I have a Scala pre-examination question which I cannot get through. Probably someone could help.
Does the expression on the right conforms to the declared type, and why?
(a) val x1: B => D = (b: A) => new D
(b) val x2: A => C => D = (a: A) => (b: D) => new C
(c) val x3: (D => B) => A = (db: D => A) => new B
The class hierarchy:
class A
class B extends A
class C
class D extends C
You don't need us for this. Just paste the code into the REPL.
scala> class A
defined class A
scala> class B extends A
defined class B
scala> class C
defined class C
scala> class D extends C
defined class D
scala> val x1: B => D = (b: A) => new D
x1: B => D = <function1>
scala> val x2: A => C => D = (a: A) => (b: D) => new C
<console>:10: error: type mismatch;
found : C
required: D
val x2: A => C => D = (a: A) => (b: D) => new C
^
scala> val x3: (D => B) => A = (db: D => A) => new B
x3: (D => B) => A = <function1>
I guess that there is no correct expression amongst them.
Mb there is an answer, which has more information then needed.
when we write T => F -- it means a function type, sugar for a trait Function1[-P, +R] { def apply(p: P): R }
when we write (t: T) => new F it means a lambda function, which has a type T => F, or (alias, sugar) Function1[T, F], as you can notice t is an application argument.
we have relations on types, one of them is sub typing, so you need an explanation of function sub typing in this task.
Let's consider why code, provided here by #ChrisMartin, compiling.
So function, is a trait:
trait Function1[-P, +R] {
def apply(p: P): R
}
That means that function is contravariant (+) by argument and covariant (-) by result.
Let F: P → R, p ∈ P => F(p) ∈ R, and let F' : P' → R' : P' ⊃ P, R' ⊂ R'. Then, p ∈ P => p ∈ P' => F'(p) ∈ R'=> F'(p) ∈ R.
That means that F' function is a special case of F function on its domain. In other words some function from a super type of P to the sub type of R is a subtype of function from P to R.
In your examples:
B => D is a more general function then A => D (A is a super type of B) (check: implicitly[(A => D) <:< (B => D)])
A => C => D is a more specific function then A => D => C (check: implicitly[(A => C => D) <:< (A => D => C)])
(D => B) => A is a more general function then (D => A) => B (check: implicitly[((D => A) => B) <:< ((D => B) => A)])
Related
I am wondering whether the following is possible in Scala:
Given some vector x = (x_1, x_2, ..., x_n) in R^n and a function f that maps R^n to R, I would like to replicate this concept in Scala. The idea of Scala's partial function/currying should hold here (i.e. when applying a single value x_i, return a function that is defined only for a subset of its input domain). For example, when n = 2 define f(x, y) = sin(x + y), then trivially, f(2, y) = sin(2 + y).
However, the dimension (n > 0) may vary from case to case and may even be provided in input.
Partial application for n = 2 is:
def leftPartialFunction(f: (Double, Double) => Double)(x: Double): Double => Double = f(x, _)
but how can this be generalized for arbitrary n?
For example, how can I apply the function in position i?
Something like this I assume would not work:
def partialFunction(f: IndexedSeq[Double] => Double)(xi: Double): IndexedSeq[Double] => Double = .... // cannot work well with indexed seq as they are not "disjoint"
Try the following implementation of partialFunction:
import shapeless.{::, HList, HNil, Nat, Succ}
import shapeless.ops.function.{FnFromProduct, FnToProduct}
import shapeless.ops.hlist.{At, Drop, Prepend, Take}
def partialFunction[N <: Nat, F, X,
L <: HList, Y, L1 <: HList, L2 <: HList, L3 <: HList
](i: N)(f: F)(xi: X)(implicit
fnToProduct: FnToProduct.Aux[F, L => Y],
at: At.Aux[L, N, X],
take: Take.Aux[L, N, L1],
drop: Drop.Aux[L, Succ[N], L2],
prepend: Prepend.Aux[L1, L2, L3],
fnFromProduct: FnFromProduct[L3 => Y],
take1: Take.Aux[L3, N, L1],
drop1: Drop.Aux[L3, N, L2],
prepend1: Prepend.Aux[L1, X :: L2, L],
): fnFromProduct.Out =
fnFromProduct(l3 => fnToProduct(f)(prepend1(take1(l3), xi :: drop1(l3))))
Testing:
import shapeless.Nat._1
val f: (Int, Boolean, Double) => String = (i, b, d) => s"i=$i, b=$b, d=$d"
f(1, true, 2.0) // i=1, b=true, d=2.0
val f1 = partialFunction(_1)(f)(true)
f1: ((Int, Double) => String)
f1(1, 2.0) // i=1, b=true, d=2.0
You can also write partialFunction(Nat(1))(f)(true) instead of partialFunction(_1)(f)(true).
