Let's define an operator `|> = (flip $)`, which has a type signature `a -> (a -> b) -> b`, which, by your reasoning, should force sequencing too. Except it doesn't: Laziness is a bitch to work with, and `a |> f |> g` evaluates to `g(f(a))`, and g is evaluated first, which then evaluates f(a) on demand. If you consider `1 |> (+1) |> const 4`, this will evaluate to 4 without ever performing the addition.
Instead, let's compare this to the monadic equivalent. `(return 1) >>= return . (+ 1) >>= return . (const 4)`. The sequence that's being forced here isn't that (+ 1) is evaluated, it's that the `return` in `return 1` is evaluated (producing a 1) before `return . (+ 1)` is ever touched, and that _that_ return is evaluated (presumably producing an unevaluated thunk for (1 + 1)) before `return . (const 4)` is touched.
In IO world, this corresponds to guaranteeing that the IO operations happen in sequence irrespective of the order in which the pure computations they wrap are evaluated in.
> the `return` in `return 1` is evaluated (producing a 1) before `return . (+ 1)` is ever touched, and that _that_ return is evaluated (presumably producing an unevaluated thunk for (1 + 1)) before `return . (const 4)` is touched.
That's not quite right either. The former returns need never be evaluated.
Right — I should've said "that's the mechanism through which the implementation of `>>=` can force sequencing". `>>=` for Identity doesn't, most of the other common ones do, in some sense or another.
Right! `>>=` can force sequencing. It doesn't for Identity, Writer.Lazy or State.Lazy, but it can. OK, but also Arrow and Monoid can enforce a (different sort of) sequencing relationship.
Neither does monad force a sequencing relationship nor is it the only thing than can force a sequencing relationship so I can only conclude that "What a Monad does is to force sequencing by creating a data dependency" is indeed one of those misconceptions!
Indeed. But bad_user made a claim about monads in general
("What a Monad does is to force sequencing by creating a data dependency.") so all I need to do to refute it is give an example of a particular monad where the claim doesn't hold.
Monads force a denotation of sequencing. Of course you can write a language implementation where the sequencing of operation does not represent the sequencing of denotation (just as you can write a "compiler" that compiles all programs to Hello World), but, well, that's on you.
What about the Identity monad? Does that "force sequencing"? What "sequencing relationship" does it create?
If you are convinced that the Identity monad does "force sequencing" then you are also convinced that lazy evaluation "forces sequencing" (because the Identity monad is just lazy evaluation). That would be a fairly controversion conviction. If you are not convinced that the Identity monad "forces sequencing" then can you explain in more detail the meaning of your claim "What a Monad does is to force sequencing by creating a data dependency.".
It's in the signature. `b` has a causal dependency on `a` because `a` is on the left hand side of the arrow and `b` is on the right hand side of the arrow. That's it. Functional programming 101. There's no way around this.
The operational semantics and details of thunks are not relevant here. Tricks with `undefined` are not relevant here. This is just the meaning of the arrow type.
Because the values have extra "stuff" surrounding them. That's why it's `m a` and not just `a`: the `m` holds the invisible context that enriches the `a`.
Although your example is exactly what's happening in the identity monad, because there is no extra stuff.
Monads have been called "executable semicolons", because you can define a function to be executed each time the program passes from one statement to the next.
