So I'm trying to write a function that, given two lists of integers, adds the ith even number of each list and returns them in another list. In case one of the list doesn't have an ith even number, a 0 is considered. For example, if the lists are [1,2,1,4,6] and [2,2], it returns [4,6,6] ([2+2,4+2,6+0]). I have the following code:
addEven :: [Int] -> [Int] -> [Int]
addEeven [] [] = []
addEeven (x:xs) [] = filter (\g -> g `mod`2 == 0) (x:xs)
addEven [] (y:ys) = filter (\g -> g `mod` 2 == 0) (y:ys)
addEven (x:xs) (y:ys) = (a + b):(addEven as bs)
where
(a:as) = filter (\g -> g `mod` 2 == 0) (x:xs)
(b:bs) = filter (\g -> g `mod` 2 == 0) (y:ys)
When I run that with the previous example, I get:
[4,6*** Exception: ex.hs:(4,1)-(8,101): Non-exhaustive patterns in function addEven
I really can't see what I'm missing, since it doesn't work with any input I throw at it.
A filter might eliminate elements, hence filter (\g -> gmod2 == 0) is not said to return any elements, and thus the patterns (a:as) and (b:bs) might fail.
That being said, I think you make the problem too complex here. You can first define a helper function that adds two elements of a list:
addList :: Num a => [a] -> [a] -> [a]
addList (x:xs) (y:ys) = (x+y) : addList xs ys
addList xs [] = xs
addList [] ys = ys
Then we do the filter on the two parameters, and make a function addEven that looks like:
addEven :: Integral a => [a] -> [a] -> [a]
addEven xs ys = addList (filter even xs) (filter even ys)
or with on :: (b -> b -> c) -> (a -> b) -> a -> a -> c:
import Data.Function(on)
addEven :: Integral a => [a] -> [a] -> [a]
addEven = addList `on` filter even
While using filter is very instinctive in this case, perhaps using filter twice and then summing up the results might be slightly ineffficient for large lists. Why don't we do the job all at once for a change..?
addMatches :: [Int] -> [Int] -> [Int]
addMatches [] [] = []
addMatches [] ys = filter even ys
addMatches xs [] = filter even xs
addMatches xs ys = first [] xs ys
where
first :: [Int] -> [Int] -> [Int] -> [Int]
first rs [] ys = rs ++ filter even ys
first rs (x:xs) ys = rs ++ if even x then second [x] xs ys
else first [] xs ys
second :: [Int] -> [Int] -> [Int] -> [Int]
second [r] xs [] = [r] ++ filter even xs
second [r] xs (y:ys) = if even y then first [r+y] xs ys
else second [r] xs ys
λ> addMatches [1,2,1,4,6] [2,2]
[4,6,6]
Related
I am practicing some Haskell exam paper questions, and have come across the following
Define a Haskell function weaveHunks which takes an int and
two lists and weaves them together in hunks of the given size.
Be sure to declare its type signature.
Example:
weaveHunks 3 "abcdefghijklmno" "ABCDEFGHIJKLMNO"
=> "abcABCdefDEFghiGHIjklJKLmnoMNO"
I have found the following on Stack Overflow, which is just too weave two lists together but only in chunks of 1
weaveHunks :: [a] -> [a] -> [a]
weaveHunks xs [] = xs
weaveHunks [] ys = ys
weaveHunks (x:xs) (y:ys) = x : y : weaveHunks xs ys
I am having problems adjusting this to take chunks fo n size, I am very new to Haskell but this is what I have so far
weaveHunks :: Int -> [a] -> [a] -> [a]
weaveHunks n xs [] = xs
weaveHunks n [] ys = ys
weaveHunks n xs ys = (take n xs) : (take n ys) : weaveHunks n (drop n xs) (drop n ys)
I am getting an error on the last line
(Couldn't match type a' with[a]')
Is (drop n xs) not a list?
You're very close!
