I have to implement the Map function using only the foldRight, foldLeft and unfold. This means that I have to loop through every element in the list and apply a function f to it.
I have declared my own list as follow:
abstract class IntList
case class Nil() extends IntList
case class Cons(h: Int, t: IntList) extends IntList
And I've implemented the foldRight, foldLeft and unfold functions.
and the implementation of the new map function:
def map(ls: IntList, f: Int => Int): IntList = // ??
I've been thinking for a while now, but I don't have a clue where to begin. I may not use recursion in the map function. I'm pretty sure that I have to combine the power of fold and unfold together. Unfold returns a IntList, which is the return type of map. But I'm not sure what I have to give with this function.
Anyone has a clue? :)
Match the types, fill in the arguments to match.
For instance, if you are going to use foldRight, then B must be IntList, because that's the type returned by map. Now fill in the arguments to foldRight with whatever values you have that match the types.
[In reply to previous comments.]
I don't know which exact variant of unfold you are given. Assuming it's something like this (in Ocaml, sorry, don't have Scala installed right now):
(* unfold : ('a -> ('b * 'a) option) -> 'a -> 'b list *)
let rec unfold f x =
match f x with
| None -> []
| Some (y, x') -> y :: unfold f x'
Then a solution for map is the following:
let map f = unfold (function [] -> None | x::xs -> Some (f x, xs))
Hope that helps.
Related
I was reading the original paper about data types a la carte and decided to try to implement the idea in Scala (I know it's already implemented in many functional libraries). Unfortunately I found the original paper is hard to comprehend and I stuck somewhere in the beginning. Then I found another paper that was easier to understand and I managed to rewrite Haskell code from the paper into Scala, you can find it here. However I still struggling to understand a few moments:
A quote from the second paper
Orignal Expr data type
data Expr = Val Int | Add Expr Expr
New type signature:
data Arith e = Val Int | Add e e
For any functor f, its induced recursive datatype, Fix f, is defined as the least fixpoint of f, implemented as follows:
data Fix f = In (f (Fix f))
Now that we have tied the recursive knot of a signature,
Fix Arith is a language equivalent to the original Expr datatype
which allowed integer values and addition.
What does it mean exactly "we have tied the recursive knot of a signature" and what does it mean Fix Arith is a language equivalent to the original Expr ?
The actual type of In is In :: f (Fix f) -> Fix f
If we try to construct a value using In construct and Val 1 variable we'll get the following result:
> :t In(Val 1)
> In(Val 1) :: Fix Arith
Scala encoding of the same data types:
sealed trait Arith[A]
case class Val[A](x: Int) extends Arith[A]
case class Add[A](a: A, b: A) extends Arith[A]
trait Fix[F[_]]
case class In[F[_]](exp: F[Fix[F]]) extends Fix[F]
fold function
The fold function has the following signature and implementation
Haskell:
fold :: Functor f => (f a -> a) -> Fix f -> a
fold f (In t) = f (fmap (fold f) t)
Scala variant I came up with
def fold[F[_] : Functor, A](f: F[A] => A): Fix[F] => A = {
case In(t) =>
val g: F[Fix[F]] => F[A] = implicitly[Functor[F]].lift(fold(f))
f(g(t))
}
The thing that I'm curious about is that in my Scala version function g has the following type F[Fix[F]] => F[A] but the type of variable t after pattern matching is LaCarte$Add with value Add(In(Val(1)),In(Val(2))), how it happens that it's valid to apply function g to LaCarte$Add ? Also, I'd very appreciate if you can help me to understand fold function ?
Quote from the paper:
The first argument of fold is an f-algebra, which provides
the behavior of each constructor associated with a given signature f.
What does it mean exactly “we have tied the ‘recursive knot’ of a signature”?
The original Expr datatype is recursive, referring to itself in its own definition:
data Expr = Val Int | Add Expr Expr
The Arith type “factors out” the recursion by replacing recursive calls with a parameter:
data Arith e = Val Int | Add e e
The original Expr type can have any depth of nesting, which we want to support with Arith as well, but the maximum depth depends on what type we choose for e:
Arith Void can’t be nested: it can only be a literal value (Val n) because we can’t construct an Add, because we can’t obtain a value of type Void (it has no constructors)
Arith (Arith Void) can have one level of nesting: the outer constructor can be an Add, but the inner constructors can only be Lit.
