I am building some product monads from Control.Monad.Product package. For reference, the product monad type is:
newtype Product g h a = Product { runProduct :: (g a, h a) }
Its monad instance is:
instance (Monad g, Monad h) => Monad (Product g h) where
return a = Product (return a, return a)
Product (g, h) >>= k = Product (g >>= fst . runProduct . k, h >>= snd . runProduct . k)
Product (ga, ha) >> Product (gb, hb) = Product (ga >> gb, ha >> hb)
source: http://hackage.haskell.org/packages/archive/monad-products/3.0.1/doc/html/src/Control-Monad-Product.html
Problem I
I build a simple monad that is a product of two State Int Monads, However, when I try to access the underlying state next:
ss :: Product (State Int) (State Int) Int
ss = do
let (a,b) = unp $ P.Product (get,get) :: (State Int Int,State Int Int)
return 404
You see get just creates another State Int Int, and I am not sure how to actually get the value of the underlying state, how might I do that? Note I could potentially runState a and b to get the underlying value, but this solution doesn't seem very useful since the two states' initial values must be fixed a priori.
Question II.
I would really like to be able to create a product monad over states of different types, ie:
ss2 :: Product (State Int) (State String) ()
ss2 = do
let (a,b) = unp $ P.Product (get,get) :: (State Int Int,State Int String)
return ()
But I get this type error:
Couldn't match expected type `String' with actual type `Int'
Expected type: (State Int Int, State String String)
Actual type: (StateT Int Identity Int,
StateT String Identity Int)
Because I presume the two get must return the same type, which is an unfortunate restriction. Any thoughts on how to get around this?
The solution is to use a state monad with a product of your states:
m :: State (Int, String) ()
Then you can run an operation that interacts with one of the two fields of the product using zoom and _1/_2 from the lens library, like this:
m = do
n <- zoom _1 get
zoom _2 $ put (show n)
To learn more about this technique, you can read my blog post on lenses which goes into much more detail.
It can't be done the way you want. Suppose there would be a way how to get the current state out of the left monad. Then you'd have a function of type
getLeft :: Product (State a) (State b) a
which is isomorphic to (State a a, State b a).
Now we can choose to throw away the left part and run only the right part:
evalState (snd (runProduct getLeft)) () :: a
So we get an inhabitant of an arbitrary type a.
In other words, the two monads inside Product are completely independent. They don't influence each other and can be run separately. Therefore we can't take a value out of one and use it in another (or both of them).
Related
I've seen the Maybe and Either functor (and applicative) used in code and that made sense, but I have a hard time coming up with an example of the State functor and applicative. Maybe they are not very useful and only exist because the State monad requires a functor and an applicative? There are plenty of explanations of their implementations out there but not any examples when they are used in code, so I'm looking for illustrations of how they might be useful on their own.
I can think of a couple of examples off the top of my head.
First, one common use for State is to manage a counter for the purpose of making some set of "identifiers" unique. So, the state itself is an Int, and the main primitive state operation is to retrieve the current value of the counter and increment it:
-- the state
type S = Int
newInt :: State S Int
newInt = state (\s -> (s, s+1))
The functor instance is then a succinct way of using the same counter for different types of identifiers, such as term- and type-level variables in some language:
type Prefix = String
data Var = Var Prefix Int
data TypeVar = TypeVar Prefix Int
where you generate fresh identifiers like so:
newVar :: Prefix -> State S Var
newVar s = Var s <$> newInt
newTypeVar :: Prefix -> State S TypeVar
newTypeVar s = TypeVar s <$> newInt
The applicative instance is helpful for writing expressions constructed from such unique identifiers. For example, I've used this approach pretty frequently when writing type checkers, which will often construct types with fresh variables, like so:
typeCheckAFunction = ...
let freshFunctionType = ArrowType <$> newTypeVar <*> newTypeVar
...
Here, freshFunctionType is a new a -> b style type with fresh type variables a and b that can be passed along to a unification step.
