Monadic profunctors and grammars

[echatav]

This paper by Li-yao Xia, Dominic Orchard, and Meng Wang

Composing Bidirectional Programs Monadically https://link.springer.com/chapter/10.1007/978-3-030-17184-1_6

conceives of monadic profunctors which specialize monoidal profunctors.

type Monoidal p = (Profunctor p, forall x. Applicative (p x))
type Monadic p = (Profunctor p, forall x. Monad (p x))

For parsers, the extra power in Monad is recognized to be context-sensitivity.

For the grammatical printer/parser types, only Parsor is Monadic, not Printor. But, the paper suggests a version of a printer type which is monadic, which in this library's style, and with a punned name (or not), could be:

newtype Lintor s f a b = Lintor {runLintor :: a -> f (b, s -> s)}

deriving via ReaderT a (WriterT (Endo s) f)
  instance Monad f => Monad (Lintor s f a)

We could define context sensitive grammars as

type GrammarM a = forall p. (Grammatical p, Monadic p) => p a a

with generators yielded from Parsor and Lintor. What are nice combinators for Monadic profunctors? Should we export overloaded operators for qualified do notation or do Prelude's work out-of-the box for Monadic grammars? What is an equivalent to regexBNF invariant context free grammars for invariant context sensitive grammars?

[echatav]

@aspiwack, thanks for the tip ^

[aspiwack]

You're welcome.

[echatav]

@Lysxia Any thoughts on this ^ ?