Game of Life kata: branchless Grid

In my previous log I have tackled Conway's Game Of Life.

In conclusion, I have stated that we could implement the full kata (i.e. even the grid) without branches (no if/else/case/pattern matching, so no sum types).

As a reminder, we ended up with this:

-- ...
describe "Reproduction (three live neighbours)" $ do
  it "Any dead cell with exactly three live neighbours becomes a live cell" $
    reproduction.next Dead `shouldBe` Alive
  it "Any live cell with three live neighbours lives on to the next generation" $
    reproduction.next Alive `shouldBe` Alive
describe "Overpopulation (more than three live neighbours)" $ do
  it "Any live cell with more than three live neighbours dies" $
    overpopulation.next Alive `shouldBe` Dead
describe "Survive (two live neighbours)" $ do
  it "Any Dead cell with fewer than three live neighbours stays dead on to the next generation" $
    survive.next Dead `shouldBe` Dead
  it "Any dead cell with exactly three live neighbours becomes a live cell" $
    survive.next Alive `shouldBe` Alive
describe "Underpopulation (zero or one live neighbours)" $ do
  it "Any Dead cell with fewer than three live neighbours stays dead on to the next generation" $
    underpopulation.next Dead `shouldBe` Dead
  it "Any live cell with fewer than two live neighbours dies" $
    underpopulation.next Alive `shouldBe` Dead
-- ...

data Cell
  = Alive
  | Dead
  deriving stock (Eq, Show)

newtype Neighbours
  = Neighbours { getNeighbours :: Int }
  deriving newtype (Eq, Ord, Show, Num)

newtype Neighbourhood
  = Neighbourhood { next :: Cell -> Cell }

reproduction :: Neighbourhood
reproduction = Neighbourhood $ const Alive

overpopulation :: Neighbourhood
overpopulation = Neighbourhood $ const Dead

survive :: Neighbourhood
survive = Neighbourhood id

underpopulation :: Neighbourhood
underpopulation = Neighbourhood $ const Dead

Let's start with the neighbourhood selection.

If I have to draft things up, I would come up with an implementation like this:

neighbourhood :: Int -> Neighbourhood
neighbourhood =
  \case
    0 -> underpopulation
    1 -> underpopulation
    2 -> survive
    3 -> reproduction
    _ -> overpopulation

But there are two problems:

1. There is branching
2. I have no direct way to test it directly (I could test the function with its behavior, but it's not handy)

Note: here is the limit of the kata, in production code I would keep the here-above implementation while testing the functions.

Let's tackle the testability issue through refactoring adding the neighbourhood name:

data Neighbourhood = Neighbourhood
  { name :: String
  , next :: Cell -> Cell
  }

reproduction :: Neighbourhood
reproduction =
  Neighbourhood
    { name = "reproduction"
    , next = const Alive
    }

overpopulation :: Neighbourhood
overpopulation =
  Neighbourhood
    { name = "overpopulation"
    , next = const Dead
    }

survive :: Neighbourhood
survive =
  Neighbourhood
    { name = "survive"
    , next = id
    }

underpopulation :: Neighbourhood
underpopulation =
  Neighbourhood
    { name = "underpopulation"
    , next = const Dead
    }

Then, instead of taking a number for the function, let's take a Cells list

-- ...
it "No alive neighbours should be 'underpopulation'" $
  (neighbourhood []).name `shouldBe` "underpopulation"
-- ...

neighbourhood :: [Cell] -> Neighbourhood
neighbourhood = const underpopulation

Then we need to test survive (2):

-- ...
it "Two alive neighbours should be 'survive'" $
  (neighbourhood [Alive, Alive]).name `shouldBe` "survive"
-- ...

data Neighbourhood = Neighbourhood
  { name :: String
  , next :: Cell -> Cell
  , nextNeighbourhood :: Neighbourhood
  }

underpopulation0 :: Neighbourhood
underpopulation0 =
  Neighbourhood
    { name = "underpopulation"
    , next = const Dead
    , nextNeighbourhood = underpopulation1
    }

underpopulation1 :: Neighbourhood
underpopulation1 =
  Neighbourhood
    { name = "underpopulation"
    , next = const Dead
    , nextNeighbourhood = survive
    }

survive :: Neighbourhood
survive =
  Neighbourhood
    { name = "survive"
    , next = id
    , nextNeighbourhood = reproduction
    }

reproduction :: Neighbourhood
reproduction =
  Neighbourhood
    { name = "reproduction"
    , next = const Alive
    , nextNeighbourhood = overpopulation
    }

overpopulation :: Neighbourhood
overpopulation =
  Neighbourhood
    { name = "overpopulation"
    , next = const Dead
    , nextNeighbourhood = overpopulation
    }

neighbourhood :: [Cell] -> Neighbourhood
neighbourhood = foldr (const (.nextNeighbourhood)) underpopulation0

So, a lot of things went on here:

• For each Neighbourhood, there is a nextNeighbourhood specified (which is the Neighbourhood when there is one more Alive Cell, it is what's called Church Encoding)
• I have split underpopulation into underpopulation0 and underpopulation1 to represent 0 and 1 Alive Cell

There's have few more coverage tests:

it "No alive neighbours should be 'underpopulation'" $
  (neighbourhood []).name `shouldBe` "underpopulation"
it "One alive neighbours should be 'underpopulation'" $
  (neighbourhood [Alive]).name `shouldBe` "underpopulation"
it "Two alive neighbours should be 'survive'" $
  (neighbourhood [Alive, Alive]).name `shouldBe` "survive"
it "Three alive neighbours should be 'reproduction'" $
  (neighbourhood [Alive, Alive, Alive]).name `shouldBe` "reproduction"
it "Four alive neighbours should be 'overpopulation'" $
  (neighbourhood [Alive, Alive, Alive, Alive]).name `shouldBe` "overpopulation"

Good, now, we are able to break things dealing with Dead Cell.

