Eugene Zhulenev

Working on a Tensorflow at Google Brain

Type-Level Instant Insanity in Scala

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This post is a Scala version of Haskell Type-Level Instant Insanity by Conrad Parker

This post shows an implementation of Instant Insanity puzzle game at compile time, using powerful Scala type system. This post is based on amazing article by Conrad Parker in the Monad Reader Issue 8. Original article is around 20 pages long, this post is much more concise version of it. Original article is very well written and easy to understand, this post should help with jumping from Scala to Haskell code for people who are not familiar with Haskell language.

Textbook Implementation

Instant Insanity puzzle formulated as:

It consists of four cubes, with faces coloured blue, green, red or white. The problem is to arrange the cubes in a vertical pile such that each visible column of faces contains four distinct colours.

“Classic” solution in scala can be found here, this solution stacks the cubes one at a time, trying each possible orientation of each cube.

I’m going to show how to translate this solution into Scala Type System.

Type-Level Implementation

When I say “Type-Level Implementation”, I mean that I’m going to find a solution to the problem without creating any “values”, just operating with types, and I’ll do it using Scala compiler only, without running any code.

⊥ (Bottom)

I’m working in the type system, so I don’t need any value for any of the variables, just the type. In scala there is one special type, that is subtype of every other type - Nothing. There exist no instances of this type, but that’s ok because I don’t need one.

I’m introducing two functions that will make code look closer to Haskell version, and will save me some typing.

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def undefined[T]: T = ???
def ⊥[T]: T = undefined

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scala> :type ⊥[Int]
Int

scala> :type ⊥[Seq[String]]
Seq[String]

Simple Types

There are four possible colors. Rather then encoding them as values of type Char, I’m introducing new types.

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trait R   // Red
trait G   // Green
trait B   // Blue
trait W   // White

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scala> :type ⊥[R]
R

Parametrized Types

A cube is a thing that can have six faces. In Scala Type System, I use the keyword trait to introduce such a thing:

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trait Cube[u, f, r, b, l, d]

I can’t get a type of a Cube because there is not such thing, Cube exist only when it’s applied to type parameters, namely u, f, r, b, l, d. Applying concrete types to Cube will create concrete result type. One way to think about Cube is that it’s like a function, but at the type level.

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scala> :type ⊥[Cube]
<console>:17: error: trait Cube takes type parameters
       ⊥[Cube]

scala> :type ⊥[Cube[R, R, R, R, R, R]] // a red cube
Cube[R,R,R,R,R,R]

Type Aliases

Now I can define the actual cubes in our puzzle as results of applying Cube function to concrete types.

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type CubeRed = Cube[R, R, R, R, R, R]
type CubeBlue = Cube[B, B, B, B, B, B]

type Cube1 = Cube[B, G, W, G, B, R]
type Cube2 = Cube[W, G, B, W, R, R]
type Cube3 = Cube[G, W, R, B, R, R]
type Cube4 = Cube[B, R, G, G, W, W]

Transformation Functions

Now I need to encode following three functions from textbook solution at a type-level.

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// Rotate a cube 90 degrees over its Z-axis, leaving up and down in place.
def rot: Cube => Cube = { case Seq(u, f, r, b, l, d) => Seq(u, r, b, l, f, d) }

// Twist a cube around the axis running from the upper-front-right
// corner to the back-left-down corner.
def twist: Cube => Cube = { case Seq(u, f, r, b, l, d) => Seq(f, r, u, l, d, b) }

// Exchange up and down, front and left, back and right.
def flip: Cube => Cube = { case Seq(u, f, r, b, l, d) => Seq(d, l, b, r, f, u) }

I’m going to group all of them into single Transform trait. Implicit transform function can generate an instance of Transform for any type u, f, r, b, l, d. I’m not providing any implementation for any of them, because I never going to call them.