Second, another use of State is to manage a seed for random number generation. For example, if you want a low-quality but ultra-fast LCG generator for something, you can write:
lcg :: Word32 -> Word32
lcg x = (a * x + c)
where a = 1664525
c = 1013904223
-- monad for random numbers
type L = State Word32
randWord32 :: L Word32
randWord32 = state $ \s -> let s' = lcg s in (s', s')
The functor instance can be used to modify the Word32 output using a pure conversion function:
randUniform :: L Double
randUniform = toUnit <$> randWord32
where toUnit w = fromIntegral w / fromIntegral (maxBound `asTypeOf` w)
while the applicative instance can be used to write primitives that depend on multiple Word32 outputs:
randUniform2 :: L (Double, Double)
randUniform2 = (,) <$> randUniform <*> randUniform
and expressions that use your random numbers in a reasonably natural way:
-- area of a random triangle, say
a = areaOf <$> (Triangle <$> randUniform2 <*> randUniform2 <$> randUniform2)
What is the simplest Haskell library that allows composition of stateful functions?
We can use the State monad to compute a stock's exponentially-weighted moving average as follows:
import Control.Monad.State.Lazy
import Data.Functor.Identity
type StockPrice = Double
type EWMAState = Double
type EWMAResult = Double
computeEWMA :: Double -> StockPrice -> State EWMAState EWMAResult
computeEWMA α price = do oldEWMA <- get
let newEWMA = α * oldEWMA + (1.0 - α) * price
put newEWMA
return newEWMA
However, it's complicated to write a function that calls other stateful functions.
For example, to find all data points where the stock's short-term average crosses its long-term average, we could write:
computeShortTermEWMA = computeEWMA 0.2
computeLongTermEWMA = computeEWMA 0.8
type CrossingState = Bool
type GoldenCrossState = (CrossingState, EWMAState, EWMAState)
checkIfGoldenCross :: StockPrice -> State GoldenCrossState String
checkIfGoldenCross price = do (oldCrossingState, oldShortState, oldLongState) <- get
let (shortEWMA, newShortState) = runState (computeShortTermEWMA price) oldShortState
let (longEWMA, newLongState) = runState (computeLongTermEWMA price) oldLongState
let newCrossingState = (shortEWMA < longEWMA)
put (newCrossingState, newShortState, newLongState)
return (if newCrossingState == oldCrossingState then
"no cross"
else
"golden cross!")
Since checkIfGoldenCross calls computeShortTermEWMA and computeLongTermEWMA, we must manually wrap/unwrap their states.
Is there a more elegant way?
If I understood your code correctly, you don't share state between the call to computeShortTermEWMA and computeLongTermEWMA. They're just two entirely independent functions which happen to use state internally themselves. In this case, the elegant thing to do would be to encapsulate runState in the definitions of computeShortTermEWMA and computeLongTermEWMA, since they're separate self-contained entities:
computeShortTermEWMA start price = runState (computeEWMA 0.2 price) start
All this does is make the call site a bit neater though; I just moved the runState into the definition. This marks the state a local implementation detail of computing the EWMA, which is what it really is. This is underscored by the way GoldenCrossState is a different type from EWMAState.
In other words, you're not really composing stateful functions; rather, you're composing functions that happen to use state inside. You can just hide that detail.
More generally, I don't really see what you're using the state for at all. I suppose you would use it to iterate through the stock price, maintaining the EWMA. However, I don't think this is necessarily the best way to do it. Instead, I would consider writing your EWMA function over a list of stock prices, using something like a scan. This should make your other analysis functions easier to implement, since they'll just be list functions as well. (In the future, if you need to deal with IO, you can always switch over to something like Pipes which presents an interface really similar to lists.)