Arith (Arith (Arith Void)) can have two levels
And so on
What Fix Arith gives us is a way to talk about the fixed point Arith (Arith (Arith …)) with no limit on the depth.
This is just like how we can replace a recursive function with a non-recursive function and recover the recursion with the fixed-point combinator:
factorial' :: (Integer -> Integer) -> Integer -> Integer
factorial' recur n = if n <= 1 then 1 else n * recur (n - 1)
factorial :: Integer -> Integer
factorial = fix factorial'
factorial 5 == 120
What does it mean Fix Arith is a language equivalent to the original Expr?
The language (grammar) that Fix Arith represents is equivalent to the language that Expr represents; that is, they’re isomorphic: you can write a pair of total functions Fix Arith -> Expr and Expr -> Fix Arith.
How it happens that it’s valid to apply function g to LaCarte$Add?
I’m not very familiar with Scala, but it looks like Add is a subtype of Arith, so the parameter of g of type F[Fix[F]] can be filled with a value of type Arith[Fix[Arith]] which you get by matching on the In constructor to “unfold” one level of recursion.
Is there any way to do something like
first = {x:0}
second = {x:1,y:1}
both = [first, second]
such that both is inferred as {x::Int | r} or something like that?
I've tried a few things:
[{x:3}] :: Array(forall r. {x::Int|r}) -- nope
test = Nil :: List(forall r. {x::Int|r})
{x:1} : test -- nope
type X r = {x::Int | r}
test = Nil :: List(X) -- nope
test = Nil :: List(X())
{x:1} : test
{x:1, y:1} : test -- nope
Everything I can think of seems to tell me that combining records like this into a collection is not supported. Kind of like, a function can be polymorphic but a list cannot. Is that the correct interpretation? It reminds me a bit of the F# "value restriction" problem, though I thought that was just because of CLR restrictions whereas JS should not have that issue. But maybe it's unrelated.
Is there any way to declare the list/array to support this?
What you're looking for is "existential types", and PureScript just doesn't support those at the syntax level the way Haskell does. But you can roll your own :-)
One way to go is "data abstraction" - i.e. encode the data in terms of operations you'll want to perform on it. For example, let's say you'll want to get the value of x out of them at some point. In that case, make an array of these:
type RecordRep = Unit -> Int
toRecordRep :: forall r. { x :: Int | r } -> RecordRep
toRecordRep {x} _ = x
-- Construct the array using `toRecordRep`
test :: Array RecordRep
test = [ toRecordRep {x:1}, toRecordRep {x:1, y:1} ]
-- Later use the operation
allTheXs :: Array Int
allTheXs = test <#> \r -> r unit
If you have multiple such operations, you can always make a record of them:
type RecordRep =
{ getX :: Unit -> Int
, show :: Unit -> String
, toJavaScript :: Unit -> Foreign.Object
}
toRecordRep r =
{ getX: const r.x
, show: const $ show r.x
, toJavaScript: const $ unsafeCoerce r
}
(note the Unit arguments in every function - they're there for the laziness, assuming each operation could be expensive)
But if you really need the type machinery, you can do what I call "poor man's existential type". If you look closely, existential types are nothing more than "deferred" type checks - deferred to the point where you'll need to see the type. And what's a mechanism to defer something in an ML language? That's right - a function! :-)
newtype RecordRep = RecordRep (forall a. (forall r. {x::Int|r} -> a) -> a)
toRecordRep :: forall r. {x::Int|r} -> RecordRep
toRecordRep r = RecordRep \f -> f r
test :: Array RecordRep
test = [toRecordRep {x:1}, toRecordRep {x:1, y:1}]
allTheXs = test <#> \(RecordRep r) -> r _.x
The way this works is that RecordRep wraps a function, which takes another function, which is polymorphic in r - that is, if you're looking at a RecordRep, you must be prepared to give it a function that can work with any r. toRecordRep wraps the record in such a way that its precise type is not visible on the outside, but it will be used to instantiate the generic function, which you will eventually provide. In my example such function is _.x.
