sizedgrid
A way of working with grids in Haskell with size encoded at the type level.
Quick tutorial
The core datatype of this library is Grid (cs :: '[k]) (a :: *)
. cs
is a type level list of coordinate types. We could use a single type level number here, but by using different types we can say what happened when we move outside the bounds of a grid. There are three different coordinate types provided.

Ordinal n
: An ordinal can be an integral number between 0 and n  1. As numbers outside the grid are not possible, this has the most restrictive API. One can convert between an Ordinal and a number of ordinalToNum and numToOrdinal.

HardWrap n
: Like Oridnal, HardWrap can only hold intergral numbers between 0 and n  1, but it allows a more permissive API by clamping values outside of its range. It is an instance of Semigroup
and Monoid
, where mempty
is 0 and <>
is addition.

Periodic n
: This is the most permissive. When a value is generated outside the given range, it wraps that around using modular arithmetic. Is is an instance of Semigroup
and Monoid
like HardWrap
, but also of AdditiveGroup
allowing negation.
HardWrap
and Periodic
are both instances of AffineSpace
, with their Diff
being Integer
. This means there are many occasions where one doesn't have to work directly with these values (which can be cumbersome) and can instead work with their differences as regular numbers.
The last type value of Grid
is the type of each element.
The other main type is Coord cs
, where cs
is, again, a type level list of coordinate types. For example, Coord '[Periodic 3, HardWrap 4]
is a coordinate in a 3 by 4 2D space. The different types (Periodic
and HardWrap
) tell how to handle combining theses different numbers. Coord cs
is an instance of Semigroup
, Monoid
and AdditiveGroup
as long as each of the coordinates is also an instance of that typeclass. Coord
is also an instance of of AffineSpace
, where Diff
is a ntuple, meaning we can pattern match and do all sorts of nice things.
For working directly with Coord
s, one can construct them with singleCoord
and appendCoord
and consume and update them with coordHead
and coordTail
. They are also instances of FieldN
from lens, allowing one to directly update or get a certain dimension.
There is a deliberately small number of functions that work over Grid
: we instead opt for using typeclasses to create the required functionality. Grid
is an instance of the following types (with some required constraints):
Functor
: Update all values in the grid with the same function
Applicative
: As the size of the grid is statically known, pure
just creates a grid with the same element at each point. <*>
combines the grids point wise.
Monad
: I'm not sure if there is much of a need for this, but an instance exists.
Foldable
: Combine each element of the grid
Traverse
: Apply an applicative function over the grid
IndexedFunctor
, IndexedFoldable
and IndexedTraversable
: Like Functor
, Foldable
and Traversable
, but with access to the position at each point. These are from the lens package
Distributive
: Like Traversable
, but the other way round. Allows us to put a functor inside the grid
Representable
: Grid cs a
is isomorphic Coord cs > a
, so we can tabulate
and index
to make this conversion
We also have a FocusedGrid
type, which is like Grid
but has a certain focused position. This means that we lose many instances, but we gain Comonad
and ComonadStore
.
When dealing with areas around Coord
s, we can use moorePoints
and vonNeumanPoints
to generate Moore and von Neuman neighbourhoods. Note that these include the center point.
We introduce two new typeclasses: IsCoord
and IsGrid
. IsGrid
has gridIndex
, which allows us to get a single element of the grid and lenses to convert between FocusedGrid
and Grid
. IsCoord
has CoordSized
, which is the size of the coord and an iso to convert between Ordinal
and the Coord
.
Example  Game of Life
As is traditional for anything with grids and comonads in Haskell, we can reimplement Conway's Game of Life.
This is a literate Haskell file, so we start by turning on some language extensions, importing our library and some other utilities.
{# LANGUAGE MultiWayIf #}
{# LANGUAGE TypeFamilies #}
{# LANGUAGE FlexibleContexts #}
{# LANGUAGE TypeOperators #}
{# LANGUAGE MonoLocalBinds #}
{# LANGUAGE DataKinds #}
import SizedGrid
import Control.Comonad
import Control.Lens
import Control.Comonad.Store
import Data.AdditiveGroup
import Data.AffineSpace
import Data.Distributive
import Data.Functor.Rep
import Data.Semigroup (Semigroup(..))
import GHC.TypeLits
import qualified GHC.TypeLits as GHC
import System.Console.ANSI
We create a datatype for alive or dead.
data TileState = Alive  Dead deriving (Eq,Show)
We encode the rules of the game via a step function.
type Rule = TileState > [TileState] > TileState
gameOfLife :: Rule
gameOfLife here neigh =
let aliveNeigh = length $ filter (== Alive) neigh
in if  here == Alive && aliveNeigh `elem` [2,3] > Alive
 here == Dead && aliveNeigh == 3 > Alive
 otherwise > Dead
We can then write a function to apply this to every point in a grid.
applyRule ::
( All IsCoordLifted cs
, All Monoid cs
, All Semigroup cs
, All AffineSpace cs
, All Eq cs
, AllDiffSame Integer cs
, AllSizedKnown cs
, IsGrid cs (grid cs)
)
=> Rule
> grid cs TileState
> grid cs TileState
applyRule rule = over asFocusedGrid $
extend $ \fg > rule (extract fg) $ map (\p > peek p fg) $
filter (/= pos fg) $ moorePoints (1 :: Integer) $ pos fg
We can create a simple drawing function to display it to the screen.
displayTileState :: TileState > Char
displayTileState Alive = '#'
displayTileState Dead = '.'
displayGrid :: (KnownNat (x GHC.* y), KnownNat x, KnownNat y) =>
Grid '[f x, g y] TileState > String
displayGrid = unlines . collapseGrid . fmap displayTileState
Let's create a glider, and watch it move!
glider ::
( KnownNat (CoordNat x GHC.* CoordNat y)
, Semigroup x
, Semigroup y
, Monoid x
, Monoid y
, IsCoordLifted x
, IsCoordLifted y
, AffineSpace x
, AffineSpace y
, Diff x ~ Integer
, Diff y ~ Integer
)
=> Coord '[x,y]
> Grid '[x,y] TileState
glider offset = pure Dead
& gridIndex (offset .+^ (0,1)) .~ Alive
& gridIndex (offset .+^ (1,0)) .~ Alive
& gridIndex (offset .+^ (1,1)) .~ Alive
& gridIndex (offset .+^ (0,1)) .~ Alive
& gridIndex (offset .+^ (1,1)) .~ Alive
We can now make our glider run!
run =
let start :: Grid '[Periodic 10, Periodic 10] TileState
start = glider (mempty .+^ (3,3))
doStep grid = do
clearScreen
putStrLn $ displayGrid grid
_ < getLine
doStep $ applyRule gameOfLife grid
in doStep start
main = return ()