functional programming, permaculture, math

Bounded Space Automata

Rule 110

After reading Chris Penner's blog post on Conway's Game Of Life Using Representable and Comonads, I decided to implement a similar solution for the 1 dimensional automata Rule 110. Being such a simple algorithm I thought the implementation would be trivial. However, it turns out that working with arbitarily bounded spaces in the Store Comonad requires a clever use of dependent types to apply memoizaition as described in Chris' blog.

This blog post walks through my experience attempting to use Store, Representable in Haskell to implement Rule 110, difficulty finding a suitable Representable instance, and then a pivot to Idris where I was able to solve the problem using the finite set type Fin n.

The Rule 110 algorithm takes a vector of boolean values and applies a simple set of rules for transforming each index based upon its neighbors. You can think of it as a 1 dimensional variant of Conway's Game of Life, but with the twist that the first and last element of the vector are considered neighbors. In other words, the automata in Rule 110 exists on a circle rather then a line segment.

Here is the rubric for generating the new state for a given element:

Current Pattern 111 110 101 100 011 010 001 000
New State 0 1 1 0 1 1 1 0

This post assumes no prior knowledge of Comonads, Representable Functors, Fin n. The first few sections are intended to introduce these concepts by focusing on the actual implementations of their APIs.

The Store Type and its API

The Store type can be thought of as a way to query some indexable state along with a cursor that queries into the store at some index.

data Store s a = Store (s -> a) s

State is held implicitly in your query function and, as we shall see later, can be transformed through composition.

As an example, we can construct a Store whose implicit state is an infinite list of Booleans which we index into with an Integer:

initialStore :: Store Int Bool
initialStore = Store query 0
      query :: Int -> Bool
      query i = (cycle [True, False]) !! abs i

The Store API then allows us to do things like query arbitrary indices, shift around our cursor, and through its Functor and Comonad instances to perform transformations on the implicit state held inside the query function.

Lets walk through the implementation of this API to gain a better intuition.

First we need a getter to lookup the current index:

pos :: Store s a -> s
pos (Store query s) = s

Next, we want a way to query the Store:

peek :: s -> Store s a -> a
peek s (Store query _) = query s

peeks :: (s -> s) -> Store s a -> a
peeks f (Store query s) = query (f s)

peek disregards the current cursor and instead applies a new cursor to the store's query function. peeks uses a function s -> s to modify the current cursor and then query the store with the new cursor.

We would also like a way to set and shift our cursor index:

seek :: s -> Store s a -> Store s a
seek s (Store query _) = Store query s

seeks :: (s -> s) -> Store s a -> Store s a
seeks f (Store query s) = Store query (f s)

This covers are basic getters and setters, but what if we wanted a way to query multiple indices into our store?

We might write this function:

peekMany :: [s] -> Store s a -> [a]
peekMany xs store = fmap (flip peek store) xs

This function looks pretty good, but notice we are using fmap. We can take advantage of polymorphism here allow use use of any Functor:

peekFunctor :: Functor f => f s -> Store s a -> f a
peekFunctor fs store = fmap (flip peek store) fs

Now we can use any Functor we want, but notice how we aren't using our cursor at all. This is a clue that we might be able to generalize this function further.

It would be great if we could modify peekFunctor to use our cursor value without losing the current behavior of the function.

If we replace the f s parameter with a function s -> f s then we could apply the current cursor to our function and generate an f s to generate our f a output. Then if we want the precise behavior of peekFunctor we can simply apply const (x :: f s) and ignore the current cursor!

experiment :: Functor f => (s -> f s) -> Store s a -> f a
experiment f store = fmap (flip peek store) (f (pos store))

To recap our Store API we have:

pos :: Store s a -> a
peek :: s -> Store s a -> a
peeks :: (s -> s) Store s a -> a
seek :: s -> Store s a -> a
seeks :: (s -> s) Store s a -> a
experiment :: Functor f => (s -> f s) -> Store s a -> f a

Transformation of State

The state of the system is stored implicitely inside the s -> a query function rather then in some data structure. Because of this, the only way to modify the state would be to modify the query function itself.

We can do this by first querying the current store to get an a value and then apply some a -> b function to the value to produce a modified version of the state at that index.

