I'm not a Haskell programmer, but I'm curious about the following questions.
Informal function specification:
Let MapProduct be a function that takes a function called F and multiple lists. It returns a list containing the results of calling F with one argument from each list in each possible combination.
Example:
Call MapProduct with F being a function that simply returns a list of its arguments, and two lists. One of the lists contains the integers 1 and 2, the other one contains the strings "a" and "b". It should return a list that contains the lists: 1 and "a", 1 and "b", 2 and "a", 2 and "b".
Questions:
How is MapProduct implemented?
What is the function's type? What is F's type?
Can one guess what the function does just by looking at its type?
Can you handle inhomogeneous lists as input? (e.g. 1 and "a" in one of the input lists)
What extra limitation (if any) do you need to introduce to implement MapProduct?
How is MapProduct implemented?
.
Prelude> :m + Control.Applicative
Prelude Control.Applicative> (,,) <$> [1,2,3] <*> ["a","b","c"] <*> [0.8, 1.2, 4.4]
[(1,"a",0.8),(1,"a",1.2),...,(3,"c",4.4)]
What is the function's type? What is F's type?
The type of F depends on the list you want to apply. <$> here is fmap, and (<*>) :: f(a->b) -> f a -> f b where f = [] here.
Can you handle inhomogeneous lists as input? (e.g. 1 and "a" in one of the input lists)
There is no such thing as an heterogeneous list. But you can simulate a heterogeneous list for a specific context with existential types. And then you can just use the method above to do the MapProduct.
*Main Control.Applicative> :i SB
data ShowBox where
SB :: forall s. (Show s) => s -> ShowBox
-- Defined at v.hs:1:35-36
*Main Control.Applicative> [SB 2, SB "a", SB 6.4]
[2,"a",6.4]
*Main Control.Applicative> (,) <$> [SB 2, SB "a", SB 6.4] <*> [SB 'z', SB 44]
[(2,'z'),(2,44),("a",'z'),("a",44),(6.4,'z'),(6.4,44)]
It is possible to define a function mapProduct that works for any arity of function:
{-# LANGUAGE FlexibleInstances, TypeFamilies #-}
module MapProduct (
mapProduct
) where
import Control.Monad
newtype ProdFuncList a b = ProdFuncList [ a -> b ]
class MapProdResult p where
type MapProdArg p
apNext :: ProdFuncList x (MapProdArg p) -> [x] -> p
instance (MapProdResult b) => MapProdResult ([a] -> b) where
type MapProdArg ([a] -> b) = (a -> MapProdArg b)
apNext (ProdFuncList fs) = apNext . ProdFuncList . ap fs
instance MapProdResult [b] where
type MapProdArg [b] = b
apNext (ProdFuncList fs) = ap fs
mapProduct :: (MapProdResult q) => (a -> MapProdArg q) -> [a] -> q
mapProduct f = apNext (ProdFuncList [f])
Here it is in action:
> :l MapProduct.hs
[1 of 1] Compiling MapProduct ( MapProduct.hs, interpreted )
Ok, modules loaded: MapProduct.
> mapProduct (+10) [1..4] :: [Int]
[11,12,13,14]
> mapProduct (*) [1..4] [10..12] :: [Int]
[10,11,12,20,22,24,30,33,36,40,44,48]
> mapProduct (\a b c -> a:b:c:[]) "bcs" "ao" "dnt" :: [String]
["bad","ban","bat","bod","bon","bot","cad","can","cat","cod","con","cot","sad","san","sat","sod","son","sot"]
The downside of this approach is that you'll most likely have to type annotate the result (as shown in the examples above). It would be much more idiomatic to simply use fmap and ap directly:
> :m + Control.Monad
> (+10) `fmap` [1..4]
[11,12,13,14]
> (*) `fmap` [1..4] `ap` [10..12]
[10,11,12,20,22,24,30,33,36,40,44,48]
> (\a b c -> a:b:c:[]) `fmap` "bcs" `ap` "ao" `ap` "dnt"
["bad","ban","bat","bod","bon","bot","cad","can","cat","cod","con","cot","sad","san","sat","sod","son","sot"]
This requires no type annotations, and is fully general over all monads, not just [].
(The MapProduct module above could easily be generalized over all monads as well. I didn't so as to make it clearly solve the original question.)
The function that you describe is closely related to the zipWithN functions. It will have the same type - it will just result in bigger result lists. Now the problem is that there is no way to express "a function that takes N arguments of the types t_1, ..., t_n" or "n lists of the types [t_1],...,[t_n]" (or "an n-tuple of type ([t_1], ..., [t_n]") ) in haskell's type system (without extensions like template haskell). This is why there is not one zipWith function, but one for each number of arguments lists that is supported.
So to answer your questions:
It is implemented by defining a function mapProductN for every number N that you want to support. For N=2 it would look like this:
mapProduct f l1 l2 = [f x1 x2 | x1 <- l1, x2 <- x2]
Or as a general blueprint (i.e. pseudo-code) how to define the functions for any N:
mapProduct f l1 ... ln = [f x1 ... xn | x1 <- l1, ..., xn <- ln]
As I said it's the same as the types of the zipWith functions, i.e.:
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d]
zipWith4 :: (a -> b -> c -> d -> e) -> [a] -> [b] -> [c] -> [d] -> [e]
Since f is the first argument to the function, the type of the first argument is f's type (so for n=2 it'd be a -> b -> c)
Well since it has the same type as zipWith and zipWith does something else, that'd be a no.