-- ...
it "Two alive neighbours and three dead should be 'survive'" $
  (neighbourhood [Dead, Alive, Dead, Alive, Dead]).name `shouldBe` "survive"
-- ...

neighbourhood :: [Cell] -> Neighbourhood
neighbourhood = foldr go underpopulation0
  where go =
          \case
            Alive -> (.nextNeighbourhood)
            Dead -> id

Here we are:

• On Dead Cell, keep the Neighbourhood
• On Alive Cell, go to the nextNeighbourhood

But, but, but, conditions are back with the parttern matching, we have to rework the Cell (using Church Encoding again):

newtype Cell = Cell (forall a. forall a. a -> a -> a)

instance Show Cell where
  show = runCell "Dead" "Alive"

instance Eq Cell where
  x == y = runCell (Left ()) (Right ()) x == runCell (Left ()) (Right ()) y

runCell :: a -> a -> Cell -> a
runCell d a (Cell f) = f d a

alive :: Cell
alive = Cell $ \_ a -> a

dead :: Cell
dead = Cell const

I have to admit that's the first time I have written Eq and Show instances for a function-based type (and I'm glad it went so well).

Finally we can rewrite neighbourhood:

neighbourhood :: [Cell] -> Neighbourhood
neighbourhood = foldr (runCell id (.nextNeighbourhood)) underpopulation0

Note: at this point I find the code way more brittle than a case/pattern matching. That's what happen when you have to parameters with the same type but which position is important (i.e. are not interchangeable without changing the output).

Let's quickly improve that:

newtype WhenDead a = WhenDead a

newtype WhenAlive a = WhenAlive a

runCell :: WhenDead a -> WhenAlive a -> Cell -> a
runCell (WhenDead d) (WhenAlive a) (Cell f) = f d a

neighbourhood :: [Cell] -> Neighbourhood
neighbourhood = foldr (runCell (WhenDead id) (WhenAlive (.nextNeighbourhood))) underpopulation0

A bit verbose, but less error prone.

We can now finally proceed to the Grid.

For simplicity reasons, instead of an infinite Grid, we'll have a finite one.

Then we'll start with a simple design:

• Grid will be opaque (meaning the constructor won't be exported, though it'll be a Map)
• Providing going back-and-worth Grid through a Set Pos for Alive Cell

For the tests, I have chosen the blinker on a 3x3 Grid for our test cases:

describe "Grid" $ do
  let verticalBlinkerCells = [Pos 1 0, Pos 1 1, Pos 1 2]
  it "Vertical blinker should become horizontal" $
    aliveCells (nextGrid $ mkGrid (Pos 2 2) verticalBlinkerCells)
      `shouldBe` [Pos 0 1, Pos 1 1, Pos 2 1]
  it "Vertical blinker should become vertical after two generations" $
    aliveCells (nextGrid $ nextGrid $ mkGrid (Pos 2 2) verticalBlinkerCells)
      `shouldBe` verticalBlinkerCells

Let's start with types:

data Pos = Pos
  { posX :: Int,
    posY :: Int
  }
  deriving stock (Eq, Ord, Show)

newtype Grid
  = Grid { getGrid :: Map.Map Pos Cell }
  deriving stock (Eq, Show)

Then the helpers:

mkGrid :: Pos -> Set.Set Pos -> Grid
mkGrid limits alives =
  Grid $
    Map.fromList
    [ let p = Pos x y in (p, if Set.member p alives then alive else dead)
      | x <- [0 .. limits.posX]
      , y <- [0 .. limits.posY]
    ]

aliveCells :: Grid -> Set.Set Pos
aliveCells =
  Set.fromList
  . mapMaybe (\(p, c) -> runCell (WhenDead Nothing) (WhenAlive $ Just p) c)
  . Map.toList
  . getGrid

And finally the function to evolve the Grid:

nextGrid :: Grid -> Grid
nextGrid (Grid grid) =
  Grid $
    Map.mapWithKey (\p -> (neighbourhood $ neighbours p).next) grid
  where neighbours (Pos x y) =
          [ Map.findWithDefault dead (Pos (x + dx) (y + dy)) grid
            | dx <- [(-1) .. 1]
            , dy <- [(-1) .. 1]
            , dx /= 0 || dy /= 0
          ]

And we're done.

Sure, it's not perfect, especially it will be slow on wide Grid with few Alive Cell, but I think it's acceptable for a code kata.