We can do this by pattern matching on our Store then composing an a -> b function with our query function:

updateStoreState :: (a -> b) -> Store s a -> Store s b
updateStoreState f (Store query s) = Store (f . query) s

updateStoreState allows you to modify the query result for all possible values inside the Store's implicit state. This signature shoud look familiar as it is fmap and Store is in fact a Functor.

instance Functor (Store s) where
    fmap :: (a -> b) -> Store s a -> Store s b
    fmap f (Store query s) = Store (f . query) s

Now we can model successive transformations of the state through applications of fmap. For example, using the Store Int Bool example from earlier, we might want to apply not :: Bool -> Bool to invert the state of our system:

initialStore :: Store Int Bool
initialStore = Store query 0
      query i = (cycle [True, False]) !! i

newState :: Store Int Bool
newState = fmap not initialStore

Using equational reasoning we can translate that into:

newState = fmap not initialStore
         = fmap not (Store query s)
         = Store (not . query) s

Using this technique to model 3 manipulations of a system shows how each modification to the implicit state builds up a larger composed query function:

newState :: (a -> b) -> (b -> c) -> (c -> d) -> Store s a -> Store s b
newState h g f store = fmap f (fmap g (fmap h store))
                     = fmap f (fmap g (fmap h (Store query s)))
                     = fmap f (fmap g (Store (h . query) s))
                     = fmap f (Store (g . h . query) s)
                     = Store (f . g . h . query) s

While elegant, modeling state transformations as function composition means that every time we query an index in a Store, we must recalculate every single previous transformation going back to the original Store query. Without caching these intermediate computations this will get very expensive.

Luckily, Chris Penner showed us a fantastic solution for this using Representable Functors. We will look at Representable shortly, but for now lets ignore the performance issue and focus on the tools we need to naively implement Rule 110.

Extending our API with Comonad

Not only is Store a Functor, it is also a Comonad.

Comonads are the dual of Monads. Where Monads introduce some effect to an argument via the form a -> m b, Comonads introduce a notion of querying a structure for data (co-effects):

-- Monadic
a -> m a
repeat :: a -> [a]
-- Comonadic
w a -> a
length :: [a] -> Int

The Comonad Typeclass has three functions:

class Functor w => Comonad w where
    extract   :: w a -> a
    duplicate :: w a -> w (w a)
    extend    :: (w a -> b) -> w a -> w b

Notice the inverse relation to return, join, and (=<<) from Monad:

extract :: w a -> a
return  :: a -> m a

duplicate :: w a     -> w (w a)
join      :: m (m a) -> m a

extend :: (w a -> b) -> w a -> w b
(=<<)  :: (a -> m b) -> m a -> m b

If a defining characteristic of Comonads in Haskell is to provide a mechanism for querying some structure for data, then we can almost definitionally say that Store is a Comonad. Store is literally a mechnanism for querying a structure to produce data!

instance Comonad (Store s) where
    extract :: Store s a -> a
    extract (Store query s) = query s

    extend :: (Store s a -> b) -> Store s a -> Store s b
    extend f (Store query s) = Store (\s' -> f (Store query s')) s

extract applies the current cursor to the query function and extend chains state transforming queries.

extract is fairly trivial in this case, but extend is a little trickier. It helps to think about in relation to our fmap implementation.

Like fmap, it uses a function to modify our implicit state, but where fmap composed a pure a -> b function with our query, extend creates a new query function by applying your entire store to a comonadic action. This allows us to bring into scope the entire current store when modifying particular points in the store.

extend is extremely powerful and allows us to do really interesting things like create windowing functions and perform kernel convolution. It allows us to modify every single individual points in "parallel" using the entire state as context.

One interesting example of extend is to perform a moving average on some time series data.

First we need a Store modeling time sequenced data. We will use Int for our Index and it will represent a single unit of time in a data stream. We want some fairly dynamic data source so I chose the Fibbonaci sequence. At each point in time (each index) we get the next Fibbonaci number.

fibStore :: Store Int Int
fibStore = store query 0
    query :: Int -> Int
    query 0 = 0
    query 1 = 1
    query n = query (n - 1) + query (n - 2)

Now, if we want to calculate a window starting from a given cursor want some way to query for the subsequent points in time. experiement will work perfectly here:

window :: Store Int a -> [a]
window store = experiment (\s -> [s..s+10]) store

I chose to fix the window at 10 units of time arbitrarily.

Now notice the shape of window is Store Int a -> [a]. That looks a lot like the comonadic action for extend: Store s a -> b. We can use extend to apply window over the enter store:

windowedStore :: Store Int [Int]
windowedStore = extend window fibStore

Now if we peek at any index in the store we see a window of the subsequent Fibonacci numbers!