Haskell doesn't allow inhomogeneous lists without an extension.
There's an upper limit on the number of lists unless you're willing to spend the time writing infinite versions of mapProduct.
Related
Shouldn’t this definition be allowed in a lazy language like Haskell in which functions are curried?
apply f [] = f
apply f (x:xs) = apply (f x) xs
It’s basically a function that applies the given function to the given list of arguments and is very easily done in Lisp for example.
Are there any workarounds?
It is hard to give a static type to the apply function, since its type depends on the type of the (possibly heterogeneous) list argument. There are at least two ways one way to write this function in Haskell that I can think of:
Using reflection
We can defer type checking of the application until runtime:
import Data.Dynamic
import Data.Typeable
apply :: Dynamic -> [Dynamic] -> Dynamic
apply f [] = f
apply f (x:xs) = apply (f `dynApp` x) xs
Note that now the Haskell program may fail with a type error at runtime.
Via type class recursion
Using the semi-standard Text.Printf trick (invented by augustss, IIRC), a solution can be coded up in this style (exercise). It may not be very useful though, and still requires some trick to hide the types in the list.
Edit: I couldn't come up with a way to write this, without using dynamic types or hlists/existentials. Would love to see an example
I like Sjoerd Visscher's reply, but the extensions -- especially IncoherentInstances, used in this case to make partial application possible -- might be a bit daunting. Here's a solution that doesn't require any extensions.
First, we define a datatype of functions that know what to do with any number of arguments. You should read a here as being the "argument type", and b as being the "return type".
data ListF a b = Cons b (ListF a (a -> b))
Then we can write some (Haskell) functions that munge these (variadic) functions. I use the F suffix for any functions that happen to be in the Prelude.
headF :: ListF a b -> b
headF (Cons b _) = b
mapF :: (b -> c) -> ListF a b -> ListF a c
mapF f (Cons v fs) = Cons (f v) (mapF (f.) fs)
partialApply :: ListF a b -> [a] -> ListF a b
partialApply fs [] = fs
partialApply (Cons f fs) (x:xs) = partialApply (mapF ($x) fs) xs
apply :: ListF a b -> [a] -> b
apply f xs = headF (partialApply f xs)
For example, the sum function could be thought of as a variadic function:
sumF :: Num a => ListF a a
sumF = Cons 0 (mapF (+) sumF)
sumExample = apply sumF [3, 4, 5]
However, we also want to be able to deal with normal functions, which don't necessarily know what to do with any number of arguments. So, what to do? Well, like Lisp, we can throw an exception at runtime. Below, we'll use f as a simple example of a non-variadic function.
f True True True = 32
f True True False = 67
f _ _ _ = 9
tooMany = error "too many arguments"
tooFew = error "too few arguments"
lift0 v = Cons v tooMany
lift1 f = Cons tooFew (lift0 f)
lift2 f = Cons tooFew (lift1 f)
lift3 f = Cons tooFew (lift2 f)
fF1 = lift3 f
fExample1 = apply fF1 [True, True, True]
fExample2 = apply fF1 [True, False]
fExample3 = apply (partialApply fF1 [True, False]) [False]
Of course, if you don't like the boilerplate of defining lift0, lift1, lift2, lift3, etc. separately, then you need to enable some extensions. But you can get quite far without them!
Here is how you can generalize to a single lift function. First, we define some standard type-level numbers:
{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, FlexibleContexts, TypeFamilies, UndecidableInstances #-}
data Z = Z
newtype S n = S n
Then introduce the typeclass for lifting. You should read the type I n a b as "n copies of a as arguments, then a return type of b".
class Lift n a b where
type I n a b :: *
lift :: n -> I n a b -> ListF a b
instance Lift Z a b where
type I Z a b = b
lift _ b = Cons b tooMany
instance (Lift n a (a -> b), I n a (a -> b) ~ (a -> I n a b)) => Lift (S n) a b where
type I (S n) a b = a -> I n a b
lift (S n) f = Cons tooFew (lift n f)
And here's the examples using f from before, rewritten using the generalized lift:
fF2 = lift (S (S (S Z))) f
fExample4 = apply fF2 [True, True, True]
fExample5 = apply fF2 [True, False]
fExample6 = apply (partialApply fF2 [True, False]) [False]
No, it cannot. f and f x are different types. Due to the statically typed nature of haskell, it can't take any function. It has to take a specific type of function.
Suppose f is passed in with type a -> b -> c. Then f x has type b -> c. But a -> b -> c must have the same type as a -> b. Hence a function of type a -> (b -> c) must be a function of type a -> b. So b must be the same as b -> c, which is an infinite type b -> b -> b -> ... -> c. It cannot exist. (continue to substitute b -> c for b)
Here's one way to do it in GHC. You'll need some type annotations here and there to convince GHC that it's all going to work out.
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE IncoherentInstances #-}
class Apply f a r | f -> a r where
apply :: f -> [a] -> r
instance Apply f a r => Apply (a -> f) a r where
apply f (a:as) = apply (f a) as
instance Apply r a r where
apply r _ = r
test = apply ((+) :: Int -> Int -> Int) [1::Int,2]
apply' :: (a -> a -> a) -> [a] -> a
apply' = apply
test' = apply' (+) [1,2]
This code is a good illustration of the differences between static and dynamic type-checking. With static type-checking, the compiler can't be sure that apply f really is being passed arguments that f expects, so it rejects the program. In lisp, the checking is done at runtime and the program might fail then.
