Functional equivalent of if (p(f(a), f(b)) a else b

Question

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?"

Answer 1

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

Answer 2

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.

Answer 3

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

Answer 4

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 (_ < _).

Answer 5

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.

Answer 6

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