Note, however, that herein lies the problem: the row r is literally not known when you get to work with an element of the array, so you can't do anything with it. Like, at all. All you can do is get the x field, because its existence is hardcoded in the signatures, but besides the x - you just don't know. And that's by design: if you want to put anything into the array, you must be prepared to get anything out of it.
Now, if you do want to do something with the values after all, you'll have to explain that by constraining r, for example:
newtype RecordRep = RecordRep (forall a. (forall r. Show {x::Int|r} => {x::Int|r} -> a) -> a)
toRecordRep :: forall r. Show {x::Int|r} => {x::Int|r} -> RecordRep
toRecordRep r = RecordRep \f -> f r
test :: Array RecordRep
test = [toRecordRep {x:1}, toRecordRep {x:1, y:1}]
showAll = test <#> \(RecordRep r) -> r show
Passing the show function like this works, because we have constrained the row r in such a way that Show {x::Int|r} must exist, and therefore, applying show to {x::Int|r} must work. Repeat for your own type classes as needed.
And here's the interesting part: since type classes are implemented as dictionaries of functions, the two options described above are actually equivalent - in both cases you end up passing around a dictionary of functions, only in the first case it's explicit, but in the second case the compiler does it for you.
Incidentally, this is how Haskell language support for this works as well.
Folloing #FyodorSoikin answer based on "existential types" and what we can find in purescript-exists we can provide yet another solution.
Finally we will be able to build an Array of records which will be "isomorphic" to:
exists tail. Array { x :: Int | tail }
Let's start with type constructor which can be used to existentially quantify over a row type (type of kind #Type). We are not able to use Exists from purescript-exists here because PureScript has no kind polymorphism and original Exists is parameterized over Type.
newtype Exists f = Exists (forall a. f (a :: #Type))
We can follow and reimplement (<Ctrl-c><Ctrl-v> ;-)) definitions from Data.Exists and build a set of tools to work with such Exists values:
module Main where
import Prelude
import Unsafe.Coerce (unsafeCoerce)
import Data.Newtype (class Newtype, unwrap)
newtype Exists f = Exists (forall a. f (a :: #Type))
mkExists :: forall f a. f a -> Exists f
mkExists r = Exists (unsafeCoerce r :: forall a. f a)
runExists :: forall b f. (forall a. f a -> b) -> Exists f -> b
runExists g (Exists f) = g f
Using them we get the ability to build an Array of Records with "any" tail but we have to wrap any such a record type in a newtype before:
newtype R t = R { x :: Int | t }
derive instance newtypeRec :: Newtype (R t) _
Now we can build an Array using mkExists:
arr :: Array (Exists R)
arr = [ mkExists (R { x: 8, y : "test"}), mkExists (R { x: 9, z: 10}) ]
and process values using runExists:
x :: Array [ Int ]
x = map (runExists (unwrap >>> _.x)) arr
Scala offers a List#flatten method for going from List[Option[A]] to List[A].
scala> val list = List(Some(10), None)
list: List[Option[Int]] = List(Some(10), None)
scala> list.flatten
res11: List[Int] = List(10)
I attempted to implement it in Haskell:
flatten :: [Maybe a] -> [a]
flatten xs = map g $ xs >>= f
f :: Maybe a -> [Maybe a]
f x = case x of Just _ -> [x]
Nothing -> []
-- partial function!
g :: Maybe a -> a
g (Just x) = x
However I don't like the fact that g is a partial, i.e. non-total, function.
Is there a total way to write such flatten function?
Your flatten is the same as catMaybes (link) which is defined like this:
catMaybes :: [Maybe a] -> [a]
catMaybes ls = [x | Just x <- ls]
The special syntax Just x <- ls in a list comprehension means to draw an element from ls and discard it if it is not a Just. Otherwise assign x by pattern matching the value against Just x.
A slight modification of the code you have will do the trick:
flatten :: [Maybe a] -> [a]
flatten xs = xs >>= f
f :: Maybe a -> [a]
f x = case x of Just j -> [j]
Nothing -> []
If we extract the value inside of the Just constructor in f, we avoid g altogether.
Incidentally, f already exists as maybeToList and flatten is called catMaybes, both in Data.Maybe.