I learn scala in university and I cannot understand how to use map, flatmap and Option. Here's couple functions from my lab. I know how to implement first but I have no idea how to deal with second? So, the question: how to implement second function without changing it's signature (using map and flatmap)?
def testCompose[A, B, C, D](f: A => B)
(g: B => C)
(h: C => D): A => D = h compose g compose f
def testMapFlatMap[A, B, C, D](f: A => Option[B])
(g: B => Option[C])
(h: C => D): Option[A] => Option[D] = // help
_.flatMap(f).flatMap(g).map(h)
because:
_ - receive an Option[A]
flatMap(f) - peek inside, return Option[B] (flatMap() won't re-wrap it)
flatMap(g) - peek inside, return Option[C] (flatMap() won't re-wrap it)
map(h) - peek inside, return D (map() will re-wrap it)
def map2[A,B,C] (a: Par[A], b: Par[B]) (f: (A,B) => C) : Par[C] =
(es: ExecutorService) => {
val af = a (es)
val bf = b (es)
UnitFuture (f(af.get, bf.get))
}
def map3[A,B,C,D] (pa :Par[A], pb: Par[B], pc: Par[C]) (f: (A,B,C) => D) :Par[D] =
map2(map2(pa,pb)((a,b)=>(c:C)=>f(a,b,c)),pc)(_(_))
I have map2 and need to produce map3 in terms of map2. I found the solution in GitHub but it is hard to understand. Can anyone put a sight on it and explain map3 and also what this does (())?
On a purely abstract level, map2 means you can run two tasks in parallel, and that is a new task in itself. The implementation provided for map3 is: run in parallel (the task that consist in running in parallel the two first ones) and (the third task).
Now down to the code: first, let's give name to all the objects created (I also extended _ notations for clarity):
def map3[A,B,C,D] (pa :Par[A], pb: Par[B], pc: Par[C]) (f: (A,B,C) => D) :Par[D] = {
def partialCurry(a: A, b: B)(c: C): D = f(a, b, c)
val pc2d: Par[C => D] = map2(pa, pb)((a, b) => partialCurry(a, b))
def applyFunc(func: C => D, c: C): D = func(c)
map2(pc2d, pc)((c2d, c) => applyFunc(c2d, c)
}
Now remember that map2 takes two Par[_], and a function to combine the eventual values, to get a Par[_] of the result.
The first time you use map2 (the inside one), you parallelize the first two tasks, and combine them into a function. Indeed, using f, if you have a value of type A and a value of type B, you just need a value of type C to build one of type D, so this exactly means that partialCurry(a, b) is a function of type C => D (partialCurry itself is of type (A, B) => C => D).
Now you have again two values of type Par[_], so you can again map2 on them, and there is only one natural way to combine them to get the final value.
The previous answer is correct but I found it easier to think about like this:
def map3[A, B, C, D](a: Par[A], b: Par[B], c: Par[C])(f: (A, B, C) => D): Par[D] = {
val f1 = (a: A, b: B) => (c: C) => f(a, b, c)
val f2: Par[C => D] = map2(a, b)(f1)
map2(f2, c)((f3: C => D, c: C) => f3(c))
}
Create a function f1 that is a version of f with the first 2 arguments partially applied, then we can map2 that with a and b to give us a function of type C => D in the Par context (f1).
Finally we can use f2 and c as arguments to map2 then apply f3(C => D) to c to give us a D in the Par context.
Hope this helps someone!
OK, so I'm trying to implement the basics of lambda calculus. Here it goes.