By using the : operator to prepend the hunks, you're expressing that take n xs is one element of the result list, take n ys the next, and so on. But actually in both cases it's multiple elements you're prepending. That's the [a] that should actually be just a.
The solution is to use the ++ operator instead, which prepends an entire list rather than just a single element.
This is the full solution as I'd write it:
weaveHunks :: Int -> [a] -> [a] -> [a]
weaveHunks _ xs [] = xs
weaveHunks _ [] ys = ys
weaveHunks n xs ys = xHunk ++ yHunk ++ weaveHunks n xRemain yRemain
where [(xHunk, xRemain), (yHunk, yRemain)] = splitAt n <$> [xs,ys]
As #leftaroundabout said, since your appending lists of type [a], you need to use ++ instead of :. With this in mind, your code would then look like this:
weaveHunks :: Int -> [a] -> [a] -> [a]
weaveHunks _ xs [] = xs
weaveHunks _ [] ys = ys
weaveHunks n xs ys = (take n xs) ++ (take n ys) ++ weaveHunks n (drop n xs) (drop n ys)
If your interested, you can also use library functions to do this task:
import Data.List.Split
weaveHunks :: Int -> [a] -> [a] -> [a]
weaveHunks n xs ys = concat $ zipWith (++) (chunksOf n xs) (chunksOf n ys)
Note: chunksOf is from Data.List.Split, which splits the list into sublists of length n, so the type of this function is Int -> [a] -> [[a]]. zipWith zips two lists based on a condition, in this case concatenation ++. concat turns a list of [[a]] into [a].
How would I define an append function that takes two lists and output 1 list.
So plusplus [1,2,3] [4,5,6] should return [1,2,3,4,5,6].
I have the following so far:
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] [] = []
plusplus [] [x] = [x]
plusplus [x] [] = [x]
plusplus (x:xs) (y:ys) = x: plusplus xs (y:ys)
I want to define my own ++ function, but with recursion. The code above gives an error when I actually execute it with two lists.
This is an example of an error I get with the command:
plusplus [1,2,3] [4,5,6]
[1,2,3*** Exception: baby.hs:(...): Non-exhaustive patterns in function plusplus
ok basically the only problem you have here is a missing case:
pluspluse (x:xs) [] = ...
is missing (and will cause an error)
but you can clean up quite a bit:
The first two cases are basically the same: whatever the second input is will be the result - so you can just it those a name (say ys) and write:
plusplus [] ys = ys
(can you see how this includes your first two cases? - what is ys in those?)
your 3rd line one will be covered by the correct last case (see below) - to see this remember that [x] == x:[]
I'll give you the complete answer in a few minutes - but you should try to figure it out:
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] ys = ys
plusplus (x:xs) ? = ??
try to find the correct things to replace ? and ?? with
you almost had it - and the answer can be:
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] ys = ys
plusplus (x:xs) ys = x: plusplus xs ys
as you can see you never have to touch the second list - but need to recreate all of the first list (to link to the second)
We know that plusplus = flip (foldr (:)). Here's the definition of foldr:
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr step = let fold acc [] = acc
fold acc (x:xs) = step x (fold acc xs)
in fold
On specializing, we get:
foldr :: (a -> b -> b) -> b -> [a] -> b
| | | | | |
| | | | | |
(:) :: (a -> [a] -> [a]) | | |
| | |
foldr (:) :: [a] -> [a] -> [a]
Here's the definition of the specialized function:
fold :: [a] -> [a] -> [a]
fold acc [] = acc
fold acc (x:xs) = x : fold acc xs
Ofcourse, we need to flip the arguments:
plusplus :: [a] -> [a] -> [a]
plusplus [] ys = ys
plusplus (x:xs) ys = x : plusplus xs ys
That's the reason why plusplus is defined the way it is.
Thanks guys! With the help of Carsten spotting my silly mistake.
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] [] = []
plusplus (x:xs) [] = x:xs
plusplus [] (x:xs) = x:xs
plusplus (x:xs) (y:ys) = x: plusplus xs (y:ys)
This works.