One could quite easily write a simple recursive function which goes through a list and rejects all the Nothings from the Maybe monad. Here's how I'd do it as a recursive sequence:
flatten :: [Maybe a] -> [a]
flatten [] = []
flatten (Nothing : xs) = flatten xs
flatten (Just x : xs) = x : flatten xs
However, it may be clearer to write it as a fold:
flatten :: [Maybe a] -> [a]
flatten = foldr go []
where go Nothing xs = xs
go (Just x) xs = x : xs
Or, we could use a blindingly elegant solution thanks to #user2407038, which I'd recommend playing around with in GHCi to work out the individual functions' jobs:
flatten :: [Maybe a] -> [a]
flatten = (=<<) (maybe [] (:[])
And it's faster, folded brother:
flatten :: [Maybe a] -> [a]
flatten = foldr (maybe id (:))
Your solution is halfway there. My suggestion if to rewrite your function f to use pattern matching (like my temporary go function), and enclose it in a where statement to keep relevant functions in one place. You've got to remember the differences in function syntax within scala and Haskell.
The big problem you're having is you don't know the differences I've mentioned. Your g function can use pattern matching with multiple patterns:
g :: Maybe a -> [a]
g (Just x) = [x]
g Nothing = []
There you go: your g function is now what you call 'complete', though more accurately, it would be said to have exhaustive patterns.
You can find more about function syntax here.
Is there a concise, idiomatic way how to express function iteration? That is, given a number n and a function f :: a -> a, I'd like to express \x -> f(...(f(x))...) where f is applied n-times.
Of course, I could make my own, recursive function for that, but I'd be interested if there is a way to express it shortly using existing tools or libraries.
So far, I have these ideas:
\n f x -> foldr (const f) x [1..n]
\n -> appEndo . mconcat . replicate n . Endo
but they all use intermediate lists, and aren't very concise.
The shortest one I found so far uses semigroups:
\n f -> appEndo . times1p (n - 1) . Endo,
but it works only for positive numbers (not for 0).
Primarily I'm focused on solutions in Haskell, but I'd be also interested in Scala solutions or even other functional languages.
Because Haskell is influenced by mathematics so much, the definition from the Wikipedia page you've linked to almost directly translates to the language.
Just check this out:
Now in Haskell:
iterateF 0 _ = id
iterateF n f = f . iterateF (n - 1) f
Pretty neat, huh?
So what is this? It's a typical recursion pattern. And how do Haskellers usually treat that? We treat that with folds! So after refactoring we end up with the following translation:
iterateF :: Int -> (a -> a) -> (a -> a)
iterateF n f = foldr (.) id (replicate n f)
or point-free, if you prefer:
iterateF :: Int -> (a -> a) -> (a -> a)
iterateF n = foldr (.) id . replicate n
As you see, there is no notion of the subject function's arguments both in the Wikipedia definition and in the solutions presented here. It is a function on another function, i.e. the subject function is being treated as a value. This is a higher level approach to a problem than implementation involving arguments of the subject function.
Now, concerning your worries about the intermediate lists. From the source code perspective this solution turns out to be very similar to a Scala solution posted by #jmcejuela, but there's a key difference that GHC optimizer throws away the intermediate list entirely, turning the function into a simple recursive loop over the subject function. I don't think it could be optimized any better.
To comfortably inspect the intermediate compiler results for yourself, I recommend to use ghc-core.
In Scala:
Function chain Seq.fill(n)(f)
See scaladoc for Function. Lazy version: Function chain Stream.fill(n)(f)
Although this is not as concise as jmcejuela's answer (which I prefer), there is another way in scala to express such a function without the Function module. It also works when n = 0.
def iterate[T](f: T=>T, n: Int) = (x: T) => (1 to n).foldLeft(x)((res, n) => f(res))
To overcome the creation of a list, one can use explicit recursion, which in reverse requires more static typing.
def iterate[T](f: T=>T, n: Int): T=>T = (x: T) => (if(n == 0) x else iterate(f, n-1)(f(x)))
There is an equivalent solution using pattern matching like the solution in Haskell:
def iterate[T](f: T=>T, n: Int): T=>T = (x: T) => n match {
case 0 => x
case _ => iterate(f, n-1)(f(x))
}
Finally, I prefer the short way of writing it in Caml, where there is no need to define the types of the variables at all.
let iterate f n x = match n with 0->x | n->iterate f (n-1) x;;
let f5 = iterate f 5 in ...