My numbers:
def zero[Z](s: Z => Z)(z: Z): Z = z
def one[Z](s: Z => Z)(z: Z): Z = s(z)
def two[Z](s: Z => Z)(z: Z): Z = s(s(z))
Partially (actually, non) applied version of them is smth like that:
def z[Z]: (Z => Z) => (Z => Z) = zero _
Before I continue I define some types:
type FZ[Z] = Z => Z
type FFZ[Z] = FZ[Z] => FZ[Z]
Fine, succ function goes like (Application order should be exactly like that! I took the definition here):
def succ[Z](w: FFZ[Z])(y: FZ[Z])(x: Z): Z = y((w(y))(x))
And the unapplied version of it gets as scary as:
def s[Z]: FFFZ[Z] = successor _
Beg your pardon, here is the missing types:
type FFFZ[Z] = FFZ[Z] => FFZ[Z]
type FFFFZ[Z] = FFFZ[Z] => FFFZ[Z]
But I'm stuck at the add function. If conformed to types and definition (taken here as well) it goes like
def add[Z](a: FFFFZ[Z])(b: FFZ[Z]): FFZ[Z] =
(a(s))(b)
But I want a to be a common number of type FFZ[Z].
So -- how can I define addition?
It's totally possible to implement Church numerals in Scala. Here is one such rather straight-forward implementation:
object ChurchNumerals {
type Succ[Z] = Z => Z
type ChNum[Z] = Succ[Z] => Z => Z
def zero[Z]: ChNum[Z] =
(_: Succ[Z]) => (z: Z) => z
def succ[Z] (num: ChNum[Z]): ChNum[Z] =
(s: Succ[Z]) => (z: Z) => s( num(s)(z) )
// a couple of church constants
def one[Z] : ChNum[Z] = succ(zero)
def two[Z] : ChNum[Z] = succ(one)
// the addition function
def add[Z] (a: ChNum[Z]) (b: ChNum[Z]) =
(s: Succ[Z]) => (z: Z) => a(s)( b(s)(z) )
def four[Z] : ChNum[Z] = add(two)(two)
// test
def church_to_int (num: ChNum[Int]): Int =
num((x: Int) => x + 1)(0)
def fourInt: Int = church_to_int(four)
def main(args: Array[String]): Unit = {
println(s"2 + 2 = ${fourInt}")
}
}
Compiles and prints:
$ scala church-numerals.scala
2 + 2 = 4
If I were to explain Church numerals from scratch I'd add more commentaries. But taking the context into account, I'm not sure on what to comment in this case. Please feel free to ask and I'll add more explanations.
I have coded Numerals, Booleans and Pairs: https://github.com/pedrofurla/church/blob/master/src/main/scala/Church.scala following Church's style.
One thing I noticed is that using the curried function syntax was much easier than using multiple argument lists. Some of the interesting snippets
type NUM[A] = (A => A) => A => A
def succ [A]: NUM[A] => NUM[A] = m => n => x => n(m(n)(x))
def zero [A]: NUM[A] = f => x => x
def one [A]: NUM[A] = f => x => f(x)
def two [A]: NUM[A] = f => x => f(f(x))
def three [A]: NUM[A] = f => x => f(f(f(x)))
def plus [A]: (NUM[A]) => (NUM[A]) => NUM[A] = m => n => f => x => m(f)(n(f)(x))
Now for printing them out (very similar to Antov Trunov's solution):
def nvalues[A] = List(zero[A], one[A], two[A], three[A])
val inc: Int => Int = _ + 1
def num: (NUM[Int]) => Int = n => n(inc)(0)
def numStr: (NUM[String]) => String = n => n("f (" + _ + ") ")("z")
Some output:
scala> println(nvalues map num)
List(0, 1, 2, 3)
scala> println(nvalues map numStr) // Like this better :)
List(z, f (z) , f (f (z) ) , f (f (f (z) ) ) )
I'm guessing that there must be a better functional way of expressing the following:
def foo(i: Any) : Int
if (foo(a) < foo(b)) a else b
So in this example f == foo and p == _ < _. There's bound to be some masterful cleverness in scalaz for this! I can see that using BooleanW I can write:
p(f(a), f(b)).option(a).getOrElse(b)
But I was sure that I would be able to write some code which only referred to a and b once. If this exists it must be on some combination of Function1W and something else but scalaz is a bit of a mystery to me!
EDIT: I guess what I'm asking here is not "how do I write this?" but "What is the correct name and signature for such a function and does it have anything to do with FP stuff I do not yet understand like Kleisli, Comonad etc?"