So for input:
plusplus [1,2,3] [4,5,6]
Output is:
[1,2,3,4,5,6]
This is really a riff on Carsten's answer, but I think it's worth going into and won't fit into a comment. One of the keys to avoiding non-exhaustive pattern errors in function definitions is to approach writing functions in a more mechanical, formulaic manner. The definition of the list type is something like this:
data [a] = [] | a : [a]
So list types have two data constructors, [] and :. So a good first thing to try when starting your function is to write two equations, one for each of the constructors. Carsten's template (which I'm mostly copying now) is following that pattern:
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] ys = _ -- Equation for empty list
plusplus (x:xs) ys = _ -- Equation for list cells
Now since we've written one pattern for each of the constructors of the type, we're now guaranteed that our patterns are exhaustive. So we stare at what we got a bit, and now we see the solution:
plusplus :: [Int] -> [Int] -> [Int]
plusplus [] ys = ys
plusplus (x:xs) ys = x : plusplus xs ys
Let's try another one! Signature:
-- | Merge two lists by alternating elements of one with the
-- other. After either list runs out, just continue with the
-- remaining elements of the other.
interleave :: [a] -> [a] -> [a]
interleave xs ys = _
Let's expand this to two equations by splitting the xs variable according to the two constructors, and fill in the easy parts on the right hand side:
interleave :: [a] -> [a] -> [a]
interleave [] ys = ys -- this one was easy
interleave (x:xs) ys = x : _ -- we know it has to start with `x`
Now we can go two ways. The first one is we could take the second equation and split it into two cases for ys:
interleave :: [a] -> [a] -> [a]
interleave [] ys = ys
interleave (x:xs) [] = x : _
interleave (x:xs) (y:ys) = x : _
And filling in those blanks is easy:
interleave :: [a] -> [a] -> [a]
interleave [] ys = ys
interleave (x:xs) [] = x : xs
interleave (x:xs) (y:ys) = x : y : interleave xs ys
The second way is, instead of splitting ys on the constructors' cases, to notice that this works:
interleave :: [a] -> [a] -> [a]
interleave [] ys = ys
interleave (x:xs) ys = x : interleave ys xs
So by going about it systematically based on the constructors we know for sure that all combinations are covered.
here's my question:
How to extract the same elements from two equal length lists to another list?
For example: given two lists [2,4,6,3,2,1,3,5] and [7,3,3,2,8,8,9,1] the answer should be [1,2,3,3]. Note that the order is immaterial. I'm actually using the length of the return list.
I tried this:
sameElem as bs = length (nub (intersect as bs))
but the problem is nub removes all the duplications. The result of using my function to the former example is 3 the length of [1,3,2] instead of 4 the length of [1,3,3,2]. Is there a solution? Thank you.
Since the position seems to be irrelevant, you can simply sort the lists beforehand and then traverse both lists:
import Data.List (sort)
intersectSorted :: Ord a => [a] -> [a] -> [a]
intersectSorted (x:xs) (y:ys)
| x == y = x : intersectSorted xs ys
| x < y = intersectSorted xs (y:ys)
| x > y = intersectSorted (x:xs) ys
intersectSorted _ _ = []
intersect :: Ord a => [a] -> [a] -> [a]
intersect xs ys = intersectSorted (sort xs) (sort ys)
Note that it's also possible to achieve this with a Map:
import Data.Map.Strict (fromListWith, assocs, intersectionWith, Map)
type Counter a = Map a Int
toCounter :: Ord a => [a] -> Counter a
toCounter = fromListWith (+) . flip zip (repeat 1)
intersectCounter :: Ord a => Counter a -> Counter a -> Counter a
intersectCounter = intersectionWith min
toList :: Counter a -> [a]
toList = concatMap (\(k,c) -> replicate c k) . assocs
intersect :: Ord a => [a] -> [a] -> [a]
intersect xs ys = toList $ intersectCounter (toCounter xs) (toCounter ys)
You could write a function for this. There is probably a more elegant version of this involving lambda's or folds, but this does work for your example:
import Data.List
same (x:xs) ys = if x `elem` ys
then x:same xs (delete x ys)
else same xs ys
same [] _ = []
same _ [] = []
The delete x ys in the then-clause is important, without that delete command items from the first list that occur at least once will be counted every time they're encountered.
