Why are side-effects modeled as monads in Haskell?

Question

Could anyone give some pointers on why the impure computations in Haskell are modelled as monads?

I mean monad is just an interface with 4 operations, so what was th

Answer 1

As you say, Monad is a very simple structure. One half of the answer is: Monad is the simplest structure that we could possibly give to side-effecting functions and be able to use them. With Monad we can do two things: we can treat a pure value as a side-effecting value (return), and we can apply a side-effecting function to a side-effecting value to get a new side-effecting value (>>=). Losing the ability to do either of these things would be crippling, so our side-effecting type needs to be "at least" Monad, and it turns out Monad is enough to implement everything we've needed to so far.

The other half is: what's the most detailed structure we could give to "possible side effects"? We can certainly think about the space of all possible side effects as a set (the only operation that requires is membership). We can combine two side effects by doing them one after another, and this will give rise to a different side effect (or possibly the same one - if the first was "shutdown computer" and the second was "write file", then the result of composing these is just "shutdown computer").

Ok, so what can we say about this operation? It's associative; that is, if we combine three side effects, it doesn't matter which order we do the combining in. If we do (write file then read socket) then shutdown computer, it's the same as doing write file then (read socket then shutdown computer). But it's not commutative: ("write file" then "delete file") is a different side effect from ("delete file" then "write file"). And we have an identity: the special side effect "no side effects" works ("no side effects" then "delete file" is the same side effect as just "delete file") At this point any mathematician is thinking "Group!" But groups have inverses, and there's no way to invert a side effect in general; "delete file" is irreversible. So the structure we have left is that of a monoid, which means our side-effecting functions should be monads.

Is there a more complex structure? Sure! We could divide possible side effects into filesystem-based effects, network-based effects and more, and we could come up with more elaborate rules of composition that preserved these details. But again it comes down to: Monad is very simple, and yet powerful enough to express most of the properties we care about. (In particular, associativity and the other axioms let us test our application in small pieces, with confidence that the side effects of the combined application will be the same as the combination of the side effects of the pieces).

Answer 2

Could anyone give some pointers on why the unpure computations in Haskell are modeled as monads?

Well, because Haskell is pure. You need a mathematical concept to distinguish between unpure computations and pure ones on type-level and to model programm flows in respectively.

This means you'll have to end up with some type IO a that models an unpure computation. Then you need to know ways of combining these computations of which apply in sequence (>>=) and lift a value (return) are the most obvious and basic ones.

With these two, you've already defined a monad (without even thinking of it);)

In addition, monads provide very general and powerful abstractions, so many kinds of control flow can be conveniently generalized in monadic functions like sequence, liftM or special syntax, making unpureness not such a special case.

See monads in functional programming and uniqueness typing (the only alternative I know) for more information.