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

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Effect Systems in Haskell - Part I
Sanchayan Maity
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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 compose1

  • transformers/mtl has limitations

    • Monad transformer stacks are rigid
    • Doesn't allow handling something like Reader Int (Reader String) due to functional dependencies
    class Monad m => MonadReader r m | m -> r
    • More than a few effects in stack become unwieldy
    • n-square instances problem

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.

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

Huh, what did that mean

  • Define a monad in terms of return, fmap and join, rather than return and (>>=).
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.
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.34
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

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

and

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.
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

Questions?