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trait Transform[u, f, r, b, l, d] {
  def rot: Cube[u, f, r, b, l, d] => Cube[u, r, b, l, f, d]
  def twist: Cube[u, f, r, b, l, d] => Cube[f, r, u, l, d, b]
  def flip: Cube[u, f, r, b, l, d] => Cube[d, l, b, r, f, u]
}

implicit def transform[u, f, r, b, l, d] = new Transform[u, f, r, b, l, d] {
  def rot: (Cube[u, f, r, b, l, d]) => Cube[u, r, b, l, f, d] = ⊥
  def twist: (Cube[u, f, r, b, l, d]) => Cube[f, r, u, l, d, b] = ⊥
  def flip: (Cube[u, f, r, b, l, d]) => Cube[d, l, b, r, f, u] = ⊥
}

def rot[u, f, r, b, l, d](cube: Cube[u, f, r, b, l, d])
    (implicit t: Transform[u, f, r, b, l, d]) = t.rot(cube)
def twist[u, f, r, b, l, d](cube: Cube[u, f, r, b, l, d])
    (implicit t: Transform[u, f, r, b, l, d]) = t.twist(cube)
def flip[u, f, r, b, l, d](cube: Cube[u, f, r, b, l, d])
    (implicit t: Transform[u, f, r, b, l, d]) = t.flip(cube)

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scala> :type rot(⊥[Cube1])
Cube[B,W,G,B,G,R]

scala> :type twist(flip(rot(⊥[Cube1])))
Cube[G,B,R,W,B,G]

Type-Level Boolean Algebra

So far we’ve seen how to construct simple types, and perform type transformations of one parametrized type into a differently parametrized type.

For solving a puzzle I’ll need some rudimentary boolean algebra at a type-level:

  • encode true/false in types
  • provide boolean and operator
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trait True
trait False

trait And[l, r, o]

implicit object ttt extends And[True, True, True]
implicit object tff extends And[True, False, False]
implicit object ftf extends And[False, True, False]
implicit object fff extends And[False, False, False]

def and[l, r, o](l: l, r: r)(implicit and: And[l, r, o]): o = ⊥

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scala> :type and(⊥[True], ⊥[False])
False

scala> :type and(⊥[True], ⊥[True])
True

Type-Level Lists

Lists at a type-level can be defined as following atoms:

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trait Nil
trait :::[x, xs]

To make lists useful I’ll need concatenate function, that should be also encoded at a type-level.

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trait ListConcat[l1, l2, l]

implicit def nilConcat[l]: ListConcat[Nil, l, l] = ⊥
implicit def notNilConcat[x, xs, ys, zs](implicit
  lc: ListConcat[xs, ys, zs]
): ListConcat[x ::: xs, ys, x ::: zs] = ⊥

def listConcat[xs, ys, zs](l1: xs, l2: ys)(implicit
  lc: ListConcat[xs, ys, zs]
): zs = ⊥

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scala> :type listConcat(⊥[R ::: Nil], ⊥[G ::: W ::: Nil])
:::[R,:::[G,:::[W,Nil]]]

As you can see now I can concatenate two lists at type-level: [R] concat [G, W] yields [R, G, W].

Applyable Type Functions

For this puzzle I need to be able to do things like flip each of the cubes in a list, which sounds exactly like map at the type level.

First step is abstracting application of a type-level function, so I introduce Apply and supported operations:

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trait Rotation
trait Twist
trait Flip

trait Apply[f, a, b]

implicit def apRotation[u, f, r, b, l, d]
  : Apply[Rotation, Cube[u, f, r, b, l, d], Cube[u, r, b, l, f, d]] = ⊥

implicit def apTwist[u, f, r, b, l, d]
  : Apply[Twist, Cube[u, f, r, b, l, d], Cube[f, r, u, l, d, b]] = ⊥

implicit def apFlip[u, f, r, b, l, d]
  : Apply[Flip, Cube[u, f, r, b, l, d], Cube[d, l, b, r, f, u]] = ⊥

def ap[t, u, f, r, b, l, d, o](r: t, c1: Cube[u, f, r, b, l, d])(implicit
  ap: Apply[t, Cube[u, f, r, b, l, d], o]
): o = ⊥

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scala> :type ap(⊥[Rotation], ⊥[Cube1])
Cube[B,W,G,B,G,R]

Map and Filter

Now I can create a function that recurses over a list and Applys another function f to each element. This is type-level equivalent of the map function.