λ> peek 4 $ extend window fibStore

A First Attempt at an Algorithm

With the tools now available to us, we can make a first attempt at our Rule 110 algorithm.

The first step is to load our initial state into the Store:

type Index = Int

initializeStore :: [Bool] -> Store Index Bool
initializeStore xs = Store query 0
      query :: Index -> Bool
      query i = xs !! i

We are modeling our initial state as a list and using an unsafe list lookup function for our query. This isn't ideal, but we are just trying to put together a rough draft.

Next we need a way to query an index and its neighbors. Just like our windowing function, we can use experiment here.

neighbors :: Store Index Bool -> [Bool]
neighbors = experiment _lookupIndices

_lookupIndices is a type hole we need to fill. By starting from neighborValues we let GHC tell us what shape we need for lookupIndices:

Found hole: _lookupIndices :: Index -> [Index]

We wan't something like this:

lookupIndices :: Int -> [Int]
lookupIndices s = [s-1, s, s+1]

But which accounts for the fact that our automata lives on a circle not a line. We need to be able to identify the first and last index and use that information to select the correct neighbors.

The first, simplest solution is to pass in the length of the list as a value:

type Index = Int
type Size = Int

lookupIndices :: Size -> Index -> [Index]
lookupIndices size i
 | i == 0       = [size - 1, 0, 1]
 | i == size -1 = [i - 1, i, 0]
 | otherwise    = [i - 1, i, i + 1]

neighbors :: Size -> Store Int Bool -> [Bool]
neighbors size = experiment (lookupIndices size)

With a mechanism to lookup the state for our index and its neighbors, we next need to use that information to calculate the next state of at our index. We can do this by casing on the output of neighborValues:

Now we need to case on the state of the neighbors and apply our rubric to determine the new state at our index:

newState :: Size -> Store Index Bool -> Bool
newState size store =
  case neighbors size store of
    [False, False, False] -> False
    [True, False, False]  -> False
    [True, True, True]    -> False
    _ -> True

Lastly, we need a way to apply this transformation to the entire store to create the next generation of the automata. extend to the rescue!

nextGen :: Size -> Store Index Bool -> Store Index Bool
nextGen size = extend (newState size)

Lets use equational reasoning to take a closer look at what happens when you call nextGen:

nextGen size store = extend (newState size) store
                   = extend (newState size) (Store query s)
                   = Store (\s' -> (newState size) (Store query s')) s

And calling nextGen twice:

nextGen size (nextGen size store) = extend (newState size) (extend (newState size) store)
                                  = extend (newState size) (extend (newState size) (Store query s))
                                  = extend (newState size) (Store (\s' -> (newState size) (Store query s')) s)
                                  = Store (\s'' -> (newState size) (Store (\s' -> (newState size) (Store query s')) s'')) s

Its a little hard to grok, but if you squint your eyes a bit you can see that we are building up our query function by chaining calls of newState on the store. So whenever you query an index, the composed query function will apply newState a bunch of times to your store.

The last step to a working implementation is a function to view a store as a list. This isn't really a part of the algorithm per se, but we do want a way to view our results!

viewStore :: Size -> Store Index Bool -> [Bool]
viewStore size store = experiment (const [0..size]) store

To run the simulation we can use a recursive function in IO to repeatedely print the result of viewStore and then call nextGen to update the state:

runSimulation :: Size -> Store Index Bool -> IO ()
runSimulation size store = do
    print $ viewStore size store
    runSimulation size $ nextGen size store

This implementation does work, but if you try running it you will see that it has major performance issue. Extending a new generation means growing a chain of calls to newState.

This ever growing query function has to be calculated in full every time you peek into an index. We do that for every index at every generation when we call runSimulation.

Its actually worse then that though! To calculate the new state at each index we also have to peek at its neighbors. So that means for every index we are repeating the same huge query 3 times!

What we need is a way to memoize all these calculations.

Representable Functors

A Functor f is representable if it has a corresponding type Rep f that indexes f completely. There must be a valid index into f for every value of Rep f and at the same time we must be able to construct a container where each element in the container is produced from its Rep f index.

Another more formal way of stating this is that there must exist an isomorphism between f a and Rep f -> a. This isomorphism is witnessed by the tabulate and index functions from the Representable typeclass:

class Functor f => Representable f where
  type Rep f :: *
  tabulate :: (Rep f -> a) -> f a
  index    :: f a -> (Rep f -> a)

It's hard to see exactly how this would be useful to us, but there is a clever trick we can play with Representable to get memoization 'for free.'

tabulate will takes some function that produces a values from Rep f values and then constructors a Representable f containing an a value for every possible Rep f value.

index lets you use a Rep f to query a Representable f for a values. One nice property of Representable is that if you have a lawful instance then index must be a safe function without the need for Maybe!

