From f7bc4d9a10c14726c01e500ad67cd71009395ef9 Mon Sep 17 00:00:00 2001 From: taco Date: Tue, 1 Oct 2024 12:40:38 +1000 Subject: [PATCH 1/3] add randmonad --- _chapters/randmonad.md | 192 ++++++++++++++++++ .../images/chapterImages/randmonad/bind.png | Bin 0 -> 191852 bytes .../chapterImages/randmonad/randMonad.png | Bin 0 -> 141351 bytes 3 files changed, 192 insertions(+) create mode 100644 _chapters/randmonad.md create mode 100644 assets/images/chapterImages/randmonad/bind.png create mode 100644 assets/images/chapterImages/randmonad/randMonad.png diff --git a/_chapters/randmonad.md b/_chapters/randmonad.md new file mode 100644 index 00000000..3c6b43cc --- /dev/null +++ b/_chapters/randmonad.md @@ -0,0 +1,192 @@ +--- +layout: chapter +title: "Foldable and Traversable" +--- + +# Rand Monad + +```haskell +newtype Rand a = Rand { next :: Seed -> (Seed, a) } +``` + +`Rand` is a newtype wrapper around a function `Seed -> (Seed, a)`. +It represents a computation that, given a starting Seed, produces: + +1. A new updated `Seed`. +2. A value of type `a`. + +![Rand Monad](/assets/images/chapterImages/randmonad/randMonad.png) + +## Functor + +Definition of `Functor`. The `Functor` instance for `Rand` allows you to map a function over the result of a random computation. + +```haskell +instance Functor Rand where + fmap :: (a -> b) -> Rand a -> Rand b + fmap f (Rand g) = Rand h + where + -- The function inside rand + -- Apply f to the `value` a + h seed = (newSeed, f a) + where + (newSeed, a) = g seed +``` + +fmap constructs a new Rand value, Rand h, by: + +1. Taking a function `f` and a random computation `Rand g`. +2. Defining a new function `h` that, given an initial Seed, runs `g` to get `(newSeed, a)`. +3. Returning `(newSeed, f a)`, where `f a` is the transformed value. + +After applying fmap f, we have a new random computation that takes the same Seed as input and produces a transformed value (f a), while maintaining the same mechanics of randomness (i.e., correctly passing and updating the Seed state). + +We can also be a bit more succinct, by making use of `fmap` instances + +```haskell +fmap :: (a -> b) -> Rand a -> Rand b +fmap f r = Rand $ (f <$>)<$> next r +``` + +## Applicative + +```haskell +pure :: a -> Rand a +pure x = Rand (,x) -- Return the input seed and the value + + +(<*>) :: Rand (a -> b) -> Rand a -> Rand b +left <*> right = Rand h +where + h s = (s'', f v) -- Need to return a function of type (Seed -> (Seed, Value)) + where + (s', f) = next left s -- Get the next seed and function from the left Rand + (s'', v) = next right s' -- Get the next seed and value from the right Rand +``` + +`<*>` constructs a new Rand value Rand h by: + +1. Extracting a function `f` from the left computation using the initial seed. +2. Using the new seed to extract a value `v` from the right computation. +3. Returning a new seed and the result of applying `f` to `v`. +4. This allows us to apply a random function to a random value in a sequence while maintaining proper state management of the Seed. + +## Monad + +```haskell +instance Monad Rand where + (>>=) :: Rand a -> (a -> Rand b) -> Rand b + r >>= f = Rand $ \s -> + let (s1, val) = next r s + in next (f val) s1 +``` + +`r >>= f` creates a new Rand computation by: + +1. Running the first computation `r` with the initial seed `s`. +2. Extracting the value `val` and the new seed `s1`. +3. Using `val` to determine the next random computation `f val`. +4. Running `f val` with the updated seed `s1` to produce the final result. + +![Bind](/assets/images/chapterImages/randmonad/bind.png) + +### `Get` and `Put` + +`put` is used to set the internal state (the `Seed`) of the `Rand` monad. There is no value yet, hence we use the unit (`()`) + +`put` allows