There is really no need to use any monad at all for these simple functions. You're (ab)using the State monad to calculate a one-off result in computeEWMA when there is no state involved. The only line that is actually important is the formula for EWMA, so let's pull that into it's own function.
ewma :: Double -> Double -> Double -> Double
ewma a price t = a * t + (1 - a) * price
If you inline the definition of State and ignore the String values, this next function has almost the exact same signature as your original checkIfGoldenCross!
type EWMAState = (Bool, Double, Double)
ewmaStep :: Double -> EWMAState -> EWMAState
ewmaStep price (crossing, short, long) =
(crossing == newCrossing, newShort, newLong)
where newCrossing = newShort < newLong
newShort = ewma 0.2 price short
newLong = ewma 0.8 price long
Although it doesn't use the State monad, we're certainly dealing with state here. ewmaStep takes a stock price, the old EWMAState and returns a new EWMAState.
Now putting it all together with scanr :: (a -> b -> b) -> b -> [a] -> [b]
-- a list of stock prices
prices = [1.2, 3.7, 2.8, 4.3]
_1 (a, _, _) = a
main = print . map _1 $ scanr ewmaStep (False, 0, 0) prices
-- [False, True, False, True, False]
Because fold* and scan* use the cumulative result of previous values to compute each successive one, they are "stateful" enough that they can often be used in cases like this.
In this particular case, you have a y -> (a, y) and a z -> (b, z) that you want to use to compose a (x, y, z) -> (c, (x, y, z)). Having never used lens before, this seems like a perfect opportunity.
In general, we can promote a stateful operations on a sub-state to operate on the whole state like this:
promote :: Lens' s s' -> StateT s' m a -> StateT s m a
promote lens act = do
big <- get
let little = view lens big
(res, little') = runState act little
big' = set lens little' big
put big'
return res
-- Feel free to golf and optimize, but this is pretty readable.
Our lens a witness that s' is a sub-state of s.
I don't know if "promote" is a good name, and I don't recall seeing this function defined elsewhere (but it's probably already in lens).
The witnesses you need are named _2 and _3 in lens so, you could change a couple of lines of code to look like:
shortEWMA <- promote _2 (computeShortTermEWMA price)
longEWMA <- promote _3 (computeLongTermEWMA price)
If a Lens allows you to focus on inner values, maybe this combinator should be called blurredBy (for prefix application) or obscures (for infix application).
With a little type class magic, monad transformers allow you to have nested transformers of the same type. First, you will need a new instance for MonadState:
{-# LANGUAGE
UndecidableInstances
, OverlappingInstances
#-}
instance (MonadState s m, MonadTrans t, Monad (t m)) => MonadState s (t m) where
state f = lift (state f)
Then you must define your EWMAState as a newtype, tagged with the type of term (alternatively, it could be two different types - but using a phantom type as a tag has its advantages):
data Term = ShortTerm | LongTerm
type StockPrice = Double
newtype EWMAState (t :: Term) = EWMAState Double
type EWMAResult = Double
type CrossingState = Bool
Now, computeEWMA works on an EWMASTate which is polymorphic in term (the afformentioned example of tagging with phantom types), and in monad:
computeEWMA :: (MonadState (EWMAState t) m) => Double -> StockPrice -> m EWMAResult
computeEWMA a price = do
EWMAState old <- get
let new = a * old + (1.0 - a) * price
put $ EWMAState new
return new
For specific instances, you give them monomorphic type signatures:
computeShortTermEWMA :: (MonadState (EWMAState ShortTerm) m) => StockPrice -> m EWMAResult
computeShortTermEWMA = computeEWMA 0.2
computeLongTermEWMA :: (MonadState (EWMAState LongTerm) m) => StockPrice -> m EWMAResult
computeLongTermEWMA = computeEWMA 0.8
Finally, your function:
checkIfGoldenCross ::
( MonadState (EWMAState ShortTerm) m
, MonadState (EWMAState LongTerm) m
, MonadState CrossingState m) =>
StockPrice -> m String
checkIfGoldenCross price = do
oldCrossingState <- get
shortEWMA <- computeShortTermEWMA price
longEWMA <- computeLongTermEWMA price
let newCrossingState = shortEWMA < longEWMA
put newCrossingState
return (if newCrossingState == oldCrossingState then "no cross" else "golden cross!")
The only downside is you have to explicitly give a type signature - in fact, the instance we introduced at the beginning has ruined all hopes of good type errors and type inference for cases where you have multiple copies of the same transformer in a stack.