Note that the output is not sorted, since you were only interested in the length of the resulting list.
import Data.List (delete)
mutuals :: Eq a => [a] -> [a] -> [a]
mutuals [] _ = []
mutuals (x : xs) ys | x `elem` ys = x : mutuals xs (delete x ys)
| otherwise = mutuals xs ys
gives
mutuals [2,4,6,3,2,1,3,5] [7,3,3,2,8,8,9,1] == [2,3,1,3]
So I am working on a function that detects if two binary trees have the same numbers in them.
So what I've come up with is the following which works just fine but the problem is that I am using total of 5 functions. Is there another way of detecting if two BT's have the same elements with just one function ? Here is my solution so far which seems to work just fine.
flatten :: BinTree a -> [a]
flatten Empty = []
flatten (Node l x r) = flatten l ++ [x] ++ flatten r
splt :: Int -> [a] -> ([a], [a])
splt 0 xs = ([], xs)
splt _ [] = ([],[])
splt n (x:xs) = (\ys-> (x:fst ys, snd ys)) (splt (n-1) xs)
merge :: Ord a => [a] -> [a] -> [a]
merge xs [] = xs
merge [] ys = ys
merge (x:xs) (y:ys) = if (x > y) then y:merge (x:xs) ys else x:merge xs(y:ys)
msort :: Ord a => [a] -> [a]
msort [] =[]
msort (x:[]) = (x:[])
msort xs = (\y -> merge (msort (fst y)) (msort (snd y))) (splt (length xs `div` 2) xs)
areTreesEqual :: (Ord a) => BinTree a -> BinTree a-> Bool
areTreesEqual Empty Empty = True
areTreesEqual Empty a = False
areTreesEqual a Empty = False
areTreesEqual a b = msort (flatten (a) ) == msort (flatten (b))
How about
import Data.MultiSet as Set
equal a b = accum a == accum b
where accum Empty = Set.empty
accum (Node l x r) = Set.insert x (accum l `Set.union` accum r)
Sets are lovely for unordered comparison and multisets will make sure that
1 /= 1
1 1
Eg, that duplicate numbers are counted properly. If this isn't a big concern, than swap MultiSet for Set. I think 3 lines is pretty decent for this sort of thing.
I'm doing a bit of self study on functional languages (currently using Haskell). I came across a Haskell based assignment which requires defining map and filter in terms of foldr. For the life of me I'm not fully understanding how to go about this.
For example when I define a map function like:
map' :: (a -> b) -> [a] -> [b]
map' f [] = []
map' f (x:xs) = foldr (\x xs -> (f x):xs) [] xs
I don't know why the first element of the list is always ignored. Meaning that:
map' (*2) [1,2,3,4]
results in [4,6,8] instead of [2,4,6,8]
Similarly, my filter' function:
filter' :: (a -> Bool) -> [a] -> [a]
filter' p [] = []
filter' p (x:xs) = foldr (\x xs -> if p x then x:xs else xs ) [] xs
when run as:
filter' even [2,3,4,5,6]
results in [4,6] instead of [2,4,6]
Why would this be the case? And how SHOULD I have defined these functions to get the expected results? I'm assuming something is wrong with my lambda expressions...
I wish I could just comment, but alas, I don't have enough karma.
The other answers are all good ones, but I think the biggest confusion seems to be stemming from your use of x and xs.
