example Say I have a dynamical process f: α → α. I can iterate it to my heart’s happiness: f ∘ f: α → α, f ∘ f ∘ f: α → α and so on. Soon I should want to add some effects to it, like say non-determinism with ndf: α → [α] or a progress bar with iof: α → IO α. So far so good, ndf <=< ndf: α → [α] and…

[image] anon58422685: So how do we check the equations? The first thing you can do is to try to find a counterexample; I’ve found this one: import Control.Monad newtype IOL a = IOL { unIOL :: IO [a] } instance Functor IOL where fmap f = IOL . fmap (fmap f) . unIOL instance Applicative IOL…

Yep, same thing with Writer and [ ] — they break S₄. Since IO works as a writer here, it is also broken. I did not find this because my Writer example was with a commutative monoid, which makes it a commutative monad. If IO sorted its output, it would also be somewhat like a commutative monad, and i…

Just found this article — Monads and Effects by Nick Benton, John Hughes and Eugenio Moggi — that seems to be the first source for monad transformers. (Dated 2002.) Hopefully it will answer some of my questions. Also, this article — Layer by Layer – Combining Monads by Fredrik Dahlqvist, Louis Par…

copied from fp discord: the question about monad composition is a very interesting one. Nicholas Wu wrote a paper Effect Handlers In Scope that does something similar to that and it turns out the semantics are not good (perhaps because he didn’t follow the distributive laws). people have invented …

Another article Composing Monads Using Coproducts by Christoph Lüth and Neil Ghani offers another way to put monads together: sum in the category of monadic functors and natural transformations that commute with join and pure. It is somewhat technical, and I have never seen anything like this put to…

[image] anon58422685: What is IO, anyway? [image] Should Haskell be rebranded away from "Pure Functional Programming" What a surprise: it’s a question about I/O in Haskell - I’m assuming none of these helped. Let’s try something different and see how Ag…

Another way of motivating the distributive laws is deriving them by breaking down the composed monads (that is, those which arise by mere functor composition) in terms of either the Kleisli or the Eilenberg-Moore adjunctions. I have discussed that in Do monad transformers, generally speaking, arise …

Cool! I wish you wrote that in mathematical notation instead of suggestive Haskell notation. I recall Stack Overflow does not support Latex…

Side note: I’ve just noticed the Proxy/Const () counterexample you mention at the opening post demonstrates a violation of law # 2 for the compose-on-the-inside, Traversable-based transformers. I’ve updated the SO answer to mention that.

summary of the laws required for the composition of monads to be a monad notation Capital letters like F and N stand for functors. In Haskell they are usually represented as type constructors with a Functor instance. Application of a functor F to an object X is denoted as function application FX, …

Whence monad transformers? Where are distributive laws?

Learn

atravers June 25, 2023, 10:43am 7

Semantics of fixIO by Levent Erkok, John Launchbury and Andrew Moran.
Tackling the Awkward Squad: … by Simon Peyton Jones (from section 3.2 on page 20 onwards).

Maintaining haskell programs