Just in case it's not in Scalaz:
def x[T,R](f : T => R)(p : (R,R) => Boolean)(x : T*) =
x reduceLeft ((l, r) => if(p(f(l),f(r))) r else l)
scala> x(Math.pow(_ : Int,2))(_ < _)(-2, 0, 1)
res0: Int = -2
Alternative with some overhead but nicer syntax.
class MappedExpression[T,R](i : (T,T), m : (R,R)) {
def select(p : (R,R) => Boolean ) = if(p(m._1, m._2)) i._1 else i._2
}
class Expression[T](i : (T,T)){
def map[R](f: T => R) = new MappedExpression(i, (f(i._1), f(i._2)))
}
implicit def tupleTo[T](i : (T,T)) = new Expression(i)
scala> ("a", "bc") map (_.length) select (_ < _)
res0: java.lang.String = a
I don't think that Arrows or any other special type of computation can be useful here. Afterall, you're calculating with normal values and you can usually lift a pure computation that into the special type of computation (using arr for arrows or return for monads).
However, one very simple arrow is arr a b is simply a function a -> b. You could then use arrows to split your code into more primitive operations. However, there is probably no reason for doing that and it only makes your code more complicated.
You could for example lift the call to foo so that it is done separately from the comparison. Here is a simiple definition of arrows in F# - it declares *** and >>> arrow combinators and also arr for turning pure functions into arrows:
type Arr<'a, 'b> = Arr of ('a -> 'b)
let arr f = Arr f
let ( *** ) (Arr fa) (Arr fb) = Arr (fun (a, b) -> (fa a, fb b))
let ( >>> ) (Arr fa) (Arr fb) = Arr (fa >> fb)
Now you can write your code like this:
let calcFoo = arr <| fun a -> (a, foo a)
let compareVals = arr <| fun ((a, fa), (b, fb)) -> if fa < fb then a else b
(calcFoo *** calcFoo) >>> compareVals
The *** combinator takes two inputs and runs the first and second specified function on the first, respectively second argument. >>> then composes this arrow with the one that does comparison.
But as I said - there is probably no reason at all for writing this.
Here's the Arrow based solution, implemented with Scalaz. This requires trunk.
You don't get a huge win from using the arrow abstraction with plain old functions, but it is a good way to learn them before moving to Kleisli or Cokleisli arrows.
import scalaz._
import Scalaz._
def mod(n: Int)(x: Int) = x % n
def mod10 = mod(10) _
def first[A, B](pair: (A, B)): A = pair._1
def selectBy[A](p: (A, A))(f: (A, A) => Boolean): A = if (f.tupled(p)) p._1 else p._2
def selectByFirst[A, B](f: (A, A) => Boolean)(p: ((A, B), (A, B))): (A, B) =
selectBy(p)(f comap first) // comap adapts the input to f with function first.
val pair = (7, 16)
// Using the Function1 arrow to apply two functions to a single value, resulting in a Tuple2
((mod10 &&& identity) apply 16) assert_≟ (6, 16)
// Using the Function1 arrow to perform mod10 and identity respectively on the first and second element of a `Tuple2`.
val pairs = ((mod10 &&& identity) product) apply pair
pairs assert_≟ ((7, 7), (6, 16))
// Select the tuple with the smaller value in the first element.
selectByFirst[Int, Int](_ < _)(pairs)._2 assert_≟ 16
// Using the Function1 Arrow Category to compose the calculation of mod10 with the
// selection of desired element.
val calc = ((mod10 &&& identity) product) ⋙ selectByFirst[Int, Int](_ < _)
calc(pair)._2 assert_≟ 16
Well, I looked up Hoogle for a type signature like the one in Thomas Jung's answer, and there is on. This is what I searched for:
(a -> b) -> (b -> b -> Bool) -> a -> a -> a
Where (a -> b) is the equivalent of foo, (b -> b -> Bool) is the equivalent of <. Unfortunately, the signature for on returns something else:
(b -> b -> c) -> (a -> b) -> a -> a -> c
This is almost the same, if you replace c with Bool and a in the two places it appears, respectively.
So, right now, I suspect it doesn't exist. It occured to me that there's a more general type signature, so I tried it as well:
(a -> b) -> ([b] -> b) -> [a] -> a
This one yielded nothing.