Lets try to come up with a valid instance of Representable.

The most obvious first choice for Functor is [], but what would we use for Rep f? Int doesn't work because you can't have a negative index into []. Nat almost works, but then what happes if the list is empty? Unfortunately there is no Representable instance for [].

NonEmpty solves those problems, but it is still possible to have a NonEmpty without an element for every Nat.

What if our Functor was an infinite stream?

newtype Stream a = Cons a (Strema a)
data Nat = Z | S Nat

instance Functor Stream where
    fmap :: (a -> b) -> Stream a -> Stream b
    fmap f (Stream a as) = Cons (f a) (fmap f as)

instance Representable Stream where
    type Rep f = Nat

    tabulate :: (Nat -> a) -> Stream a
    tabulate f = Cons (f Z) (tabulate (f . S))

    index    :: Stream a -> Nat -> a
    index (Cons a as) Z = a
    index (Cons _ as) (S n) = index as n

Yup! Our First Representable. Here is another somewhat trivial instance:

data Identity a = Identity a

instance Functor Identity where
    fmap f (Identity a) = Identity (f a)

instance Representable Identity where
    type Rep f = ()

    tabulate :: (() -> a) -> Identity a
    tabulate f = Identity $ f ()

    index :: Identity a -> () -> a
    index (Identity a) () = a

Since Identity can only hold a single a and () is inhabited by a single value, this instance lawful and total. :)

If our Store had a single element Identity would be perfect to represent queries into it and likewise, if it had an infinite amount of elements Stream might be perfect. However, we are looking for something in between.

What we need is some sort of bounded data structure, but in order to make it Representable we also need a corresponding bounded indexing type Rep f. Identity and () is a great example of a such a bounded Representable, but its bound at a fixed size of 1 element. We would like some Functor f and Rep f where we can fix the bound at whatever size we wish.

Another way of saying this is that we want is a family of Representable functors and a family of corresponding Rep f types, one pairing for each possible size boundary.

We can do this using Vect n a and Fin n a. Vect is a fixed length vector whose length is encoded with a type level Nat. Fin is a finite natural whose maximum value is encoded with a type level Nat.

data Vect (n :: Nat) a where
    VNil  :: Vect Z a
    VCons :: a -> Vect n a -> Vect (S n) a

data Fin (n :: Nat) where
    FZ :: Fin (S n)
    FS :: Fin n -> Fin (S n)

And like that, we have entered the world of dependent types. A world very messy and confusing in Haskell. It is now time to switch over to Idris, but don't worry all the implementations up to this point are identical modulo a few small syntax changes.

Here is our Representable interface (Typeclass) in Idris and our instance for Vect~/~Fin:

interface Functor f => Representable (f : Type -> Type) (rep : Type) | f where
  tabulate : (rep -> a) -> f a
  index : f a -> rep -> a

Representable (Vect n) (Fin n) where
  tabulate f {n = Z} = []
  tabulate f {n = (S k)} = f FZ :: tabulate (f . FS)
  index (x :: _) FZ {n = (S k)} = x
  index (_ :: xs) (FS x) {n = (S k)} = index xs x

In Idris there is no distinction between the "type level" and "term level." Types are first class values that can be passed around and worked with like any other values.

Correspondingly, type parameters (such as the n in Vect n (Fin n) are the same as term level parameters and can be passed into functions and pattern matched on. You can see this in the definition of tabulate above where the n from Vect n is wrapped in curly braces and treated like a function parameter.

The curly braces indicate that it is an implict parameter. This means that the typechecker is able to infer the value of the parameter and the caller never has to explicitely pass a value in. This might seem like magic, but its very similar to the type inference you are used to with Haskell.

With this instance of Representable, the indexing type Fin n cannot ever produce a value greater then n - 1 and the Functor must be of size n. This is guaranteed at compile time.

This means that by choosing a different value for n, we can have a Representable instance for an arbitrary fixed length vector.

Using Representable with Store

Now that we have switched over to Idris we need to rewrite our Store type. We also need to make it to work with Representable:

data Store : (Type -> Type) -> Type -> Type -> Type where
  MkStore : rep -> f a -> Store f rep a

This version of Store contains our current index (now called rep) and our Representable type f a. We no longer have a query function. Instead we are going to store our state as data in the Representable Functor and then use the index function to query it.