Then a small helper function:
runState3 :: StateT a (StateT b (State c)) x -> a -> b -> c -> ((a , b , c) , x)
runState3 sa a b c = ((a' , b', c'), x) where
(((x, a'), b'), c') = runState (runStateT (runStateT sa a) b) c
and:
>runState3 (checkIfGoldenCross 123) (shortTerm 123) (longTerm 123) True
((EWMAState 123.0,EWMAState 123.0,False),"golden cross!")
>runState3 (checkIfGoldenCross 123) (shortTerm 456) (longTerm 789) True
((EWMAState 189.60000000000002,EWMAState 655.8000000000001,True),"no cross")
Let's say I have these two functions:
errorm :: ( MonadError String m ) => Bool -> m Int
errorm cond = if cond then return 1 else throwError "this is an error"
errorms :: ( MonadError String m ) => Bool -> m String
errorms cond = if cond then return "works" else throwError "does not work"
As you can see, one returns a string in the safe case, while the other returns an int
I now want to use them together within another monad. Trivially:
errErr :: MonadError String m => Bool -> Bool -> m (Int, String)
errErr b1 b2 = do
a <- errorm b1
b <- errorms b2
return (a,b)
The function signature here is derived by the GHC, and I am not sure how to use this function. I tried this:
runErrorT ( runErrorT ( errErr1 True True ) ) -- should give me (Right 1, Right "works")
But instead it gives me:
Ambiguous type variable `e0' in the constraint:
(Error e0) arising from a use of `errErr1'
Probable fix: add a type signature that fixes these type variable(s)
In the first argument of `runErrorT', namely `(errErr1 True True)'
In the first argument of `runErrorT', namely
`(runErrorT (errErr1 True True))'
In the expression: runErrorT (runErrorT (errErr1 True True))
In general, this is just one instance of my problem. I feel like I am not grasping how exactly to stack two monadT that are of the same class, but have different type parameters. Another example might be stacking the pair of a functions:
f :: ( MonadState Int m ) => m ()
g :: ( MonadState String m ) => m ()
---------------------------------------------------- Update ----------------------------------------------------
Per Daniel's comment below, I added a concrete instance of functions f and g from above. But thanks to Tikhon's answer, I think I figured it out.
type Counter = Int
type Msg = String
incr :: (MonadState Counter m) => Counter -> m ()
incr i = modify (+i)
addMsg :: ( MonadState Msg m ) => Msg -> m()
addMsg msg = modify ( ++ msg )
incrMsg:: (MonadTrans t, MonadState Msg m, MonadState Counter (t m)) => t m ()
incrMsg = do
lift . addMsg $ "one"
incr 1
return ()
incrMsgt = runIdentity $ runStateT ( runStateT incrMsg 1 ) "hello" :: (((), Int), String)
In this particular case, you do not need to stack two transformers--since both are MonadError String, they can be used as the same monad. You can use errorm and errorms together just like you would use two values in any other monad.
As a more concrete explanation, ignore the transformers for a second: you can imagine that the values are just Either String Int and Either String String. Clearly, you can just use them together. This is why you only need one runErrorT at the end rather than two: both values are in the same monad.
Now, what about your actual question: how do you stack two monad transformers? It works just like combining any two monad transformers. Two state transformers stacked upon each other look just like two different transformers stacked upon each other.
Now, it is a little tricky to use them. Depending on which one you're using, you will need to use lift differently. If you have a value in the base monad, you need to lift twice. If you have a value in the inner state monad, you will need to use it once. And if you have one in the outer level, you won't need it at all. This is just like normal transformers.
Going back to your error example, let's imagine you actually did want to stack two different error monad transformers instead of using them as one. This would mean that if you want to throw an error in the inner one, you would have to write lift (throwError "message"). If you had actually done this and had two stacked error transformers, than using runErrorT twice would have worked.