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trait Map[f, xs, zs]

implicit def mapNil[f]: Map[f, Nil, Nil] = ⊥
implicit def mapCons[f, x, z, xs, zs](implicit
  ap: Apply[f, x, z],
  map: Map[f, xs, zs]
): Map[f, x ::: xs, z ::: zs] = ⊥

def map[f, xs, zs](f: f, xs: xs)(implicit
  map: Map[f, xs, zs]
): zs = ⊥

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scala> :type map(⊥[Flip], ⊥[Cube1 ::: Cube2 ::: Nil])
:::[Cube[R,B,G,W,G,B],:::[Cube[R,R,W,B,G,W],Nil]]

Filter function is similar to Map:

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trait Filter[f, xs, zs]

implicit def filterNil[f]: Filter[f, Nil, Nil] = ⊥
implicit def filterCons[f, x, b, xs, ys, zs](implicit
  ap: Apply[f, x, b],
  f: Filter[f, xs, ys],
  apn: AppendIf[b, x, ys, zs]
): Filter[f, x ::: xs, zs] = ⊥

Filter introduced new constraint, AppendIf, which takes a boolean values b, a value x and a list ys. The given values x is appended to ys only if b is True, otherwise ys is returned unaltered:

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trait AppendIf[b, x, ys, zs]

implicit def appendIfTrue[x, ys]: AppendIf[True, x, ys, x ::: ys] = ⊥
implicit def appendIfFalse[x, ys]: AppendIf[False, x, ys, ys] = ⊥

def append[b, x, ys, zs](b: b, x: x, ys: ys)(implicit
  apn: AppendIf[b, x, ys, zs]
): zs = ⊥

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scala> :type append(⊥[True], ⊥[R], ⊥[G ::: W ::: Nil])
:::[R,:::[G,:::[W,Nil]]]

scala> :type append(⊥[False], ⊥[R], ⊥[G ::: W ::: Nil])
:::[G,:::[W,Nil]]

Sequence Comprehensions

Unfortunately sequence comprehensions can’t be directly mimiced in Scala Type System, but we can translate the meaning of a given sequence comprehension using the type-level list functions.

For example, building a list of the possible orientations of a cube involves appending a list of possible applications of flip, so we will need to be able to map over a list and append the original list. Original sequence comprehension was:

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def orientations: Cube => Seq[Cube] = { c: Cube =>
    for {
      ...
      c3 <- Seq(c2, flip(c2))
    } yield c3
  }

I create MapAppend class in order to compose Map and ListConcat:

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trait MapAppend[f, xs, zs]

implicit def mapAppendNil[f]: MapAppend[f, Nil, Nil] = ⊥
implicit def mapAppendCons[f, xs, ys, zs](implicit
  m: Map[f, xs, ys],
  lc: ListConcat[xs, ys, zs]
): MapAppend[f, xs, zs] = ⊥

Further, I’ll need to be able to do the same twice for twist and three times for rot:

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trait MapAppend2[f, xs, zs]

implicit def mapAppend2Nil[f]: MapAppend2[f, Nil, Nil] = ⊥
implicit def mapAppend2Cons[f, xs, ys, _ys, zs](implicit
  m: Map[f, xs, ys],
  ma: MapAppend[f, ys, _ys],
  lc: ListConcat[xs, _ys, zs]
): MapAppend2[f, xs, zs] = ⊥

trait MapAppend3[f, xs, zs]

implicit def mapAppend3Nil[f]: MapAppend3[f, Nil, Nil] = ⊥
implicit def mapAppend3Cons[f, xs, ys, _ys, zs](implicit
  m: Map[f, xs, ys],
  ma2: MapAppend2[f, ys, _ys],
  lc: ListConcat[xs, _ys, zs]
): MapAppend3[f, xs, zs] = ⊥

Orientation

The full sequence comprehension for generating all possible orientations of a cube build upon all combinations of rot, twist and flip:

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def orientations: Cube => Seq[Cube] = { c: Cube =>
  for {
    c1 <- Seq(c, rot(c), rot(rot(c)), rot(rot(rot(c))))
    c2 <- Seq(c1, twist(c1), twist(twist(c1)))
    c3 <- Seq(c2, flip(c2))
  } yield c3
}

I will implement Orientation as an Applyable type function. It’s defined in terms of applications of Rotation, Twist and Flip, invoked via various MapAppend functions:

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trait Orientations

implicit def orientations[c, fs, ts, zs](implicit
  ma: MapAppend[Flip, c ::: Nil, fs],
  ma2: MapAppend2[Twist, fs, ts],
  ma3: MapAppend3[Rotation, ts, zs]
): Apply[Orientations, c, zs] = ⊥

For any Cube this function generates the 24 possible orientations:

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scala> :type ap(⊥[Orientations], ⊥[Cube1])
:::[Cube[B,G,W,G,B,R],:::[Cube[R,B,G,W,G,B],
:::[Cube[G,W,B,B,R,G],:::[Cube[B,G,R,G,B,W],
:::[Cube[W,B,G,R,G,B],:::[Cube[G,R,B,B,W,G],
:::[Cube[B,W,G,B,G,R],:::[Cube[R,G,W,G,B,B],
:::[Cube[G,B,B,R,W,G],:::[Cube[B,R,G,B,G,W],
:::[Cube[W,G,R,G,B,B],:::[Cube[G,B,B,W,R,G],
:::[Cube[B,G,B,G,W,R],:::[Cube[R,W,G,B,G,B],
:::[Cube[G,B,R,W,B,G],:::[Cube[B,G,B,G,R,W],
:::[Cube[W,R,G,B,G,B],:::[Cube[G,B,W,R,B,G],
:::[Cube[B,B,G,W,G,R],:::[Cube[R,G,B,G,W,B],
:::[Cube[G,R,W,B,B,G],:::[Cube[B,B,G,R,G,W],
:::[Cube[W,G,B,G,R,B],:::[Cube[G,W,R,B,B,G],
Nil]]]]]]]]]]]]]]]]]]]]]]]]

Stacking Cubes

Given two cubes Cube[u1, f1, r1, b1, l1, d1] and Cube[u2, f2, r2, b2, l2, d2], I want to check that none of the corresponding visible faces are the same color: the front sides f1 and f2 are not equal, and the right sides r1 and r2 are not equal, and so on.

In order to do this, it’s required to define not equal relation for all four colors. Given two cubes I can apply this relations to each pair of visible faces to get four boolean values. To check that all of these are True, I will construct a list of those values and then write generic list function to check if all elements of a list are True.

Not Equal

I’m instantiating NE instance for all color combinations.

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trait NE[x, y, b]

implicit object neRR extends NE[R, R, False]
implicit object neRG extends NE[R, G, True]
implicit object neRB extends NE[R, B, True]
implicit object neRW extends NE[R, W, True]

implicit object neGR extends NE[G, R, True]
implicit object neGG extends NE[G, G, False]
implicit object neGB extends NE[G, B, True]
implicit object neGW extends NE[G, W, True]

implicit object neBR extends NE[B, R, True]
implicit object neBG extends NE[B, G, True]
implicit object neBB extends NE[B, B, False]
implicit object neBW extends NE[B, W, True]

implicit object neWR extends NE[W, R, True]
implicit object neWG extends NE[W, G, True]
implicit object neWB extends NE[W, B, True]
implicit object neWW extends NE[W, W, False]

All

Now, I define a function all to check if all elements of a list are True.

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trait All[l, b]

implicit def allNil: All[Nil, True] = ⊥
implicit def allFalse[xs]: All[False ::: xs, False] = ⊥
implicit def allTrue[b, xs](implicit all: All[xs, b]): All[True ::: xs, b] = ⊥

def all[b, xs](l: xs)(implicit all: All[xs, b]): b = ⊥

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scala> :type all(⊥[Nil])
True

scala> :type all(⊥[False ::: Nil])
False

scala> :type all(⊥[True ::: False ::: Nil])
False

scala> :type all(⊥[True ::: True ::: Nil])
True

Compatible

Now I can write the compatibility check in the Scala Type System, that corresponds original compatible funcation:

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// Compute which faces of a cube are visible when placed in a pile.
def visible: Cube => Seq[Char] = { case Seq(u, f, r, b, l, d) => Seq(f, r, b, l) }

// Two cubes are compatible if they have different colours on every
// visible face.
def compatible: (Cube, Cube) => Boolean = {
  case (c1, c2) => visible(c1).zip(visible(c2)).forall { case (v1, v2) => v1 != v2 }
}

I introduce a new Compatible class, it should check that no corresponding visible faces are the same color. It does validation by evaluating NE relationship for each pair of corresponding visible faces. 