I like pigworker's/tauli's ideas the best, but since they only gave it as a comments, I'm making a CW answer out of it.
\n f x -> iterate f x !! n
or
\n f -> (!! n) . iterate f
perhaps even:
\n -> ((!! n) .) . iterate
In the process of writing a simple RPN calculator, I have the following type aliases:
type Stack = List[Double]
type Operation = Stack => Option[Stack]
... and I have written a curious-looking line of Scala code:
val newStack = operations.foldLeft(Option(stack)) { _ flatMap _ }
This takes an initial stack of values and applies a list of operations to that stack. Each operation may fail (i.e. yields an Option[Stack]) so I sequence them with flatMap. The thing that's somewhat unusual about this (in my mind) is that I'm folding over a list of monadic functions, rather than folding over a list of data.
I want to know if there's a standard function that captures this "fold-bind" behavior. When I'm trying to play the "Name That Combinator" game, Hoogle is usually my friend, so I tried the same mental exercise in Haskell:
foldl (>>=) (Just stack) operations
The types here are:
foldl :: (a -> b -> a) -> a -> [b] -> a
(>>=) :: Monad m => m a -> (a -> m b) -> m b
So the type of my mystery foldl (>>=) combinator, after making the types of foldl and (>>=) line up, should be:
mysteryCombinator :: Monad m => m a -> [a -> m a] -> m a
... which is again what we'd expect. My problem is that searching Hoogle for a function with that type yields no results. I tried a couple other permutations that I thought might be reasonable: a -> [a -> m a] -> m a (i.e. starting with a non-monadic value), [a -> m a] -> m a -> m a (i.e. with arguments flipped), but no luck there either. So my question is, does anybody know a standard name for my mystery "fold-bind" combinator?
a -> m a is just a Kleisli arrow with the argument and result types both being a. Control.Monad.(>=>) composes two Kleisli arrows:
(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c
Think flip (.), but for Kleisli arrows instead of functions.
So we can split this combinator into two parts, the composition and the "application":
composeParts :: (Monad m) => [a -> m a] -> a -> m a
composeParts = foldr (>=>) return
mysteryCombinator :: (Monad m) => m a -> [a -> m a] -> m a
mysteryCombinator m fs = m >>= composeParts fs
Now, (>=>) and flip (.) are related in a deeper sense than just being analogous; both the function arrow, (->), and the data type wrapping a Kleisli arrow, Kleisli, are instances of Control.Category.Category. So if we were to import that module, we could in fact rewrite composeParts as:
composeParts :: (Category cat) => [cat a a] -> cat a a
composeParts = foldr (>>>) id
(>>>) (defined in Control.Category) is just a nicer way of writing as flip (.).
So, there's no standard name that I know of, but it's just a generalisation of composing a list of functions. There's an Endo a type in the standard library that wraps a -> a and has a Monoid instance where mempty is id and mappend is (.); we can generalise this to any Category:
newtype Endo cat a = Endo { appEndo :: cat a a }
instance (Category cat) => Monoid (Endo cat a) where
mempty = Endo id
mappend (Endo f) (Endo g) = Endo (f . g)
We can then implement composeParts as:
composeParts = appEndo . mconcat . map Endo . reverse
which is just mconcat . reverse with some wrapping. However, we can avoid the reverse, which is there because the instance uses (.) rather than (>>>), by using the Dual a Monoid, which just transforms a monoid into one with a flipped mappend:
composeParts :: (Category cat) => [cat a a] -> cat a a
composeParts = appEndo . getDual . mconcat . map (Dual . Endo)
This demonstrates that composeParts is a "well-defined pattern" in some sense :)
The one starting with a non-monadic value is (modulo flip)
Prelude> :t foldr (Control.Monad.>=>) return
foldr (Control.Monad.>=>) return
:: Monad m => [c -> m c] -> c -> m c
(or foldl)
(Yes, I know this doesn't answer the question, but the code layout in comments isn't satisfactory.)