This is the State Monad code I am trying to figure out
data State a = State (Int -> (a, Int))
instance Monad State where
return x = State (\c -> (x, c))
State m >>= f = State (\c ->
case m c of { (a, acount) ->
case f a of State b -> b acount})
getState = State (\c -> (c, c))
putState count = State (\_ -> ((), count))
instance Show State where -- doesn't work
show (State a) = show a -- doesn't work
I am trying to make State as instance of Show so that I can see the action of getState and putState count on the ghci prompt.
Any tutorial or link to State Monad material would be nice too.
Here's is a Show instance that can help see what's going on:
instance Show a => Show (State a) where
show (State f) = show [show i ++ " => " ++ show (f i) | i <- [0..3]]
Then you can do:
*Main> getState
["0 => (0,0)","1 => (1,1)","2 => (2,2)","3 => (3,3)"]
*Main> putState 1
["0 => ((),1)","1 => ((),1)","2 => ((),1)","3 => ((),1)"]
In Haskell, typeclasses classify only types of the same kind. Monad classifies types of kind * -> *, while Show classifies types of kind *. Your State type has kind * -> * which is why there isn't a problem with your Monad instance, but there is a problem with your Show instance. State is called a "type constructor" because it consumes a type in order to produce another type. Think of it sort of like function application at the type level. You can therefore apply a specific type and make instances of that:
instance Show (State Char) where
show _ = "<< Some State >>"
Now, this isn't a very useful instance, try something like Sjoerd's suggestion to get a more meaningful Show instance. Notice that his version uses a generic type with a constraint.
instance (Show a) => Show (State a) where
-- blah blah
The generic type is a, and the constraint is (Show a) =>, in other words, a itself must be an instance of Show.
Here's a great (and personally, my favorite) explanation of the State Monad: Learn You A Haskell. (Also a great resource for learning Haskell in general).
You may have noticed that functions aren't part of the Show typeclass in Haskell. And since State is basically just a newtype wrapper for functions of certain types, you can't make a (meaningful) State instance of Show.
Here's code using the State Monad from LYAH:
import Control.Monad.State -- first, import the state monad
pop :: State Stack Int
pop = State $ \(x:xs) -> (x,xs)
push :: Int -> State Stack ()
push a = State $ \xs -> ((),a:xs)
stackManip :: State Stack Int
stackManip = do
push 3
a <- pop
pop
And here's that code in action from ghci:
*Main> runState stackManip [1,2,3]
(1,[2,3])
The fst of the tuple is the result, and the snd of the tuple is the (modified) state.
I'm struggling to understand the exists keyword in relation to Haskell type system. As far as I know, there is no such keyword in Haskell by default, but:
There are extensions which add them, in declarations like these data Accum a = exists s. MkAccum s (a -> s -> s) (s -> a)
I've seen a paper about them, and (if I recall correctly) it stated that exists keyword is unnecessary for type system since it can be generalized by forall
But I can't even understand what exists means.
When I say, forall a . a -> Int, it means (in my understanding, the incorrect one, I guess) "for every (type) a, there is a function of a type a -> Int":
myF1 :: forall a . a -> Int
myF1 _ = 123
-- okay, that function (`a -> Int`) does exist for any `a`
-- because we have just defined it
When I say exists a . a -> Int, what can it even mean? "There is at least one type a for which there is a function of a type a -> Int"? Why one would write a statement like that? What the purpose? Semantics? Compiler behavior?
myF2 :: exists a . a -> Int
myF2 _ = 123
-- okay, there is at least one type `a` for which there is such function
-- because, in fact, we have just defined it for any type
-- and there is at least one type...
-- so these two lines are equivalent to the two lines above
Please note it's not intended to be a real code which can compile, just an example of what I'm imagining then I hear about these quantifiers.
P.S. I'm not exactly a total newbie in Haskell (maybe like a second grader), but my Math foundations of these things are lacking.
A use of existential types that I've run into is with my code for mediating a game of Clue.
