Confusion over the State Monad code on “Learn you a Haskell”

Question

I am trying to get a grasp on Haskell using the online book Learn you a Haskell for great Good.

I have, to my knowledge, been able to understand Monads so far until I hi

Answer 1

Short answer:

1. State is meant to exploit monads' features in order to simulate an imperative-like system state with local variables. The basic idea is to hide within the monad the activity of taking in the current state and returning the new state together with an intermediate result at each step (and here we have s -> (a,s).
2. Do not mistake arbitrary functions with those wrapped within the State. The former may have whatever type you want (provided that they eventually produce some State a if you want to use them in the state monad). The latter holds functions of type s -> (a,s): this is the state-passing layer managed by the monad.
3. As I said, the function wrapped within State is actually produced by means of (>>=) and return as they're defined for the Monad (State s) instance. Its role is to pass down the state through the calls of your code.

Point 3 also is the reason why the state parameter disappears from the functions actually used in the state monad.

Long answer:

The State Monad has been studied in different papers, and exists also in the Haskell framework (I don't remember the good references right now, I'll add them asap).

This is the idea that it follows: consider a type data MyState = ... whose values holds the current state of the system.

If you want to pass it down through a bunch of functions, you should write every function in such a way that it takes at least the current state as a parameter and returns you a pair with its result (depending on the state and the other input parameters) and the new (possibly modified) state. Well, this is exactly what the type of the state monad tells you: s -> (a, s). In our example, s is MyState and is meant to pass down the state of the system.

The function wrapped in the State does not take parameters except from the current state, which is needed to produce as a result the new state and an intermediate result. The functions with more parameters that you saw in the examples are not a problem, because when you use them in the do-notation within the monad, you'll apply them to all the "extra" needed parameters, meaning that each of them will result in a partially applied function whose unique remaining parameter is the state; the monad instance for State will do the rest.

If you look at the type of the functions (actually, within monads they are usually called actions) that may be used in the monad, you'll see that they result type is boxed within the monad: this is the point that tells you that once you give them all the parameters, they will actually do not return you the result, but (in this case) a function s -> (a,s) that will fit within the monad's composition laws.

The computation will be executed by passing to the whole block/composition the first/initial state of the system.

Finally, functions that do not take parameters will have a type like State a where a is their return type: if you have a look at the value constructor for State, you'll see again that this is actually a function s -> (a,s).