us to **modify** the internal state (Seed) of a random computation. + +```haskell +put :: Seed -> Rand () +put newSeed = Rand $ \_ -> (newSeed, ()) +``` + +`get` is used to retrieve the current state (the `Seed`) from the `Rand` monad. +It **does not modify** the state but instead returns the current seed as the result. This is achieved by putting the current seed in to the value part of the tuple. + +Since, when we apply transformation on the tuple, we apply the transformation according to the value! + +```haskell +get :: Rand Seed +get = Rand $ \s -> (s, s) +``` + +Using `get` and the monad instance, we can make a function to increase the seed by one. + +```haskell +incrementSeed' :: Rand Seed +incrementSeed' = get >>= \s -> pure (s + 1) +``` + +```haskell +incrementSeed :: Rand Seed +incrementSeed = do + seed <- get -- This gets the current seed + return (seed + 1) -- Increment the seed and put it in the 'state', we can do anything with the seed! +``` + +```bash +>>> next incrementSeed' 123 +(123, 124) +``` + +## Modifying a seed + +We want to modify the seed, assuming there is no value. This will simply apply a function `f` to the current seed. + +```haskell +modify :: (Seed -> Seed) -> Rand () +modify f = Rand $ \s -> (f s, ()) +``` + +We can also write this using our `get` and `put` + +```haskell +modify :: (Seed -> Seed) -> Rand () +modify f = get >>= \s -> put (f s) +``` + +This function: + +1. First retrieves the current seed (`get`). +2. Applies the function `f` to modify the seed. +3. Updates the internal state with `put`. + +This computation returns () as its result, indicating that its purpose is to update the state, not to produce a value. + +We can now write our `incrementSeed` in terms of modify + +```haskell +incrementSeed :: Rand Seed +incrementSeed = do + modify (+1) -- Use `modify` to increment the seed by 1 + get -- Return the updated seed +``` + +```haskell +nextRand :: Seed -> Seed +nextRand prevSeed = (a*prevSeed + c) `mod` m + where -- Parameters for linear congruential RNG. + a = 1664525 + c = 1013904223 + m = 2^32 + +-- | Generate a number between `l` and `u`. +genRand :: Int -> Int -> Seed -> Int +genRand l u seed = seed `mod` (u-l+1) + l +``` + +```haskell +rollDie :: Rand Int +rollDie = do + modify nextRand -- update the current seed + s <- get -- get retrieves the updated seed value s from the Rand monad's state. + pure (genRand 1 6 s) -- computes a random number and puts back in the context +``` + +We can also write this using bind notation, where we `modify nextRand` to update the seed. We then use `>>` to ignore the result (i.e., the `()`). We use get to put the seed as the value, which is then binded on to `s` and used to generate a random number. We then use pure to update the value, the seed updating is handled by our bind! + +```haskell +rollDie :: Rand Int +rollDie = modify nextRand >> get >>= \s -> pure (genRand 1 6 s) +``` diff --git a/assets/images/chapterImages/randmonad/bind.png b/assets/images/chapterImages/randmonad/bind.png new file mode 100644 index 0000000000000000000000000000000000000000..316d673f5e14caff6356f436017432aab00e02f9 GIT binary patch literal 191852 zcmZ^~WmsHIur-RiOK{ha;66AE?j9V1ySr;}celXc1P$&Gg1ZJA+}-6PdCxib*PS1G zo}S%atEyL3*F-2ONTMPUB0)evph`;tRUjZ>C?O!Am=WOKe<{fGYJUHMaTb+ULwNuA zAee-|mkC_NzqqK{o4L3fIhjJ3+u7TiGCCVOnVQ-;TiCmtL3athSEBt}NzBRA$i>p$ zj$F;s))Yd`!j+tji(J*zft-znjf0$(1Hi%x;NWuT9gcy3Acv3!il}*HAOA6r*S+&R zdpL7e*gMTQkWME+aYBP}E=dO}Cx4c0q8Ul7_OY!ItI}&z_g+7)`mW!cQDFi_I37kggka-AjS~+4`oNji zA=|O|g4n}qx?bqT3V%LqK&fIfwlQ6dV9Li9W5D2?{kQYT5 zqDcsdi9CJv@y+ z$pDHf#}4l7GfZpSR9+_+1HJ6sW3f09RG0Jt$3} zOW>+LNob*aLEf**$S(}G#60bV4j53NX6R0T(#M?mZ2|j1IiA`GA2&z}=^1LP5j;hz 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