EDIT:
Now I don't think I was that far at all. Consider, for instance, this:
Data.List.maximumBy (on compare length) ["abcd", "ab", "abc"]
The function maximumBy signature is (a -> a -> Ordering) -> [a] -> a, which, combined with on, is pretty close to what you originally specified, given that Ordering is has three values -- almost a boolean! :-)
So, say you wrote on in Scala:
def on[A, B, C](f: ((B, B) => C), g: A => B): (A, A) => C = (a: A, b: A) => f(g(a), g(b))
The you could write select like this:
def select[A](p: (A, A) => Boolean)(a: A, b: A) = if (p(a, b)) a else b
And use it like this:
select(on((_: Int) < (_: Int), (_: String).length))("a", "ab")
Which really works better with currying and dot-free notation. :-) But let's try it with implicits:
implicit def toFor[A, B](g: A => B) = new {
def For[C](f: (B, B) => C) = (a1: A, a2: A) => f(g(a1), g(a2))
}
implicit def toSelect[A](t: (A, A)) = new {
def select(p: (A, A) => Boolean) = t match {
case (a, b) => if (p(a, b)) a else b
}
}
Then you can write
("a", "ab") select (((_: String).length) For (_ < _))
Very close. I haven't figured any way to remove the type qualifier from there, though I suspect it is possible. I mean, without going the way of Thomas answer. But maybe that is the way. In fact, I think on (_.length) select (_ < _) reads better than map (_.length) select (_ < _).
This expression can be written very elegantly in Factor programming language - a language where function composition is the way of doing things, and most code is written in point-free manner. The stack semantics and row polymorphism facilitates this style of programming. This is what the solution to your problem will look like in Factor:
# We find the longer of two lists here. The expression returns { 4 5 6 7 8 }
{ 1 2 3 } { 4 5 6 7 8 } [ [ length ] bi# > ] 2keep ?
# We find the shroter of two lists here. The expression returns { 1 2 3 }.
{ 1 2 3 } { 4 5 6 7 8 } [ [ length ] bi# < ] 2keep ?
Of our interest here is the combinator 2keep. It is a "preserving dataflow-combinator", which means that it retains its inputs after the given function is performed on them.
Let's try to translate (sort of) this solution to Scala.
First of all, we define an arity-2 preserving combinator.
scala> def keep2[A, B, C](f: (A, B) => C)(a: A, b: B) = (f(a, b), a, b)
keep2: [A, B, C](f: (A, B) => C)(a: A, b: B)(C, A, B)
And an eagerIf combinator. if being a control structure cannot be used in function composition; hence this construct.
scala> def eagerIf[A](cond: Boolean, x: A, y: A) = if(cond) x else y
eagerIf: [A](cond: Boolean, x: A, y: A)A
Also, the on combinator. Since it clashes with a method with the same name from Scalaz, I'll name it upon instead.
scala> class RichFunction2[A, B, C](f: (A, B) => C) {
| def upon[D](g: D => A)(implicit eq: A =:= B) = (x: D, y: D) => f(g(x), g(y))
| }
defined class RichFunction2
scala> implicit def enrichFunction2[A, B, C](f: (A, B) => C) = new RichFunction2(f)
enrichFunction2: [A, B, C](f: (A, B) => C)RichFunction2[A,B,C]
And now put this machinery to use!
scala> def length: List[Int] => Int = _.length
length: List[Int] => Int
scala> def smaller: (Int, Int) => Boolean = _ < _
smaller: (Int, Int) => Boolean
scala> keep2(smaller upon length)(List(1, 2), List(3, 4, 5)) |> Function.tupled(eagerIf)
res139: List[Int] = List(1, 2)
scala> def greater: (Int, Int) => Boolean = _ > _
greater: (Int, Int) => Boolean
scala> keep2(greater upon length)(List(1, 2), List(3, 4, 5)) |> Function.tupled(eagerIf)
res140: List[Int] = List(3, 4, 5)
This approach does not look particularly elegant in Scala, but at least it shows you one more way of doing things.
There's a nice-ish way of doing this with on and Monad, but Scala is unfortunately very bad at point-free programming. Your question is basically: "can I reduce the number of points in this program?"
Imagine if on and if were differently curried and tupled:
def on2[A,B,C](f: A => B)(g: (B, B) => C): ((A, A)) => C = {
case (a, b) => f.on(g, a, b)
}
def if2[A](b: Boolean): ((A, A)) => A = {
case (p, q) => if (b) p else q
}
Then you could use the reader monad:
on2(f)(_ < _) >>= if2
The Haskell equivalent would be:
on' (<) f >>= if'
where on' f g = uncurry $ on f g
if' x (y,z) = if x then y else z
Or...
flip =<< flip =<< (if' .) . on (<) f
where if' x y z = if x then y else z