presentations/effects-may20-2023/effects-freer-monads.md

---
title:
- Effect Systems in Haskell - Part I
author:
- Sanchayan Maity
theme:
- default
classoption:
- aspectratio=169
---

# Agenda

- Cover two papers on Effect Systems by Oleg Kiselyov
    * Extensible Effects - An Alternative to Monad Transformers
    * Freer Monads, More Extensible Effects
- Related paper `Reflection Without Remorse`
- Some sections today's discussion isn't going to cover
    * Efficiency/Performance of the library or effect system itself
    * For the sake of time, focus more on the implementation
    * Comparison of effect system libraries or how to choose one

# What's it all about

- **Separate syntax from semantics**
- **Interpret your abstract syntax tree in various ways**
- **Not losing performance while having both**

# Why effect systems

- Monads to model effects but monads don't compose[^1]
- transformers/mtl has limitations
    * Monad transformer stacks are rigid
    * Doesn't allow handling something like `Reader Int (Reader String)` due to functional dependencies
    ```haskell
    class Monad m => MonadReader r m | m -> r
    ```

    * More than a few effects in stack become unwieldy
    * n-square instances problem

[^1]: [Composing Monads by Mark Jones and Luc Duponcheel](https://web.cecs.pdx.edu/~mpj/pubs/RR-1004.pdf)

# Effect system libraries

- `freer-simple` based on Extensible Effects and Freer Monads, More Extensible Effects by Oleg Kiselyov
- `polysemy` based on Effect Handlers in Scope by Wu, Schrijvers et al
- `fused-effects` based on Fusion for Free: Efficient Algebraic Effect Handlers by Wu, Schrijvers et al
- `cleff` based on `ReaderT IO`
- `effectful` based on `ReaderT IO`
- others?

# Free monads

Given a `Functor f`, `Free f` is a `Free` monad.

```haskell
data Free f a = Pure a
              | Free (f (Free f a))
```

A Monad is something that "computes" when monadic context is collapsed by `join :: m (m a) -> m a` (recalling that `>>=` can be defined as `x >>= y = join (fmap y x))`. This is how Monads carry context through a sequential chain of computations: because at each point in the series, the context from the previous call is collapsed with the next.

A free monad satisfies all the Monad laws, but doesn't do any collapsing (that's the computation). It just builds up a nested series of contexts. The user who creates such a free monadic value is responsible for doing something with those nested contexts, so that the meaning of such a composition can be deferred until after the monadic value has been created.[^2]

[^2]: John Wiegley on [Stack Overflow](https://stackoverflow.com/a/13388966).

# Huh, what did that mean

- Define a monad in terms of `return`, `fmap` and `join`, rather than `return` and `(>>=)`.

```haskell
m >>= f = join (fmap f m)
```

- fmap is performing substitution and join is dealing with any re-normalization.
- Done this way, `(m >>= f)` on the `Maybe` monad would first `fmap` to obtain `Just (Just a)`, `Just Nothing` or `Nothing` before flattening.
- In the `Maybe a` case, the association of binds is largely immaterial, the normalization pass fixes things up to basically the same size.
- In Free monad, the monad is purely defined in terms of substitution.

```haskell
join :: Functor f => Free f (Free f a) -> Free f a
join (Pure a) = a
join (Free as) = Free (fmap join as)
```

# Free monads performance

- Vanilla free monads don't have great performance.
- Solutions like `Codensity` monad transformer and Church encoded free monad exist.[^3][^4]

```haskell
newtype FT f m a =
FT { runFT :: forall r. (a -> m r) -> (forall x. (x -> m r) -> f x -> m r) -> m r }
```

- Think of `Codensity` as a type level construction which ensures that you end up with a right associated bind.[^5]

[^3]: Asymptotic Improvement of Computations over Free Monads - Janis Voigtländer
[^4]: [The Free and The Furious: And by 'Furious' I mean Codensity. - raichoo](https://www.youtube.com/watch?v=EiIZlX_k89Y)
[^5]: [Free Monads for less - Edward Kmett](https://ekmett.github.io/reader/2011/free-monads-for-less-2/index.html)

# Reflection without remorse

- A left associated expression is asymptotically slower than the equivalent right associated expression. $O(n^2)$ vs $O(n)$ respectively.
- What's meant by reflection? Build and observe.
- Efficient data structures give asymptotic improvement for problematic occurrences of build and observe pattern like monads and monadic reflection.