If you rewrote it as
map' :: (a -> b) -> [a] -> [b]
map' f [] = []
map' f (x:xs) = foldr (\y ys -> (f y):ys) [] xs
you would clearly see that x is not even mentioned on the right-hand side, so there's no way that it could be in the solution.
Cheers
For your first question, foldr already has a case for the empty list, so you need not and should not provide a case for it in your own map.
map' f = foldr (\x xs -> f x : xs) []
The same holds for filter'
filter' p = foldr (\x xs -> if p x then x : xs else xs) []
Nothing is wrong with your lambda expressions, but there is something wrong with your definitions of filter' and map'. In the cons case (x:xs) you eat the head (x) away and then pass the tail to foldr. The foldr function can never see the first element you already ate. :)
Alse note that:
filter' p = foldr (\x xs -> if p x then x : xs else xs) []
is equivalent (η-equivalent) to:
filter' p xs = foldr (\x xs -> if p x then x : xs else xs) [] xs
I would define map using foldr and function composition as follows:
map :: (a -> b) -> [a] -> [b]
map f = foldr ((:).f) []
And for the case of filter:
filter :: (a -> Bool) -> [a] -> [a]
filter p = foldr (\x xs -> if p x then x:xs else xs) []
Note that it is not necessary to pass the list itself when defining functions over lists using foldr or foldl.
The problem with your solution is that you drop the head of the list and then apply the map over the list and
this is why the head of the list is missing when the result is shown.
In your definitions, you are doing pattern matching for x:xs, which means, when your argument is [1,2,3,4], x is bound to 1 and xs is bound to the rest of the list: [2,3,4].
What you should not do is simply throw away x: part. Then your foldr will be working on whole list.
So your definitions should look as follows:
map' :: (a -> b) -> [a] -> [b]
map' f [] = []
map' f xs = foldr (\x xs -> (f x):xs) [] xs
and
filter' :: (a -> Bool) -> [a] -> [a]
filter' p [] = []
filter' p xs = foldr (\x xs -> if p x then x:xs else xs ) [] xs
I am new to Haskell (in fact I've found this page asking the same question) but this is my understanding of lists and foldr so far:
lists are elements that are linked to the next element with the cons (:) operator. they terminate with the empty list []. (think of it as a binary operator just like addition (+) 1+2+3+4 = 10, 1:2:3:4:[] = [1,2,3,4]
foldr function takes a function that takes two parameters. this will replace the cons operator, which will define how each item is linked to the next.
it also takes the terminal value for the operation, which can be tought as the initial value that will be assigned to the empty list. for cons it is empty list []. if you link an empty list to any list the result is the list itself. so for a sumfunction it is 0. for a multiply function it is 1, etc.
and it takes the list itself
So my solution is as follows:
filter' p = foldr (\x n -> if p x then x : n else n) []
the lambda expression is our link function, which will be used instead of the cons (:) operator. Empty list is our default value for an empty list. If predicate is satisfied we link to the next item using (:) as normal, else we simply don't link at all.
map' f = foldr (\x n -> f x : n) []
here we link f x to the next item instead of just x, which would simply duplicate the list.
Also, note that you don't need to use pattern matching, since we already tell foldr what to do in case of an empty list.
I know this question is really old but I just wanted to answer it anyway. I hope it is not against the rules.