I am not sure how much this would be helpful as I am writing this in F# but I think this can be easily done in Haskell too:
type 'a RecFunction = RecFunction of ('a -> 'a RecFunction)
let rec apply (f: 'a RecFunction) (lst: 'a list) =
match (lst,f) with
| ([],_) -> f
| ((x::xs), RecFunction z) -> apply (z x) xs
In this case the "f" in question is defined using a discriminated union which allows recursive data type definition. This can be used to solved the mentioned problem I guess.
With the help and input of some others I defined a way to achieve this (well, sort of, with a custom list type) which is a bit different from the previous answers. This is an old question, but it seems to still be visited so I will add the approach for completeness.
We use one extension (GADTs), with a list type a bit similar to Daniel Wagner's, but with a tagging function type rather than a Peano number. Let's go through the code in pieces. First we set the extension and define the list type. The datatype is polymorphic so in this formulation arguments don't have to have the same type.
{-# LANGUAGE GADTs #-}
-- n represents function type, o represents output type
data LApp n o where
-- no arguments applied (function and output type are the same)
End :: LApp o o
-- intentional similarity to ($)
(:$) :: a -> LApp m o -> LApp (a -> m) o
infixr 5 :$ -- same as :
Let's define a function that can take a list like this and apply it to a function. There is some type trickery here: the function has type n, a call to listApply will only compile if this type matches the n tag on our list type. By leaving our output type o unspecified, we leave some freedom in this (when creating the list we don't have to immediately entirely fix the kind of function it can be applied to).
-- the apply function
listApply :: n -> LApp n o -> o
listApply fun End = fun
listApply fun (p :$ l) = listApply (fun p) l
That's it! We can now apply functions to arguments stored in our list type. Expected more? :)
-- showing off the power of AppL
main = do print . listApply reverse $ "yrruC .B lleksaH" :$ End
print . listApply (*) $ 1/2 :$ pi :$ End
print . listApply ($) $ head :$ [1..] :$ End
print $ listApply True End
Unfortunately we are kind of locked in to our list type, we can't just convert normal lists to use them with listApply. I suspect this is a fundamental issue with the type checker (types end up depending on the value of a list) but to be honest I'm not entirely sure.
-- Can't do this :(
-- listApply (**) $ foldr (:$) End [2, 32]
If you feel uncomfortable about using a heterogeneous list, all you have to do is add an extra parameter to the LApp type, e.g:
-- Alternative definition
-- data FList n o a where
-- Nil :: FList o o a
-- Cons :: a -> FList f o a -> FList (a -> f) o a
Here a represents the argument type, where the function which is applied to will also have to accept arguments of all the same type.
This isn't precisely an answer to your original question, but I think it might be an answer to your use-case.
pure f <*> [arg] <*> [arg2] ...
-- example
λ>pure (\a b c -> (a*b)+c) <*> [2,4] <*> [3] <*> [1]
[7,13]
λ>pure (+) <*> [1] <*> [2]
[3]
The applicative instance of list is a lot broader than this super narrow use-case though...
λ>pure (+1) <*> [1..10]
[2,3,4,5,6,7,8,9,10,11]
-- Or, apply (+1) to items 1 through 10 and collect the results in a list
λ>pure (+) <*> [1..5] <*> [1..5]
[2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,6,7,8,9,10]
{- The applicative instance of list gives you every possible combination of
elements from the lists provided, so that is every possible sum of pairs
between one and five -}
λ>pure (\a b c -> (a*b)+c) <*> [2,4] <*> [4,3] <*> [1]
[9,7,17,13]
{- that's - 2*4+1, 2*3+1, 4*4+1, 4*3+1
Or, I am repeating argC when I call this function twice, but a and b are
different -}
λ>pure (\a b c -> show (a*b) ++ c) <*> [1,2] <*> [3,4] <*> [" look mah, other types"]
["3 look mah, other types","4 look mah, other types","6 look mah, other types","8 look mah, other types"]
So it's not the same concept, precisely, but it a lot of those compositional use-cases, and adds a few more.
I find this documentation in the basic Haskell libraries:
zip :: [a] -> [b] -> [(a, b)]
zip takes two lists and returns a list of corresponding pairs. If one input list is short, excess elements of the longer list are discarded.
zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]
zip3 takes three lists and returns a list of triples, analogous to zip.
zip4 :: [a] -> [b] -> [c] -> [d] -> [(a, b, c, d)]
The zip4 function takes four lists and returns a list of quadruples, analogous to zip.
[...snip...]
unzip :: [(a, b)] -> ([a], [b])
unzip transforms a list of pairs into a list of first components and a list of second components.
unzip3 :: [(a, b, c)] -> ([a], [b], [c])
The unzip3 function takes a list of triples and returns three lists, analogous to unzip.
unzip4 :: [(a, b, c, d)] -> ([a], [b], [c], [d])
The unzip4 function takes a list of quadruples and returns four lists, analogous to unzip.
... and so on, up to zip7 and unzip7.
Is this a fundamental limitation of Haskell's type system? Or is there a way to implement zip and unzip once, to work on different configurations of input?
This is one very useful aspect of applicatives. Check out ZipList which is just a newtype wrapper around a simple list. The reason for the wrapper is that ZipList has an applicative instance for, you guessed it, zipping lists together. Then, if you want zip7 as bs cs ds es fs gs hs, you can just do something like
(,,,,,,) <$> as <*> bs <*> cs <*> ds <*> es <*> fs <*> gs <*> hs
As you can tell, this mechanism is meant to be also for extending zipWith, which is a general case of zip. To be honest, I think we should rip out all of the zipN functions and teach people the above instead. zip itself is fine, but beyond that...