We then need to rewrite our API using index and tabulate. peek and peeks, and extract are all used for querying data so they will defined using index. extend is used for modifying our query so we know we will need to use tabulate.

Here is the full Store API in Idris:

pos : Store f rep a -> rep
pos (MkStore rep' fa) = rep'

peek : Representable f rep => rep -> Store f rep a -> a
peek rep' (MkStore _ fa) = index fa rep'

peeks : Representable f rep => (rep -> rep) -> Store f rep a -> a
peeks f (MkStore rep' fa) = index fa (f rep')

seek : Representable f rep => rep -> Store f rep a -> Store f rep a
seek rep' (MkStore _ fa) = MkStore rep' fa

seeks : Representable f rep => (rep -> rep) -> Store f rep a -> Store f rep a
seeks func (MkStore rep' fa) = MkStore (func rep') fa

experiment : (Representable f rep, Functor g) => (rep -> g rep) -> Store f rep a -> g a
experiment f s = map (`peek` s) (f (pos s))

Representable f rep => Comonad (Store f rep) where
extract (MkStore rep fa) = index fa rep
extend func (MkStore rep' fa) = MkStore rep' (tabulate (\rep'' => func (MkStore rep'' fa)))

Again we use index to do our actual lookup inside the f a. The real magic is in extend. In our previous Store, extend created a new query function using a lambda which applied the initial Store to f : Store s a -> b:

extend f (Store' g s) = Store (\s' -> f (Store g s')) s

In our new version, we do the exact same thing but then we pass the new query function into tabulate :: (Fin n -> Bool) -> Vect (Fin N) Bool, which uses that query function to generate the complete new state in a Vect (Fin n).

Lets use equational reasoning to walk through a small example of extend where we pass in a function to extract the element at the current index and apply not to it. We can use that to create a small example where we walk through all the subsitutions.

initialStore : Store (Vect (Fin 3) (Fin 3) Bool)
initialStore = MkStore FZ [True, False, True]

= extend (not . extract) initialStore
= extend (not . extract) (MkStore FZ [True,False,True])
= extend (not . extract) (MkStore FZ (tabulate (\rep' => extract (MkStore rep' fa))))
= extend (not . extract) (MkStore FZ
                                  (  (\rep'  => (not . extract) (MkStore rep' fa)) FZ
                                  :: ((\rep' => (not . extract) (MkStore rep' fa)) . FS) FZ
                                  :: ((\rep' => (not . extract) (MkStore rep' fa)) . FS . FS) FZ
                                  :: [])
= extend extract (MkStore FZ
                          (  (\FZ           => (not . extract) (MkStore rep' [True,False,True]))
                          :: (\(FS FZ)      => (not . extract) (MkStore rep' [True,False,True]))
                          :: (\(FS (FS FZ)) => (not . extract) (MkStore rep' [True,False,True]))
                          :: [])
= extend extract (MkStore FZ
                          (  (not . extract) (MkStore FZ           [True,False,True])
                          :: (not . extract) (MkStore (FS FZ)      [True,False,True])
                          :: (not . extract) (MkStore (FS (FS FZ)) [True,False,True])
                          :: [])
= extend extract (MkStore FZ
                          (  not (extract (MkStore FZ           [True,False,True]))
                          :: not (extract (MkStore (FS FZ)      [True,False,True]))
                          :: not (extract (MkStore (FS (FS FZ)) [True,False,True]))
                          :: [])
= extend extract (MkStore FZ
                          (  not (index [True,False,True] FZ)
                          :: not (index [True,False,True] (FS FZ))
                          :: not (index [True,False,True] (FS (FS FZ)))
                          :: [])
= extend extract (MkStore FZ
                          (  not True
                          :: not False
                          :: not True
                          :: [])
= extend extract (MkStore FZ
                          (  False
                          :: True
                          :: False
                          :: [])
= extend extract (MkStore FZ [False, True, False])

That example got a little big, but it demonstrates that the entire store transformation is done once and only once when you call extend. This is a huge improvement over having to recompute every prior generation every time you extend into a new generation.

Putting it all together

The last thing to do is reimplement our algorith using our new version of Store. Other then switching languages, at this point we actually don't need to modify our code a whole bunch!

The biggest changes are updating type signatures and writing some clever helper functions to work with Fin n.

