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Scala type system -> Prolog

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scalatoprolog.md
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Miles Sabin implemented a nice selection sort example in Scala's type system. See:

http://www.chuusai.com/2012/01/27/type-level-sorting-in-shapeless/

To get a better understanding how that works I ported the algorithm to Prolog.

selectleast([H|T], TM, [H|TRem]):-
    selectleast(T, TM, TRem), TM < H.

selectleast([H|T], H, T).

selectionsort(L, [M|ST]):-
    selectleast(L, M, Rem), selectionsort(Rem, ST).

selectionsort(S, S).

To test it use a Prolog interpreter:

?- selectleast([3, 1, 4, 0, 2, 5], H, T).
H = 0,
T = [3, 1, 4, 2, 5] .

?- selectionsort([3, 1, 4, 0, 2, 5], Sorted).
Sorted = [0, 1, 2, 3, 4, 5] .

The implementation is perhaps not very idiomatic Prolog code but that's not the point. The nice thing is that the translation from Scala type system syntax is completely straightforward.

Breakdown

Let's start with SelectLeast rule and convert it to Prolog.

/* hlistSelectLeast3(implicit tsl : SelectLeast[T, TM, TRem], ev : TM < H): SelectLeast[H :: T, TM, H :: TRem] */

==>

selectleast([H|T], TM, [H|TRem]):-
    selectleast(T, TM, TRem), TM < H.

The return type of 'hlistSelectLeast3' is rule's head. The implicit parameters are the predicates. Lower priority 'hlistSelectLeast1' is simply a Prolog fact.

/* hlistSelectLeast1: SelectLeast[H :: T, H, T] */

==>

selectleast([H|T], H, T).

SelectionSort is converted in a same way.

/* hlistSelectionSort2(implicit sl : SelectLeast[L, M, Rem], sr : SelectionSort[Rem, ST]): SelectionSort[L, M :: ST] */

==>

selectionsort(L, [M|ST]):-
    selectleast(L, M, Rem), selectionsort(Rem, ST).

/* hlistSelectionSort1: SelectionSort[S, S] */

==>

selectionsort(S, S).

Out of interest ... suppose you started from the idiomatic Prolog and worked back to a Scala implementation ... would it look any different?

Miles, I did a little bit Prolog some 15 years ago. So please take my view on what is idiomatic and what is not with a grain of salt :) I think some folks would prefer to make non-recursive cases more explicit. As in this implementation:

selectleast([X], X, []).

selectleast([H|T], H, [TM|TRem]):-
    selectleast(T, TM, TRem), TM > H.

selectleast([H|T], TM, [H|TRem]):-
    selectleast(T, TM, TRem), TM < H.

selectionsort([], []).

selectionsort(L, [M|ST]):-
    selectleast(L, M, Rem), selectionsort(Rem, ST).

That translates to following Shapeless code (seems to work without prioritizing the implicits, I added GT to my Shapeless build):

object SelectLeast {
  implicit def hlistSelectLeast1[H <: Nat] =
    new SelectLeast[H :: HNil, H, HNil] {
      def apply(l : H :: HNil) : (H, HNil) = (l.head, l.tail)
    }

  implicit def hlistSelectLeast2[H <: Nat, T <: HList, TM <: Nat, TRem <: HList]
    (implicit tsl : SelectLeast[T, TM, TRem], ev : TM > H) =
      new SelectLeast[H :: T, H, TM :: TRem] {
        def apply(l : H :: T) : (H, TM :: TRem) = {
          val (tm, rem) = tsl(l.tail)
          (l.head, tm :: rem)
        }
      }

  implicit def hlistSelectLeast3[H <: Nat, T <: HList, TM <: Nat, TRem <: HList]
    (implicit tsl : SelectLeast[T, TM, TRem], ev : TM < H) =
      new SelectLeast[H :: T, TM, H :: TRem] {
        def apply(l : H :: T) : (TM, H :: TRem) = {
          val (tm, rem) = tsl(l.tail)
          (tm, l.head :: rem)
        }
      }
}

object SelectionSort {
  implicit def hlistSelectionSort1 = new SelectionSort[HNil, HNil] {
    def apply(l : HNil) : HNil = l
  }

  implicit def hlistSelectionSort2[L <: HList, M <: Nat, Rem <: HList, ST <: HList]
    (implicit sl : SelectLeast[L, M, Rem], sr : SelectionSort[Rem, ST]) =
      new SelectionSort[L, M :: ST] {
        def apply(l : L) = {
          val (m, rem) = sl(l)
          m :: sr(rem)
        }
      }
}

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