Template Haskell solution
As the comments and other answers indicate, this is not a particularly satisfying answer. The one thing I was expecting someone else to implement was a TemplateHaskell version of zip and unzip. As no one has done so yet, here it is.
All it does is mechanically produce AST for zip or unzip functions. The idea behind zip is to use ZipList and behind unzip is to use foldr:
zip as ... zs === \as ... zs -> getZipList $ (, ... ,) <$> ZipList as <*> ... <*> ZipList zs
unzip === foldr (\ (a, ... ,z) ~(as, ... ,zs) -> (a:as, ... ,z:zs) ) ([], ... ,[])
The implementation looks like this.
{-# LANGUAGE TemplateHaskell #-}
module Zip (zip, unzip) where
import Prelude hiding (zip, unzip)
import Language.Haskell.TH
import Control.Monad
import Control.Applicative (ZipList(..))
-- | Given number, produces the `zip` function of corresponding arity
zip :: Int -> Q Exp
zip n = do
lists <- replicateM n (newName "xs")
lamE (varP <$> lists)
[| getZipList $
$(foldl (\a b -> [| $a <*> ZipList $(varE b) |])
[| pure $(conE (tupleDataName n)) |]
lists) |]
-- | Given number, produces the `unzip` function of corresponding arity
unzip :: Int -> Q Exp
unzip n = do
heads <- replicateM n (newName "x")
tails <- replicateM n (newName "xs")
[| foldr (\ $(tupP (varP <$> heads)) ~ $(tupP (varP <$> tails)) ->
$(tupE (zipWith (\x xs -> [| $x : $xs |])
(varE <$> heads)
(varE <$> tails))))
$(tupE (replicate n [| [] |])) |]
You can try this at GHCi:
ghci> :set -XTemplateHaskell
ghci> $(zip 3) [1..10] "abcd" [4,6..]
[(1,'a',4),(2,'b',6),(3,'c',8),(4,'d',10)]
ghci> $(unzip 3) [(1,'a',4),(2,'b',6),(3,'c',8),(4,'d',10)]
([1,2,3,4],"abcd",[4,6,8,10])
This is a zipN function that depends on the machinery of the generics-sop package:
{-# language TypeFamilies #-}
{-# language DataKinds #-}
{-# language TypeApplications #-}
import Control.Applicative
import Generics.SOP
-- "a" is some single-constructor product type, like some form of n-ary tuple
-- "xs" is a type-level list of the types of the elements of "a"
zipN :: (Generic a, Code a ~ '[ xs ]) => NP [] xs -> [a]
zipN np = to . SOP . Z <$> getZipList (hsequence (hliftA ZipList np))
main :: IO ()
main = do
let zipped = zipN #(_,_,_) ([1,2,3,4,5,6] :* ['a','b','c'] :* [True,False] :* Nil)
print $ zipped
The result:
[(1,'a',True),(2,'b',False)]
This solution has two disadvantages:
You have to wrap the argument lists in the special NP type from generics-sop that is constructed with :* and Nil.
You need to specify somehow that the result value is a list of tuples, and not a list of some other Generic-compatible type. Here, it is done with the #(_,_,_) type application.
2-ary, 3-ary.. n-ary tuples are all distinct data types, so you can't handle them uniformly directly, but you can introduce a type class that provides an interface that allows to define generic zip and unzip. Here is how it looks for generic unzip:
class Tuple t where
type Map (f :: * -> *) t
nilMap :: Proxy t -> (forall a. f a) -> Map f t
consMap :: (forall a. a -> f a -> f a) -> t -> Map f t -> Map f t
Map maps all types in a tuple type with f. nilMap constructs a Mapped tuple that contains empty values (I have no idea why Haskell requires that Proxy t there). consMap receives a function, a tuple and a Mapped tuple and zip the tuples with the function pointwise. Here is how instances look for 2- and 3-tuples:
instance Tuple (a, b) where
type Map f (a, b) = (f a, f b)
nilMap _ a = (a, a)
consMap f (x, y) (a, b) = (f x a, f y b)
instance Tuple (a, b, c) where
type Map f (a, b, c) = (f a, f b, f c)
nilMap _ a = (a, a, a)
consMap f (x, y, z) (a, b, c) = (f x a, f y b, f z c)
The gunzip itself:
gunzip :: forall t. Tuple t => [t] -> Map [] t
gunzip [] = nilMap (Proxy :: Proxy t) []
gunzip (p:ps) = consMap (:) p (gunzip ps)
This looks a lot like transpose:
transpose :: [[a]] -> [[a]]
transpose [] = repeat [] -- `gunzip` handles this case better
transpose (xs:xss) = zipWith (:) xs (transpose xss)
which it basically is, except with tuples. gunzip can be equivalently defined in terms of foldr as follows:
gunzip :: forall t. Tuple t => [t] -> Map [] t
gunzip = foldr (consMap (:)) $ nilMap (Proxy :: Proxy t) []
To define generic zip we need a type class of splittable data types (is there something like this on Hackage?).