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trait Compatible[c1, c2, b]

implicit def compatibleInstance[f1, f2, bF, r1, r2, bR, b1, b2, bB, l1, l2, bL, u1, u2, d1, d2, b](implicit
  ne1: NE[f1, f2, bF],
  ne2: NE[r1, r2, bR],
  ne3: NE[b1, b2, bB],
  ne4: NE[l1, l2, bL],
  all: All[bF ::: bR ::: bB ::: bL ::: Nil, b]
): Compatible[Cube[u1, f1, r1, b1, l1, d1], Cube[u2, f2, r2, b2, l2, d2], b] = ⊥

def compatible[c1, c2, b](c1: c1, c2: c2)(implicit
  c: Compatible[c1, c2, b]
): b = ⊥

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scala> :type compatible(⊥[Cube[R, R, R, R, R, R]], ⊥[Cube[B, B, B, B, B, B]])
True

scala> :type compatible(⊥[Cube[R, R, G, R, R, R]], ⊥[Cube[B, B, G, B, B, B]])
False

Allowed

The above Compatible class checks a cube for compatibility with another single cube. In the puzzle, a cube needs to be compatible with all the other cubes in the pile.

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// Determine whether a cube can be added to pile of cubes, without
// invalidating the solution.
def allowed: (Cube, Seq[Cube]) => Boolean = {
  case (c, cs) => cs.forall(compatible(_, c))
}

Allowed class generalize Compatible over list.

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trait Allowed[c, cs, b]

implicit def allowedNil[c]: Allowed[c, Nil, True] = ⊥
implicit def allowedCons[c, y, b1, ys, b2, b](implicit
  c: Compatible[c, y, b1],
  allowed: Allowed[c, ys, b2],
  and: And[b1, b2, b]
): Allowed[c, y ::: ys, b] = ⊥

def allowed[c, cs, b](c: c, cs: cs)(implicit a: Allowed[c, cs, b]): b = ⊥

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scala> :type allowed(⊥[CubeRed], ⊥[CubeBlue ::: CubeRed ::: Nil])
False

Solution

Now I’m ready to implement solutions:

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// Return a list of all ways of orienting each cube such that no side of
// the pile has two faces the same.
def solutions: Seq[Cube] => Seq[Seq[Cube]] = {
  case Nil => Seq(Nil)
  case c :: cs => for {
    _cs <- solutions(cs)
    _c <- orientations(c)
    if allowed(_c, _cs)
  } yield _c +: _cs
}

I will create a corresponding class Sultions, which takes a list of Cube as input, and returs a list of possible solutions, where each solution is a list of Cube in allowed orientations.

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class Solutions[cs, ss]

implicit def solutionsNil: Solutions[Nil, Nil ::: Nil] = ⊥
implicit def solutionsCons[cs, sols, c, os, zs](implicit
  s: Solutions[cs, sols],
  ap: Apply[Orientations, c, os],
  ac: AllowedCombinations[os, sols, zs]
): Solutions[c ::: cs, zs] = ⊥

The AllowedCombinations class recurses across the solutions so far, checking each against the given orientation.

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trait AllowedCombinations[os, sols, zs]
implicit def allowedCombinationsNil[os]: AllowedCombinations[os, Nil, Nil] = ⊥
implicit def allowedCombinationsCons[os, sols, as, s, bs, zs](implicit
  ac: AllowedCombinations[os, sols, as],
  mo: MatchingOrientations[os, s, bs],
  lc: ListConcat[as, bs, zs]
): AllowedCombinations[os, s ::: sols, zs] = ⊥

Finally MatchingOrientations class recurses across the orientations of the new cube, checking each against a particular solution sol.

