Haskell/Understanding monads/State
If you have programmed in any other language before, you likely wrote some functions that "kept state". For those new to the concept, a state is one or more variables that are required to perform some computation but are not among the arguments of the relevant function. Objectoriented languages like C++ make extensive use of state variables (in the form of member variables inside classes and objects). Procedural languages like C on the other hand typically use global variables declared outside the current scope or static variables in the functions to keep track of state.
In Haskell, however, such techniques are not as straightforward to apply. Doing so will require mutable variables which would mean that functions will have hidden dependencies, which is at odds with Haskell's functional purity. Fortunately, often it is possible to keep track of state in a functionally pure way. We do so by passing the state information from one function to the next, thus making the hidden dependencies explicit.
The State
type is designed to simplify this process of threading state through functions. In this chapter, we will see how it can assist us in some typical problems involving state: modelling a state machine and generating pseudorandom numbers.
State Machine
editWe will model a simple finitestate machine based on a coinoperated turnstile. Our model will be enhanced so that, in any state, it will create an output (in addition to a state transition) for each input^{[1]}.
Turnstile Example
editThe finitestate model of our turnstile is shown in this statetransition diagram:
The turnstile has two states: Locked and Unlocked. (It starts in a Locked state). There are two types of input: Coin (corresponding to someone putting a coin in the slot) and Push (corresponding to someone pushing the arm). Each input causes an output (Thank, Open or Tut) and a transition to a (new, or maybe the same) state. If someone puts a coin in when the turnstile is locked, they are thanked (yes, it can talk!) and the turnstile becomes unlocked. If they add more coins, they are thanked more but get no benefit (the turnstile simply remains unlocked with no memory of how many additional coins have been added). If someone pushes the arm when the turnstile is unlocked, the arm will open to let them through, then become locked to prevent anyone else going through. If someone pushes the arm when the turnstile is locked, it will politely tut at them but not let them through and remain locked.
Basic Model in Haskell
editWe will represent the states and outputs as follows:
data TurnstileState = Locked  Unlocked
deriving (Eq, Show)
data TurnstileOutput = Thank  Open  Tut
deriving (Eq, Show)
But what about the inputs? We can model them as functions. Here's a first attempt:
coin, push :: TurnstileState > TurnstileOutput
coin _ = Thank
push Unlocked = Open
push Locked = Tut
These correctly return the output for each input in any state, but don't give any indication of the new state. (In an imperative program, these "functions" might also update a variable to indicate the new state, but that is not an option in Haskell, nor, we claim, desirable). The answer is easy and obvious: return the new state along with the output:
coin, push :: TurnstileState > (TurnstileOutput, TurnstileState)
coin _ = (Thank, Unlocked)
push Locked = (Tut , Locked)
push Unlocked = (Open, Locked)
Sequencing Steps
editHow can we use this? One way is to list the set of outputs resulting from a sequence of inputs:
monday :: TurnstileState > ([TurnstileOutput], TurnstileState)
monday s0 =
let (a1, s1) = coin s0
(a2, s2) = push s1
(a3, s3) = push s2
(a4, s4) = coin s3
(a5, s5) = push s4
in ([a1, a2, a3, a4, a5], s5)
GHCi> monday Locked
([Thank,Open,Tut,Thank,Open],Locked)
Note that, like coin
and push
, this monday
function takes an initial state as a parameter and returns the final state alongside the list of outputs.
Exercises 


From the examples it can be seen that:
 the state (of some fixed type) is always passed from step to step, and (usually) included in the input to and output from a function: having the state as input to and output from functions allows us to chain them together as steps in bigger functions, in the same way the steps within these smaller functions are chained together;
 the (nonstate) output of a step may or may not be used in deciding subsequent steps:
hastyPerson
uses the output of the firstpush
to determine whether they need to insert a coin, butregularPerson
always does the same two steps regardless of their outputs;  the (nonstate) output of a step may or may not be used in the final (nonstate) return value from a function: the return value from
regularPerson
uses the output from each step, but the return value fromluckyPair
doesn't depend on the output from the first step;  a function may take parameters in addition to the initial state:
luckyPair
takes aBool
parameter;  the types of the (nonstate) return values can be different for different functions:
coin
returnsTurnstileOutput
,monday
returns[TurnstileOutput]
andluckyPair
returnsBool
.
But all of this code is cumbersome, tedious to write and error prone. It would be ideal if we could automate the extraction of the second member of the tuple (i.e. the new state) and feed it to the next step, whilst also allowing the function to use the (nonstate) values to make decisions about further steps and/or include in the (nonstate) result(s). This is where State
comes into the picture.
Introducing State
edit
The Haskell type State
describes functions that consume a state and produce both a result and an updated state, which are given back in a tuple.
The state function is wrapped by a data type definition which comes along with a runState
accessor so that pattern matching becomes unnecessary. For our current purposes, the State
type might be defined as:
newtype State s a = State { runState :: s > (a, s) }
Here, s
is the type of the state, and a
the type of the produced result. Calling the type State
is arguably a bit of a misnomer because the wrapped value is not the state itself but a state processor.
newtype
editNote that we defined the data type with the newtype
keyword, rather than the usual data
. newtype
can be used only for types with just one constructor and just one field. It ensures that the trivial wrapping and unwrapping of the single field is eliminated by the compiler. For that reason, simple wrapper types such as State
are usually defined with newtype
. Would defining a synonym with type
be enough in such cases? Not really, because type
does not allow us to define instances for the new data type, which is what we are about to do...