class Splittable f g where
split :: f a -> g a (f a)
E.g. for lists we have
newtype MaybeBoth a b = MaybeBoth { getMaybeBoth :: Maybe (a, b) }
instance Splittable [] MaybeBoth where
split [] = MaybeBoth Nothing
split (x:xs) = MaybeBoth (Just (x, xs))
And here is what we add to the Tuple type class:
splitMap :: (Biapplicative g, Splittable f g) => Proxy (f t) -> Map f t -> g t (Map f t)
The Biapplicative g constraint ensures that it's possible to combine g a b and g c d into g (a, c) (b, d). For 2- and 3- tuples it looks like this:
splitMap _ (a, b) = biliftA2 (,) (,) (split a) (split b)
splitMap _ (a, b, c) = biliftA3 (,,) (,,) (split a) (split b) (split c)
After providing a Biapplicative instance for MaybeBoth
instance Biapplicative MaybeBoth where
bipure x y = MaybeBoth $ Just (x, y)
MaybeBoth f <<*>> MaybeBoth a = MaybeBoth $ uncurry (***) <$> f <*> a
we can finally define gzip:
gzip :: forall t. Tuple t => Map [] t -> [t]
gzip a = maybe [] (\(p, a') -> p : gzip a') . getMaybeBoth $ splitMap (Proxy :: Proxy [t]) a
It repeteadly cuts first elements of lists in a tuple, forms a tuple from them and prepends it to the result.
It should be possible to generalize gunzip by adding a dual to Splittable (Uniteable or something like that), but I'll stop here.
EDIT: I couldn't stop.
You are right that these functions (zip2, zip3 etc.) are all instances of the same pattern and in an ideal world, they should be implementable generically. By the way, as an exercise to the reader, figure out what zip1 and zip0 should be ;).
However, it is hard to implement zipN generically, because the common pattern between all the different cases is rather non-trivial. This does not mean it's impossible to implement it generically, but you'll need some of the more advanced type system features of Haskell GHC to do it.
To be more concrete, zip2, zip3 etc. all have a different number of arguments, making this an instance of "arity-generic programming" (the arity of a function is its number of arguments). As you might expect in the world of functional programming, there is an interesting research paper that covers precisely this topic ("arity-generic programming"), and conveniently, one of their main examples is... zipWithN. It doesn't directly answer your question because it uses Agda rather than Haskell, but you might still find it interesting. In any case, similar ideas can be implemented in terms of one or more of Haskell's GHC's more advanced type-system features (TypeFamilies and DataKinds come to mind). PDF version here.
By the way, this is just about an arity-generic zipWithN. For an arity-generic zipN, you probably need some support from the compiler, particularly an arity-generic interface to the tuple constructor, which I suspect might not be in GHC. This is what I believe augustss's comment to the question and chepner's comment to Alec's answer refer to.
Is there some more powerful version of map for functions with multiple arguments? Thus the length of the reduced list would be length of original / number of arguments...
tuple a b = (a, b)
map' tuple [1, 2, 3, 4]
> [(1, 2), (3, 4)]
mul a b c = a * b * c
map' mul [1, 2, 3, 4, 5, 6]
> [3, 120]
I tried to write myself one, but it seems impossible to accept function of any number of arguments.
I would like to have generic version for variable number of arguments, because writing one myself every time takes some time and it won't use loop fusion.
The simplest solution is to just generate a helper function
for each arity that you need, e.g. for arity 3 functions:
map3 :: (a -> a -> a -> b) -> [a] -> [b]
map3 f (x : y : z : rest) = f x y z : map3 f rest
map3 f _ = []
Put these in a utility module so you can import them as needed.
If you want to get sophisticated you can generate them using
TemplateHaskell, CPP macros, or even type-classes, but be careful
about turning this simple exercise into a yak shaving party.
Your example:
interact $ unlines . fmap foo . tuple . lines
could then be expressed as:
interact $ unlines . map2 foo . lines
assuming that foo has the signature foo :: String -> String -> String
I disagree with the spirit of defining something like this, but just for the hasochism of it, I'm trying my hand at it. Here is a solution that works, but only for monomorphic types. I'm working on seeing if I can convert it to something involving type families that also doesn't need to be littered with type annotations.
{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, FlexibleContexts #-}
class Consumer f a b where
consume :: f -> [a] -> Maybe (b,[a])
instance {-# OVERLAPPABLE #-} Consumer b a b where
consume b xs = Just (b,xs)
instance {-# OVERLAPPING #-} Consumer f a b => Consumer (a -> f) a b where
consume f [] = Nothing
consume f (x:xs) = consume (f x) xs
map' :: Consumer f a b => f -> [a] -> [b]
map' f xs = case consume f xs of
Nothing -> []
Just (b,xs') -> b : map' f xs'
And testing it out
> map' ((+) :: Int -> Int -> Int) ([1,2,3,4] :: [Int]) :: [Int]
[3,7]
> map' (id :: Int -> Int) ([1,2,3,4] :: [Int]) :: [Int]
[1,2,3,4]
EDIT
After some thought, I don't think that a better solution is possible (without dependent types). The reason I have to add type annotations everywhere is that I have no way of informing GHC that the two consumer instances I've provided are actually the only instances I will ever want (are closed type classes even a thing?) and I can't etch out functional dependencies. This second problem is also why I can't rewrite this as a type family. On a side note though, annotating the function arguments can look a lot nicer now that we have TypeApplications: map' ((+) #Int) ([1..9] :: [Int]).
I am trying to combine an arbitrary number of mathematical functions that are in a list.