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trait MatchingOrientations[os, sols, zs]
implicit def matchingOrientationsNil[sol]: MatchingOrientations[Nil, sol, Nil] = ⊥
implicit def matchingOrientationsCons[os, sol, as, o, b, zs](implicit
  mo: MatchingOrientations[os, sol, as],
  a: Allowed[o, sol, b],
  apn: AppendIf[b, o ::: sol, as, zs]
): MatchingOrientations[o ::: os, sol, zs] = ⊥

If orientation is allowed, then the combination o is added to the existing solution sol, by forming the type o ::: sol.

Finally I can solve the puzzle for given cubes:

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type Cubes = (Cube1 ::: Cube2 ::: Cube3 ::: Cube4 ::: Nil)

def solutions[cs, ss](cs: cs)(implicit sol: Solutions[cs, ss]): ss = ⊥

Unfortunately I still was not able to compute solution for four cubes, it’s still running (almost 24 hours). I will update this post when it’s done. But without loss of generality I can compute all possible ways to arrange a list with one single cube.

Computing solution for two cubes takes reasonable 10 minutes, but it generates so many possible arrangements, and final type name is so long, that it breaks JVM class file size limitation.

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scala> :type solutions(⊥[Cubes])
// It's already running for 24 hours

scala> :type solutions(⊥[Cube1 ::: Nil])

    (Cube[B,G,W,G,B,R] ::: Nil)
::: (Cube[R,B,G,W,G,B] ::: Nil)
::: (Cube[G,W,B,B,R,G] ::: Nil)
::: (Cube[B,G,R,G,B,W] ::: Nil)
::: (Cube[W,B,G,R,G,B] ::: Nil)
::: (Cube[G,R,B,B,W,G] ::: Nil)
::: (Cube[B,W,G,B,G,R] ::: Nil)
::: (Cube[R,G,W,G,B,B] ::: Nil)
::: (Cube[G,B,B,R,W,G] ::: Nil)
::: (Cube[B,R,G,B,G,W] ::: Nil)
::: (Cube[W,G,R,G,B,B] ::: Nil)
::: (Cube[G,B,B,W,R,G] ::: Nil)
::: (Cube[B,G,B,G,W,R] ::: Nil)
::: (Cube[R,W,G,B,G,B] ::: Nil)
::: (Cube[G,B,R,W,B,G] ::: Nil)
::: (Cube[B,G,B,G,R,W] ::: Nil)
::: (Cube[W,R,G,B,G,B] ::: Nil)
::: (Cube[G,B,W,R,B,G] ::: Nil)
::: (Cube[B,B,G,W,G,R] ::: Nil)
::: (Cube[R,G,B,G,W,B] ::: Nil)
::: (Cube[G,R,W,B,B,G] ::: Nil)
::: (Cube[B,B,G,R,G,W] ::: Nil)
::: (Cube[W,G,B,G,R,B] ::: Nil)
::: (Cube[G,W,R,B,B,G] ::: Nil)
::: Nil

For comparison, here is the solution generated by the pure Scala version:

[GBWRBG, WGBWRR, RWRBGR, BRGGWW]
[GBRWBG, RRWBGW, RGBRWR, WWGGRB]
[GWRBBG, WBWRGR, RRBGWR, BGGWRW]
[GBBRWG, RGRWBW, RWGBRR, WRWGGB]
[GRBBWG, WWRGBR, RBGWRR, BGWRGW]
[GWBBRG, RBGRWW, RRWGBR, WGRWGB]
[GBBWRG, WRGBWR, RGWRBR, BWRGGW]
[GRWBBG, RWBGRW, RBRWGR, WGGRWB]

Conclusion

We’ve seen how to use Scala Type System as a programming language to solve a given problem, and apparently it’s as powerful as Haskell Type System.

Solving this kind of puzzles using type system is not very practical, it took me more then 24 hours to get a solution, but it shows how expressing type system can be.

Full code for Type-Level Instant Insanity is on Github

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