First we intialize the store:

initialStore : Vect (3 + k) Bool -> Store (Vect (3 + k)) (Fin (3 + k)) Bool
initialStore xs = MkStore FZ xs

Lets take a closer look at this type signature:

  1. Notice the dependent type signature Vect (3 + k) Bool. This tells

the compiler that the Vect's size must be 3 greater then any Nat. The smallest possible Nat is 0 so this means that the Vect must be at a minimum length 3.

  1. Notice how in the Store our Representable is Vect (3 + k) and our Rep f is Fin (3 + k). By sharing the same type variable k this tells the compiler that the Fin must be 3 greater then the exact same Nat as the Vect.

  2. Lastly, notice how we no longer need to pass around our Size as a term. All the information we need is now encoded in the types. :)

The next function we will need to rewrite is indices. This will require some clever tricks with Fin:

up : Fin (S k) -> Fin (S k)
up = either (const FZ) FS . strengthen

down : Fin (S k ) -> Fin (S k)
down FZ = last
down (FS k) = weaken k

indices : Fin (3 + k) -> Vect 3 (Fin (3 + k))
indices x = [down x, x, up x]

When applying FS to produce the succesor to some Fin n we produce a Fin (S n). Likewise, pattern matching and removing a FS from some Fin (S k) produces a Fin k.

This is a problem for us. We need to maintain the size of the finite set as an invariant thorughout our program. We have to use some tricks to shift our Fin values while maintaining the same size of set.

If Fin (S k) represents the finite set of naturals smaller then k, then we know morally that unwrapping FS constructors from our Fin does produce numbers that fit within Fin (S k). We just need a way to demonstrate that to the type system.

Idris provides us with weaken and strengthen to do just this:

weaken : Fin n -> Fin (S n)
strengthen : Fin (S n) -> Either (Fin (S n)) (Fin n)

weaken says that any Fin n is also a valid Fin (S n). It's like saying any number x is smaller then x + 1. This allows us to unwrap one FS from our Fin (S k) to produce a number one digit smaller that is type Fin k and then convert it back to a Fin (S k).

last is another helper function from Idris that gives you the largest valid value for Fin n. In down we use pattern matching and weaken to produce a smaller Fin n in the successor case and last in the zero case to wrap around the modulus point.

In order to increase the value of a x : Fin (S n), we need to show whether or not x is also a valid Fin n. eg., if x were 4, then it would be a member of the set of numbers 0 to 5 and the set 0 to 4 but not the set 0 to 3.

strengthen says if x : Fin (S n) is too big to be a valid Fin n then return it as Fin (S n), otherwise return it as Fin n. up is saying that if x : Fin (S n) is too big to be Fin n it must be the modulus point and we loop around to zero. Otherwise, we convert it to Fin n and apply FS to it to increment its value and keep the type as Fin (S n).

The rest of the algorithm is as you would expect. The only changes were in the type signatures and removing the explicit threading of the size:

neighbors : Store (Vect (3 + k)) (Fin (3 + k)) Bool -> Vect 3 Bool
neighbors = experiment indices

isAlive : Store (Vect (3 + k)) (Fin (3 + k)) Bool -> Bool
isAlive s =
  case neighbors s of
    [False, False, False] => False
    [True, False, False] => False
    [True, True, True] => False
    _ => True

nextGen : Store (Vect (3 + k)) (Fin (3 + k)) Bool -> Store (Vect (3 + k)) (Fin (3 + k)) Bool
nextGen = extend isAlive

The last thing we need is an updated runSimulation written in Idris:

universe : Vect k (Fin k)
universe {k = Z} = []
universe {k = (S k)} = FZ :: map FS (universe {k=k})

boolToString : Bool -> String
boolToString False = "0"
boolToString True = "1"

printState : (Vect (3 + k)) Bool -> IO ()
printState xs = do
  traverse_ (putStr . boolToString) $ toList xs
  putStrLn ""

runSimulation : Store (Vect (3 + k)) (Fin (3 + k)) Bool -> IO ()
runSimulation s {k} =
  if all id curr || all not curr
     then printState curr
     else printState curr >>= \_ => runAutomata (nextGen s)
    curr : Vect (3 + k) Bool
    curr = experiment (const universe) s

main : IO ()
main = runSimulation init
   start : Vect 14 Bool
   start = map (\i => if i == 0 then False else True) [0,0,0,1,0,0,1,1,0,1,1,1,1,1]
   init : Store (Vect 14) (Fin 14) Bool
   init = initialStore start

And there it is! A type safe, performant implementation of Rule 110 using Comonads!