I've written a compose function that chains them together:
chain :: (c -> d) -> (a -> b -> c) -> a -> b -> d
g `chain` f = \a b -> g (f a b)
So doing something like
let b = (*) `chain` (+) `chain` (-)
Creates a function that takes a b c d, and does ((a − b) + c) × d
> b 2 3 4 5
> 15
My question is, I want to be able to chain a set of these together based on an arbitrary list:
[(*),(+),(-)] would result in (*) `chain` (+) `chain` (-)
[(*),(+),(-),(*)] would result in (*) `chain` (+) `chain` (-) `chain` (*)
Is this possible? I've tried using folds but cannot get it to work since the type of the accumulator changes after each application of an element.
For example, the type of three functions chained together is:
b :: Integer -> Integer -> Integer -> Integer -> Integer
But for four it is:
b :: Integer -> Integer -> Integer -> Integer -> Integer -> Integer
If someone could point me in the right direction or knows of a solution, that would be great! Thank you.
EDIT:
My end goal is to be able to do something like this:
getZipList $ pure (function composing * and +) <*> ZipList [1,2,3] <*> ZipList [4,5,6]
<*> ZipList [7,8,9]
would result in:
[(1+4)*7, (2+5)*8, (3+6)*9]
Well, as you note yourself it can't really be possible, since different lists lengths (invisible to the type system) would result in different types. The only "solid" way do do it would be some kind of compile-time list; there are various around but none of them is really stably supported and easy to use.
The easiest alternative would be to also take the arguments as a list. i.e.,
ifxChain :: [a->a->a] -> [a] -> a
That one could easily be implemented: it's basically a zip to supply the second argument to each function, ommiting the first one. Then you fold that first parameter through the resulting list of single-parameter functions:
ifxChain fs (p0:ps) = foldl' (flip ($)) p0 $ zipWith flip fs ps
You may want to add handling of lists with non-matching lengths, should be doable reasonably safe.
If you insist on an actual variable number of function arguments (I don't think this is good!) then you need quite some type-class hackery. Such variadic functions are best known from printf.
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
class ChainedFunc f t where
chainIfxs :: [t->t->t] -> t -> f
chainIfxs fs i = chainIfxPre i id $ reverse fs
chainIfxPre :: t -> (t->t) -> [t->t->t] -> f
instance ChainedFunc t t where
chainIfxPre i f [] = f i
instance (ChainedFunc f t) => ChainedFunc (t->f) t where
chainIfxPre i fin (f:fs) x0 = chainIfxPre i (flip f x0 . fin) fs
Obviously, this is in many ways not nice at all. But, well, it works... ahm...
Main> chainIfxs [(*),(+),(-)] (2 :: Int) (3 :: Int) (4 :: Int) (5 :: Int) :: Int
15
Could this idea help you develop a general function?
Prelude> :m +Data.List
Prelude Data.List> let fs = [(-),(+),(*)]
Prelude Data.List> foldl' (\accum (i,f) -> accum `f` i) 2 (zip [3..5] fs)
15
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What are the lesser-known but useful features of the Haskell programming language. (I understand the language itself is lesser-known, but work with me. Even explanations of the simple things in Haskell, like defining the Fibonacci sequence with one line of code, will get upvoted by me.)
Try to limit answers to the Haskell core
One feature per answer
Give an example and short description of the feature, not just a link to documentation
Label the feature using bold title as the first line
User-defined control structures
Haskell has no shorthand ternary operator. The built-in if-then-else is always ternary, and is an expression (imperative languages tend to have ?:=expression, if=statement). If you want, though,
True ? x = const x
False ? _ = id
will define (?) to be the ternary operator:
(a ? b $ c) == (if a then b else c)
You'd have to resort to macros in most other languages to define your own short-circuiting logical operators, but Haskell is a fully lazy language, so it just works.
-- prints "I'm alive! :)"
main = True ? putStrLn "I'm alive! :)" $ error "I'm dead :("
Hoogle
Hoogle is your friend. I admit, it's not part of the "core", so cabal install hoogle
Now you know how "if you're looking for a higher-order function, it's already there" (ephemient's comment). But how do you find that function? With hoogle!
$ hoogle "Num a => [a] -> a"
Prelude product :: Num a => [a] -> a
Prelude sum :: Num a => [a] -> a
$ hoogle "[Maybe a] -> [a]"
Data.Maybe catMaybes :: [Maybe a] -> [a]
$ hoogle "Monad m => [m a] -> m [a]"
Prelude sequence :: Monad m => [m a] -> m [a]
$ hoogle "[a] -> [b] -> (a -> b -> c) -> [c]"
Prelude zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
The hoogle-google programmer is not able to write his programs on paper by himself the same way he does with the help of the computer. But he and the machine together are a forced not* to be reckoned with.
Btw, if you liked hoogle be sure to check out hlint!
My brain just exploded
If you try to compile this code:
{-# LANGUAGE ExistentialQuantification #-}
data Foo = forall a. Foo a
ignorefoo f = 1 where Foo a = f
You will get this error message:
$ ghc Foo.hs
Foo.hs:3:22:
My brain just exploded.
I can't handle pattern bindings for existentially-quantified constructors.
Instead, use a case-expression, or do-notation, to unpack the constructor.
In the binding group for
Foo a
In a pattern binding: Foo a = f
In the definition of `ignorefoo':
ignorefoo f = 1
where
Foo a = f
Free Theorems
Phil Wadler introduced us to the notion of a free theorem and we've been abusing them in Haskell ever since.