A different way to think about it - foldr exists because the following recursive pattern is used often:
-- Example 1: Sum up numbers
summa :: Num a => [a] -> a
summa [] = 0
summa (x:xs) = x + suma xs
Taking the product of numbers or even reversing a list looks structurally very similar to the previous recursive function:
-- Example 2: Reverse numbers
reverso :: [a] -> [a]
reverso [] = []
reverso (x:xs) = x `op` reverso xs
where
op = (\curr acc -> acc ++ [curr])
The structure in the above examples only differs in the initial value (0 for summa and [] for reverso) along with the operator between the first value and the recursive call (+ for summa and (\q qs -> qs ++ [q]) for reverso). So the function structure for the above examples can be generally seen as
-- Generic function structure
foo :: (a -> [a] -> [a]) -> [a] -> [a] -> [a]
foo op init_val [] = init_val
foo op init_val (x:xs) = x `op` foo op init_val xs
To see that this "generic" foo works, we could now rewrite reverso by using foo and passing it the operator, initial value, and the list itself:
-- Test: reverso using foo
foo (\curr acc -> acc ++ [curr]) [] [1,2,3,4]
Let's give foo a more generic type signature so that it works for other problems as well:
foo :: (a -> b -> b) -> b -> [a] -> b
Now, getting back to your question - we could write filter like so:
-- Example 3: filter
filtero :: (a -> Bool) -> [a] -> [a]
filtero p [] = []
filtero p (x:xs) = x `filterLogic` (filtero p xs)
where
filterLogic = (\curr acc -> if (p curr) then curr:acc else acc)
This again has a very similar structure to summa and reverso. Hence, we should be able to use foo to rewrite it. Let's say we want to filter the even numbers from the list [1,2,3,4]. Then again we pass foo the operator (in this case filterLogic), initial value, and the list itself. filterLogic in this example takes a p function, called a predicate, which we'll have to define for the call:
let p = even in foo (\curr acc -> if (p curr) then curr:acc else acc) [] [1,2,3,4]
foo in Haskell is called foldr. So, we've rewritten filter using foldr.
let p = even in foldr (\curr acc -> if (p curr) then curr:acc else acc) [] [1,2,3,4]
So, filter can be written with foldr as we've seen:
-- Solution 1: filter using foldr
filtero' :: (a -> Bool) -> [a] -> [a]
filtero' p xs = foldr (\curr acc -> if (p curr) then curr:acc else acc) [] xs
As for map, we could also write it as
-- Example 4: map
mapo :: (a -> b) -> [a] -> [b]
mapo f [] = []
mapo f (x:xs) = x `op` (mapo f xs)
where
op = (\curr acc -> (f curr) : acc)
which therefore can be rewritten using foldr. For example, to multiply every number in a list by two:
let f = (* 2) in foldr (\curr acc -> (f curr) : acc) [] [1,2,3,4]
So, map can be written with foldr as we've seen:
-- Solution 2: map using foldr
mapo' :: (a -> b) -> [a] -> [b]
mapo' f xs = foldr (\curr acc -> (f curr) : acc) [] xs
Your solution almost works .)
The problem is that you've got two differend bindings for x in both your functions (Inside the patternmatching and inside your lambda expression), therefore you loose track of the first Element.
map' :: (a -> b) -> [a] -> [b]
map' f [] = []
map' f (x:xs) = foldr (\x xs -> (f x):xs) [] (x:xs)
filter' :: (a -> Bool) -> [a] -> [a]
filter' p [] = []
filter' p (x:xs) = foldr (\x xs -> if p x then x:xs else xs ) [] (x:xs)
This should to the trick :). Also: you can write your functions pointfree style easily.
*Main> :{
*Main| map' :: (a -> b) -> [a] -> [b]
*Main| map' = \f -> \ys -> (foldr (\x -> \acc -> f x:acc) [] ys)
*Main| :}
*Main> map' (^2) [1..10]
[1,4,9,16,25,36,49,64,81,100]
*Main> :{
*Main| filter' :: (a -> Bool) -> [a] -> [a]
*Main| filter' = \p -> \ys -> (foldr (\x -> \acc -> if p x then x:acc else acc) [] ys)
*Main| :}
*Main> filter' (>10) [1..100]
In the above snippets acc refers to accumulator and x refers to the last element.
Everything is correct in your lambda expressions. The problem is you are missing the first element in the list. If you try,
map' f (x:xs) = foldr (\x xs -> f x:xs) [] (x:xs)
then you shouldn't miss the first element anymore. The same logic applies to filter.
filter' p (x:xs) = foldr(\ y xs -> if p y then y:xs else xs) [] (x:xs)