# Extensible effects

- Defines only one effect `Eff`
- Type level list of effects
- What does it mean to be extensible?

# Freer monads

- Improves on extensible effects
- How?
    * Relaxes the `Functor` constraint, becoming `Freer`!
    * No need for `Functor` and `Typeable` on `Union`
- `freer` and `freer-simple` are based on `Freer` monads

```haskell
data FFree f a where
    Pure :: a → FFree f a
    Impure :: f x → (x → FFree f a) → FFree f a

instance Monad (FFree f) where
    Impure fx k’ >>= k = Impure fx (k’ >>> k)
```

The construction lets this implementation choose how to perform the `fmap` operation fixed to the appropriate `output type`.

# Freer monads

- The continuation can now be accessed directly rather than via `fmap`, which has to rebuild the mapped data structure.
- The explicit continuation of `FFree` also makes it easier to change its representation.

```haskell
class Member t r where
    inj :: t v -> Union r v
    prj :: Union r v -> Maybe (t v)
```

and

```haskell
data FEFree r a where
    Pure :: a → FEFree r a
    Impure :: Union r x → (x → FEFree r a) → FEFree r a
```

# Freer monads

- `FEFree r` becomes `Eff r`, where `r` is the list of effect labels.
- The request continuation which receives the reply `x` and works towards the final answer `a`, then has the type `x → Eff r a`.

```haskell
type Arr r a b = a → Eff r b

data FTCQueue m a b where
  Leaf :: (a -> m b) -> FTCQueue m a b
  Node :: FTCQueue m a x -> FTCQueue m x b -> FTCQueue m a b

type Arrs r a b = FTCQueue (Eff r) a b

data Eff r a where
    Pure :: a → Eff r a
    Impure :: Union r x → Arrs r x a → Eff r a
```

# Resources

- [Why Free monads matter](https://www.haskellforall.com/2012/06/you-could-have-invented-free-monads.html)
- [Free monad considered harmful](https://markkarpov.com/post/free-monad-considered-harmful.html)
- [Building real-world Haskell applications using Tagless-Final and ReaderT](https://fpunfold.com/2023/01/30/final-tagless-readert.html)
- [Free monads from scratch](https://siraben.dev/2020/02/20/free-monads.html)
- [An earlier talk of my own on Free Monads](https://www.youtube.com/watch?v=fhu1UQel5eo)
- [Free Monads for less](https://ekmett.github.io/reader/2011/free-monads-for-less/index.html)
- [When to use CPS vs codensity vs reflection without remorse](https://stackoverflow.com/questions/45334985/when-to-use-cps-vs-codensity-vs-reflection-without-remorse-in-haskell)
- [ReaderT pattern is just extensible effects](https://xn--i2r.xn--rhqv96g/2022/02/03/readert-is-extensible-effects/)
- [My Effects Bibliography](https://www.dantb.dev/posts/effects-bibliography/)
- [Effects Bibliography](https://github.com/yallop/effects-bibliography)
- [Freer simple effects examples](https://git.sanchayanmaity.net/sanchayanmaity/learn-effects)
- [Continuation Passing Style](https://en.wikibooks.org/wiki/Haskell/Continuation_passing_style)
- [Existential Quantification](https://markkarpov.com/post/existential-quantification.html)

# Questions?

- Reach out on
  * Email: sanchayan@sanchayanmaity.net
  * Mastodon: https://sanchayanmaity.com/@sanchayan
  * Blog: https://sanchayanmaity.net
  * Telegram:
    - t.me/fpncr
    - t.me/SanchayanMaity