These wonderful artifacts of Hindley-Milner-style type systems help out with equational reasoning by using parametricity to tell you about what a function will not do.
For instance, there are two laws that every instance of Functor should satisfy:
forall f g. fmap f . fmap g = fmap (f . g)
fmap id = id
But, the free theorem tells us we need not bother proving the first one, but given the second it comes for 'free' just from the type signature!
fmap :: Functor f => (a -> b) -> f a -> f b
You need to be a bit careful with laziness, but this is partially covered in the original paper, and in Janis Voigtlaender's more recent paper on free theorems in the presence of seq.
Shorthand for a common list operation
The following are equivalent:
concat $ map f list
concatMap f list
list >>= f
Edit
Since more details were requested...
concat :: [[a]] -> [a]
concat takes a list of lists and concatenates them into a single list.
map :: (a -> b) -> [a] -> [b]
map maps a function over a list.
concatMap :: (a -> [b]) -> [a] -> [b]
concatMap is equivalent to (.) concat . map: map a function over a list, and concatenate the results.
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
return :: a -> m a
A Monad has a bind operation, which is called >>= in Haskell (or its sugared do-equivalent). List, aka [], is a Monad. If we substitute [] for m in the above:
instance Monad [] where
(>>=) :: [a] -> (a -> [b]) -> [b]
return :: a -> [a]
What's the natural thing for the Monad operations to do on a list? We have to satisfy the monad laws,
return a >>= f == f a
ma >>= (\a -> return a) == ma
(ma >>= f) >>= g == ma >>= (\a -> f a >>= g)
You can verify that these laws hold if we use the implementation
instance Monad [] where
(>>=) = concatMap
return = (:[])
return a >>= f == [a] >>= f == concatMap f [a] == f a
ma >>= (\a -> return a) == concatMap (\a -> [a]) ma == ma
(ma >>= f) >>= g == concatMap g (concatMap f ma) == concatMap (concatMap g . f) ma == ma >>= (\a -> f a >>= g)
This is, in fact, the behavior of Monad []. As a demonstration,
double x = [x,x]
main = do
print $ map double [1,2,3]
-- [[1,1],[2,2],[3,3]]
print . concat $ map double [1,2,3]
-- [1,1,2,2,3,3]
print $ concatMap double [1,2,3]
-- [1,1,2,2,3,3]
print $ [1,2,3] >>= double
-- [1,1,2,2,3,3]
Nestable multiline comments.
{- inside a comment,
{- inside another comment, -}
still commented! -}
Generalized algebraic data types. Here's an example interpreter where the type system lets you cover all the cases:
{-# LANGUAGE GADTs #-}
module Exp
where
data Exp a where
Num :: (Num a) => a -> Exp a
Bool :: Bool -> Exp Bool
Plus :: (Num a) => Exp a -> Exp a -> Exp a
If :: Exp Bool -> Exp a -> Exp a -> Exp a
Lt :: (Num a, Ord a) => Exp a -> Exp a -> Exp Bool
Lam :: (a -> Exp b) -> Exp (a -> b) -- higher order abstract syntax
App :: Exp (a -> b) -> Exp a -> Exp b
-- deriving (Show) -- failse
eval :: Exp a -> a
eval (Num n) = n
eval (Bool b) = b
eval (Plus e1 e2) = eval e1 + eval e2
eval (If p t f) = eval $ if eval p then t else f
eval (Lt e1 e2) = eval e1 < eval e2
eval (Lam body) = \x -> eval $ body x
eval (App f a) = eval f $ eval a
instance Eq a => Eq (Exp a) where
e1 == e2 = eval e1 == eval e2
instance Show (Exp a) where
show e = "<exp>" -- very weak show instance
instance (Num a) => Num (Exp a) where
fromInteger = Num
(+) = Plus
Patterns in top-level bindings
five :: Int
Just five = Just 5
a, b, c :: Char
[a,b,c] = "abc"
How cool is that! Saves you that call to fromJust and head every now and then.
Optional Layout
You can use explicit braces and semicolons instead of whitespace (aka layout) to delimit blocks.
let {
x = 40;
y = 2
} in
x + y
... or equivalently...
let { x = 40; y = 2 } in x + y
... instead of ...
let x = 40
y = 2
in x + y
Because layout is not required, Haskell programs can be straightforwardly produced by other programs.
seq and ($!) only evaluate enough to check that something is not bottom.
The following program will only print "there".
main = print "hi " `seq` print "there"
For those unfamiliar with Haskell, Haskell is non-strict in general, meaning that an argument to a function is only evaluated if it is needed.
For example, the following prints "ignored" and terminates with success.
main = foo (error "explode!")
where foo _ = print "ignored"
seq is known to change that behavior by evaluating to bottom if its first argument is bottom.
For example:
main = error "first" `seq` print "impossible to print"
... or equivalently, without infix ...
main = seq (error "first") (print "impossible to print")
... will blow up with an error on "first". It will never print "impossible to print".
So it might be a little surprising that even though seq is strict, it won't evaluate something the way eager languages evaluate. In particular, it won't try to force all the positive integers in the following program. Instead, it will check that [1..] isn't bottom (which can be found immediately), print "done", and exit.
main = [1..] `seq` print "done"
Operator Fixity
You can use the infix, infixl or infixr keywords to define operators associativity and precedence. Example taken from the reference:
main = print (1 +++ 2 *** 3)
infixr 6 +++
infixr 7 ***,///
(+++) :: Int -> Int -> Int
a +++ b = a + 2*b
(***) :: Int -> Int -> Int
a *** b = a - 4*b
(///) :: Int -> Int -> Int
a /// b = 2*a - 3*b
Output: -19
The number (0 to 9) after the infix allows you to define the precedence of the operator, being 9 the strongest. Infix means no associativity, whereas infixl associates left and infixr associates right.
