem to know, stands for "a list containing exactly 0 elements".
This is analogous to how you'd write list literals as expressions. I.e. if you write myList = [], myList is the empty list and if you write myList = [x], myList is a list containing exactly one element, which is the value of the variable x.
[] is the empty list, a list containing nothing.
[a] is a list containing exactly one element, and that element (not the list) will be identified as "a" in the function.
You therefore still need to consider the case where the list contains several elements.
If you just use "a" rather than "[a]", then "a" will refer to the whole list, and you can start taking it apart with the functions you have to hand.
Note that you've already dealt with the situation in which you have an empty list, so you shouldn't need to check it again in the if statement.
One thing you'll have to get used to in Haskell, which isn't very intuitive until you're reasonably experienced, is that the "namespace" for type-level things is completely separate from the namespace for value-level things. This means the same source text can have a completely different meaning when you're talking about types than when you're talking about values.
safeInit :: [a] -> Maybe [a]
Everything following the :: is talking about types. Here [a] is the list type constructor applied to the type variable a.1 So it's the type of lists (of any size) whose elements are of type a.
safeInit [a] = if length [a] <= 1
...
OTOH this equation is at the value level. Here [a] isn't a type, it's a value (on the left hand side of the = it's a pattern to be matched by the value to which safeInit was applied; on the right hand side it's just a value). At the value level the square bracket syntax isn't the list type constructor, it's syntactic sugar for writing lists, with all the elements of the lists separated by commas inside the brackets. So [a] is the value of a list containing one single element, which is denoted by the variable a. If we wanted the list to be empty we'd write []. If we wanted the list to contain just a and b we'd write [a, b], etc.
At the value level it wouldn't make much sense for [a] to be a list of any number of as, because (during any particular evaluation of this code) a is one particular value, such as 3.4. What use is an expression for a list of any number of 3.4s?
1 Just as Maybe a is the Maybe type constructor applied to the type variable a; the only thing that's special about the list type constructor is that it's called [] rather than a normal name, and gets this odd "surrounding" syntax for application rather than the usual prefix form.
There's a problem with this line:
safeInit [a] = if length [a] <= 1
On the left side of the equation, [a] will match a list with exactly one element. So the compiler sees that you have a version of safeInit for an empty list, a version of safeInit for a list with one element, but nothing for a list with more elements. That's why it's complaining about Non-exhaustive patterns.
I think that what you really want is
safeInit a = if length a <= 1
then Nothing
else Just (take (length a -1) a)
To summarise:
In a type signature, [a] stands for a list of elements of arbitrary type a.
In a pattern, [a] matches a list with exactly one element.
I'm new to Haskell. I know I can create a reverse function by doing this:
reverse :: [a] -> [a]
reverse [] = []
reverse (x:xs) = (Main.reverse xs) ++ [x]
Is there such a thing as (xs:x) (a list concatenated with an element, i.e. x is the last element in the list) so that I put the last list element at the front of the list?
rotate :: [a] -> [a]
rotate [] = []
rotate (xs:x) = [x] ++ xs
I get these errors when I try to compile a program containing this function:
Occurs check: cannot construct the infinite type: a = [a]
When generalising the type(s) for `rotate'
I'm also new to Haskell, so my answer is not authoritative. Anyway, I would do it using last and init:
Prelude> last [1..10] : init [1..10]
[10,1,2,3,4,5,6,7,8,9]
or
Prelude> [ last [1..10] ] ++ init [1..10]
[10,1,2,3,4,5,6,7,8,9]
The short answer is: this is not possible with pattern matching, you have to use a function.
The long answer is: it's not in standard Haskell, but it is if you are willing to use an extension called View Patterns, and also if you have no problem with your pattern matching eventually taking longer than constant time.
The reason is that pattern matching is based on how the structure is constructed in the first place. A list is an abstract type, which have the following structure:
data List a = Empty | Cons a (List a)
deriving (Show) -- this is just so you can print the List
When you declare a type like that you generate three objects: a type constructor List, and two data constructors: Empty and Cons. The type constructor takes types and turns them into other types, i.e., List takes a type a and creates another type List a. The data constructor works like a function that returns something of type List a. In this case you have:
Empty :: List a
representing an empty list and
Cons :: a -> List a -> List a
which takes a value of type a and a list and appends the value to the head of the list, returning another list. So you can build your lists like this:
empty = Empty -- similar to []
list1 = Cons 1 Empty -- similar to 1:[] = [1]
list2 = Cons 2 list1 -- similar to 2:(1:[]) = 2:[1] = [2,1]
This is more or less how lists work, but in the place of Empty you have [] and in the place of Cons you have (:). When you type something like [1,2,3] this is just syntactic sugar for 1:2:3:[] or Cons 1 (Cons 2 (Cons 3 Empty)).
