Applicative instance for State - order of data flow

Question

I am trying to implement Applicative instance for such a type:

newtype State s a = State {runState :: s -> (a, s)}

I have some different ideas for (<*>) function. One way to implement it that comes to my mind is

(<*>) :: State s (a -> b) -> State s a -> State s b
State f <*> State s = State $ do
    (fa, fs) <- f
    let (sa, ss) = s fs
    return (fa sa, ss)

Or

(<*>) :: State s (a -> b) -> State s a -> State s b
State f <*> State s = State $ do
    (sa, ss) <- s
    let (fa, fs) = f ss
    return (fa sa, fs)

Which one (or does even any of them) is correct and why?

They both typecheck and differ only by "state" transformations order. I cannot find any good reason to prefer one over another...

Answer 1

First off, I'd recommend not using (monadic!) do syntax for defining an applicative instance like that, because it rather obscures what's happening. Here's your definitions using only standard functional syntax:

State f <*> State s = State $ \q
     -> let (fa, fs) = f q
            (sa, ss) = s fs
        in (fa sa, ss)

and

State f <*> State s = State $ \q
     -> let (fa, fs) = f ss
            (sa, ss) = s q
        in (fa sa, fs)

This also makes it clearer that there's not really any intrinsic evaluation order in an applicative instance (unlike a monad instance).

Answer 2

Both are reasonable. Note that you can get either of them from the other one:

x <*2> y = flip ($) <$> y <*1> x

It is a library convention, though, that the "effects" are performed left-to-right. Hence, the first version looks more familiar.

Answer 3

I believe both are correct because they don't violate any Applicative laws as far as I can see. However, the first is what is actually used. I think this is because of convention: it is expected that the effects of <*> 's left-hand arguments apply first, before its right-hand argument. Compare with IO , for example, where

(,) <$> readLn <*> getLine :: IO (Int, String)

prompts for an Int first, and then reads a String. It's nice to have State behave in a similar way.