This allows you to define complex operators to do high level operations written as simple expressions.
Note that you can also use binary functions as operators if you place them between backticks:
main = print (a `foo` b)
foo :: Int -> Int -> Int
foo a b = a + b
And as such, you can also define precedence for them:
infixr 4 `foo`
Avoiding parentheses
The (.) and ($) functions in Prelude have very convenient fixities, letting you avoid parentheses in many places. The following are equivalent:
f (g (h x))
f $ g $ h x
f . g $ h x
f . g . h $ x
flip helps too, the following are equivalent:
map (\a -> {- some long expression -}) list
flip map list $ \a ->
{- some long expression -}
Pretty guards
Prelude defines otherwise = True, making complete guard conditions read very naturally.
fac n
| n < 1 = 1
| otherwise = n * fac (n-1)
C-Style Enumerations
Combining top-level pattern matching and arithmetic sequences gives us a handy way to define consecutive values:
foo : bar : baz : _ = [100 ..] -- foo = 100, bar = 101, baz = 102
Readable function composition
Prelude defines (.) to be mathematical function composition; that is, g . f first applies f, then applies g to the result.
If you import Control.Arrow, the following are equivalent:
g . f
f >>> g
Control.Arrow provides an instance Arrow (->), and this is nice for people who don't like to read function application backwards.
let 5 = 6 in ... is valid Haskell.
Infinite Lists
Since you mentioned fibonacci, there is a very elegant way of generating fibonacci numbers from an infinite list like this:
fib#(1:tfib) = 1 : 1 : [ a+b | (a,b) <- zip fib tfib ]
The # operator allows you to use pattern matching on the 1:tfib structure while still referring to the whole pattern as fib.
Note that the comprehension list enters an infinite recursion, generating an infinite list. However, you can request elements from it or operate them, as long as you request a finite amount:
take 10 fib
You can also apply an operation to all elements before requesting them:
take 10 (map (\x -> x+1) fib)
This is thanks to Haskell's lazy evaluation of parameters and lists.
Flexible specification of module imports and exports
Importing and exporting is nice.
module Foo (module Bar, blah) -- this is module Foo, export everything that Bar expored, plus blah
import qualified Some.Long.Name as Short
import Some.Long.Name (name) -- can import multiple times, with different options
import Baz hiding (blah) -- import everything from Baz, except something named 'blah'
If you're looking for a list or higher-order function, it's already there
There's sooo many convenience and higher-order functions in the standard library.
-- factorial can be written, using the strict HOF foldl':
fac n = Data.List.foldl' (*) 1 [1..n]
-- there's a shortcut for that:
fac n = product [1..n]
-- and it can even be written pointfree:
fac = product . enumFromTo 1
Equational Reasoning
Haskell, being purely functional allows you to read an equal sign as a real equal sign (in the absence of non-overlapping patterns).
This allows you to substitute definitions directly into code, and in terms of optimization gives a lot of leeway to the compiler about when stuff happens.
A good example of this form of reasoning can be found here:
http://www.haskell.org/pipermail/haskell-cafe/2009-March/058603.html
This also manifests itself nicely in the form of laws or RULES pragmas expected for valid members of an instance, for instance the Monad laws:
returrn a >>= f == f a
m >>= return == m
(m >>= f) >>= g == m >>= (\x -> f x >>= g)
can often be used to simplify monadic code.
Laziness
Ubiquitous laziness means you can do things like define
fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
But it also provides us with a lot of more subtle benefits in terms of syntax and reasoning.
For instance, due to strictness ML has to deal with the value restriction, and is very careful to track circular let bindings, but in Haskell, we can let every let be recursive and have no need to distinguish between val and fun. This removes a major syntactic wart from the language.
This indirectly gives rise to our lovely where clause, because we can safely move computations that may or may not be used out of the main control flow and let laziness deal with sharing the results.
We can replace (almost) all of those ML style functions that need to take () and return a value, with just a lazy computation of the value. There are reasons to avoid doing so from time to time to avoid leaking space with CAFs, but such cases are rare.
Finally, it permits unrestricted eta-reduction (\x -> f x can be replaced with f). This makes combinator oriented programming for things like parser combinators much more pleasant than working with similar constructs in a strict language.
This helps you when reasoning about programs in point-free style, or about rewriting them into point-free style and reduces argument noise.
Parallel list comprehension
(Special GHC-feature)
fibs = 0 : 1 : [ a + b | a <- fibs | b <- tail fibs ]
Enhanced pattern matching
Lazy patterns
Irrefutable patterns
let ~(Just x) = someExpression
See pattern matching
Enumerations
Any type which is an instance of Enum can be used in an arithmetic sequence, not just numbers:
alphabet :: String
alphabet = ['A' .. 'Z']
Including your own datatypes, just derive from Enum to get a default implementation:
data MyEnum = A | B | C deriving(Eq, Show, Enum)
main = do
print $ [A ..] -- prints "[A,B,C]"
print $ map fromEnum [A ..] -- prints "[0,1,2]"
Monads
They are not that hidden, but they are simply everywhere, even where you don't think of them (Lists, Maybe-Types) ...