When you do pattern matching, you are "de-constructing" the type. Having knowledge of how the type is structured allows you to uniquely disassemble it. Consider the function:
head :: List a -> a
head (Empty) = error " the empty list have no head"
head (Cons x xs) = x
What happens on the type matching is that the data constructor is matched to some structure you give. If it matches Empty, than you have an empty list. If if matches Const x xs then x must have type a and must be the head of the list and xs must have type List a and be the tail of the list, cause that's the type of the data constructor:
Cons :: a -> List a -> List a
If Cons x xs is of type List a than x must be a and xs must be List a. The same is true for (x:xs). If you look to the type of (:) in GHCi:
> :t (:)
(:) :: a -> [a] -> [a]
So, if (x:xs) is of type [a], x must be a and xs must be [a] . The error message you get when you try to do (xs:x) and then treat xs like a list, is exactly because of this. By your use of (:) the compiler infers that xs have type a, and by your use of
++, it infers that xs must be [a]. Then it freaks out cause there's no finite type a for which a = [a] - this is what he's trying to tell you with that error message.
If you need to disassemble the structure in other ways that don't match the way the data constructor builds the structure, than you have to write your own function. There are two functions in the standard library that do what you want: last returns the last element of a list, and init returns all-but-the-last elements of the list.
But note that pattern matching happens in constant time. To find out the head and the tail of a list, it doesn't matter how long the list is, you just have to look to the outermost data constructor. Finding the last element is O(N): you have to dig until you find the innermost Cons or the innermost (:), and this requires you to "peel" the structure N times, where N is the size of the list.
If you frequently have to look for the last element in long lists, you might consider if using a list is a good idea after all. You can go after Data.Sequence (constant time access to first and last elements), Data.Map (log(N) time access to any element if you know its key), Data.Array (constant time access to an element if you know its index), Data.Vector or other data structures that match your needs better than lists.
Ok. That was the short answer (:P). The long one you'll have to lookup a bit by yourself, but here's an intro.
You can have this working with a syntax very close to pattern matching by using view patterns. View Patterns are an extension that you can use by having this as the first line of your code:
{-# Language ViewPatterns #-}
The instructions of how to use it are here: http://hackage.haskell.org/trac/ghc/wiki/ViewPatterns
With view patterns you could do something like:
view :: [a] -> (a, [a])
view xs = (last xs, init xs)
someFunction :: [a] -> ...
someFunction (view -> (x,xs)) = ...
than x and xs will be bound to the last and the init of the list you provide to someFunction. Syntactically it feels like pattern matching, but it is really just applying last and init to the given list.
If you're willing to use something different from plain lists, you could have a look at the Seq type in the containers package, as documented here. This has O(1) cons (element at the front) and snoc (element at the back), and allows pattern matching the element from the front and the back, through use of Views.
"Is there such a thing as (xs:x) (a list concatenated with an element, i.e. x is the last element in the list) so that I put the last list element at the front of the list?"
No, not in the sense that you mean. These "patterns" on the left-hand side of a function definition are a reflection of how a data structure is defined by the programmer and stored in memory. Haskell's built-in list implementation is a singly-linked list, ordered from the beginning - so the pattern available for function definitions reflects exactly that, exposing the very first element plus the rest of the list (or alternatively, the empty list).
For a list constructed in this way, the last element is not immediately available as one of the stored components of the list's top-most node. So instead of that value being present in pattern on the left-hand side of the function definition, it's calculated by the function body onthe right-hand side.
Of course, you can define new data structures, so if you want a new list that makes the last element available through pattern-matching, you could build that. But there's be some cost: Maybe you'd just be storing the list backwards, so that it's now the first element which is not available by pattern matching, and requires computation. Maybe you're storing both the first and last value in the structures, which would require additional storage space and bookkeeping.
It's perfectly reasonable to think about multiple implementations of a single data structure concept - to look forward a little bit, this is one use of Haskell's class/instance definitions.
Reversing as you suggested might be much less efficient. Last is not O(1) operation, but is O(N) and that mean that rotating as you suggested becomes O(N^2) alghorhim.
Source:
http://www.haskell.org/ghc/docs/6.12.2/html/libraries/base-4.2.0.1/src/GHC-List.html#last
Your first version has O(n) complexity. Well it is not, becuase ++ is also O(N) operation
you should do this like
rotate l = rev l []
where
rev [] a = a
rev (x:xs) a = rev xs (x:a)
source : http://www.haskell.org/ghc/docs/6.12.2/html/libraries/base-4.2.0.1/src/GHC-List.html#reverse
In your latter example, x is in fact a list. [x] becomes a list of lists, e.g. [[1,2], [3,4]].
(++) wants a list of the same type on both sides. When you are using it, you're doing [[a]] ++ [a] which is why the compiler is complaining. According to your code a would be the same type as [a], which is impossible.
In (x:xs), x is the first item of the list (the head) and xs is everything but the head, i.e., the tail. The names are irrelevant here, you might as well call them (head:tail).
If you really want to take the last item of the input list and put that in the front of the result list, you could do something like:
rotate :: [a] -> [a]
rotate [] = []
rotate lst = (last lst):(rotate $ init lst)
N.B. I haven't tested this code at all as I don't have a Haskell environment available at the moment.