My mediation code sort of acts like a dealer. It doesn't care what the types of the players are - all it cares about is that all the players implement the hooks given in the Player typeclass.
class Player p m where
-- deal them in to a particular game
dealIn :: TotalPlayers -> PlayerPosition -> [Card] -> StateT p m ()
-- let them know what another player does
notify :: Event -> StateT p m ()
-- ask them to make a suggestion
suggest :: StateT p m (Maybe Scenario)
-- ask them to make an accusation
accuse :: StateT p m (Maybe Scenario)
-- ask them to reveal a card to invalidate a suggestion
reveal :: (PlayerPosition, Scenario) -> StateT p m Card
Now, the dealer could keep a list of players of type Player p m => [p], but that would constrict
all the players to be of the same type.
That's overly constrictive. What if I want to have different kinds of players, each implemented
differently, and run them against each other?
So I use ExistentialTypes to create a wrapper for players:
-- wrapper for storing a player within a given monad
data WpPlayer m = forall p. Player p m => WpPlayer p
Now I can easily keep a heterogenous list of players. The dealer can still easily interact with the
players using the interface specified by the Player typeclass.
Consider the type of the constructor WpPlayer.
WpPlayer :: forall p. Player p m => p -> WpPlayer m
Other than the forall at the front, this is pretty standard haskell. For all types
p that satisfy the contract Player p m, the constructor WpPlayer maps a value of type p
to a value of type WpPlayer m.
The interesting bit comes with a deconstructor:
unWpPlayer (WpPlayer p) = p
What's the type of unWpPlayer? Does this work?
unWpPlayer :: forall p. Player p m => WpPlayer m -> p
No, not really. A bunch of different types p could satisfy the Player p m contract
with a particular type m. And we gave the WpPlayer constructor a particular
type p, so it should return that same type. So we can't use forall.
All we can really say is that there exists some type p, which satisfies the Player p m contract
with the type m.
unWpPlayer :: exists p. Player p m => WpPlayer m -> p
When I say, forall a . a -> Int, it
means (in my understanding, the
incorrect one, I guess) "for every
(type) a, there is a function of a
type a -> Int":
Close, but not quite. It means "for every type a, this function can be considered to have type a -> Int". So a can be specialized to any type of the caller's choosing.
In the "exists" case, we have: "there is some (specific, but unknown) type a such that this function has the type a -> Int". So a must be a specific type, but the caller doesn't know what.
Note that this means that this particular type (exists a. a -> Int) isn't all that interesting - there's no useful way to call that function except to pass a "bottom" value such as undefined or let x = x in x. A more useful signature might be exists a. Foo a => Int -> a. It says that the function returns a specific type a, but you don't get to know what type. But you do know that it is an instance of Foo - so you can do something useful with it despite not knowing its "true" type.
It means precisely "there exists a type a for which I can provide values of the following types in my constructor." Note that this is different from saying "the value of a is Int in my constructor"; in the latter case, I know what the type is, and I could use my own function that takes Ints as arguments to do something else to the values in the data type.
Thus, from the pragmatic perspective, existential types allow you to hide the underlying type in a data structure, forcing the programmer to only use the operations you have defined on it. It represents encapsulation.
It is for this reason that the following type isn't very useful:
data Useless = exists s. Useless s
Because there is nothing I can do to the value (not quite true; I could seq it); I know nothing about its type.
UHC implements the exists keyword. Here's an example from its documentation
x2 :: exists a . (a, a -> Int)
x2 = (3 :: Int, id)
xapp :: (exists b . (b,b -> a)) -> a
xapp (v,f) = f v
x2app = xapp x2
And another:
mkx :: Bool -> exists a . (a, a -> Int)
mkx b = if b then x2 else ('a',ord)
y1 = mkx True -- y1 :: (C_3_225_0_0,C_3_225_0_0 -> Int)
y2 = mkx False -- y2 :: (C_3_245_0_0,C_3_245_0_0 -> Int)
mixy = let (v1,f1) = y1
(v2,f2) = y2
in f1 v2
"mixy causes a type error. However, we can use y1 and y2 perfectly well:"
main :: IO ()
main = do putStrLn (show (xapp y1))
putStrLn (show (xapp y2))
ezyang also blogged well about this: http://blog.ezyang.com/2010/10/existential-type-curry/