Where did the State
constructor go?
edit
In the wild, the State
type is provided by Control.Monad.Trans.State
in transformers
(and is also reexported by Control.Monad.State
in mtl
). When you use it, you will quickly notice there is no State
constructor available. The transformers
package implements the State
type in a different way. The differences do not affect how we use or understand State
, except that instead of a State
constructor, Control.Monad.Trans.State
exports a state
function,
state :: (s > (a, s)) > State s a
which does the same job. As for why the actual implementation is not the obvious one we presented above, we will get back to that a few chapters down the road.
Note
In all of the code below, State
is a (parameterised) type (that we will make instances of Monad
, etc), and state
is a function that takes an argument (of type s > (a, s)
) and returns a value of type State s a
. If you are creating your own version of State
by following this text as a tutorial (which I recommend), please add this code after the newtype
declaration:
state :: (s > (a, s)) > State s a
state = State
This will ensure the code below is valid whether you use your own State
implementation or the one provided.
Instantiating the Monad
editSo far, all we have done was to wrap a function type and give it a name. There is another ingredient, however: for every type s
, State s
can be made a Monad
instance, giving us very handy ways of using it.
To define a Monad
instance, there must also be instances for Functor
and Applicative
. As we explained previously, these superclass instances can be derived as follows from the Monad
instance that we are about to define in more detail.
import Control.Monad  you will need to put this towards the top of the file
instance Functor (State s) where
fmap = liftM
instance Applicative (State s) where
pure = return
(<*>) = ap
In a later section we will discuss the implications of State
also being a Functor
and an Applicative
in more detail. You will also get a chance to reimplement the above explicitly based on their behaviour, without simply deferring to the Monad
instance.
So let's define this instance.
instance Monad (State s) where
Note the instance is State s
, and not just State
on its own; State
can't be made an instance of Monad
, as it takes two type parameters, rather than one.
That means there are actually many different State
monads, one for each possible type of state  State String
, State Int
, State SomeLargeDataStructure
, and so forth. However, we only need to write one implementation of return
and (>>=)
; the methods will be able to deal with all choices of s
.
The return
function is implemented as:
return :: a > State s a
return x = State ( \ s > (x, s) )
Giving a value (x
) to return
produces a function which takes a state (s
) and returns it unchanged, together with the value we want to be returned. As a finishing step, the function is wrapped up with the state
function.
Exercises 


As for binding, it can be defined like this:
(>>=) :: State s a > (a > State s b) > State s b
p >>= k = q where
p' = runState p  p' :: s > (a, s)
k' = runState . k  k' :: a > s > (b, s)
q' s0 = (y, s2) where  q' :: s > (b, s)
(x, s1) = p' s0  (x, s1) :: (a, s)
(y, s2) = k' x s1  (y, s2) :: (b, s)
q = state q'
We wrote the definition above in a quite verbose way, to make the steps involved easier to pinpoint. A more compact way of writing it would be:
p >>= k = state $ \ s0 >
let (x, s1) = runState p s0  Running the first processor on s0.
in runState (k x) s1  Running the second processor on s1.
(>>=)
is given a state processor (p
) and a function (k
) that is used to create another processor from the result of the first one. The two processors are combined into a function that takes the initial state (s
) and returns the second result and the third state (i.e. the output of the second processor). Overall, (>>=)
here allows us to run two state processors in sequence, while allowing the result of the first stage to influence what happens in the second one.
One detail in the implementation is how runState
is used to undo the State
wrapping, so that we can reach the function that will be applied to the states. The type of runState p
, for instance, is s > (a, s)
.
Understanding the Bind Operator
editAnother way to understand this derivation of the bind operator >>=
is to consider once more the explicit but cumbersome way to simulate a stateful function of type a > b
by using functions of type (a, s) > (b, s)
, or, said another way: a > s > (b,s) = a > (s > (b,s))
. These classes of functions pass the state on from function to function. Note that this last signature already suggests the righthand side type in a bind operation where the abstract type is S b = (s > (b, s))
.
Now that we have seen how the types seem to suggest the monadic signatures, let's consider a much more concrete question: Given two functions f :: s > (a, s)
and g :: a > s > (b, s)
, how do we chain them to produce a new function that passes on the intermediate state?
This question does not require thinking about monads: one option is to simply use function composition. It helps our exposition if we just write it down explicitly as a lambda expression:
compose :: (s > (a,s)) > { first function }
(a > (s > (b,s))) > { second function, note type is similar to (a,s) > (b,s) }
s > (b,s) { composed function }
compose f g = \s0 > let (a1, s1) = f s0 in (g a1) s1
{This lambda expression threads both intermediate results produced by f into those required by g }
Now, if in addition to chaining the input functions, we find that the functions of signature s > (a,s)
were all wrapped in an abstract datatype Wrapped a
, and that therefore we need to call some other provided functionswrap :: (s > (a,s)) > Wrapped a
, and unwrap :: Wrapped a > (s > (a,s))
in order to get to the inner function, then the code changes slightly:
{ what happens if the type s > (a,s) is wrapped and this new type is called Wrapped a }
composeWrapped :: Wrapped a > (a > Wrapped b) > Wrapped b
composeWrapped wrappedf g = wrap (\s0 > let (a1,s1) = (unwrap wrappedf) s0 in (unwrap (g a1)) s1)
{ or, reusing compose }
composeWrapped wrappedf g = wrap (compose (unwrap wrappedf) (fmap unwrap g))
This code is the implementation of (>>=)
shown above, with wrap = state
and unwrap = runState
, so we can now see how the definition of bind given earlier is the standard function composition for this special kind of stateful function.
This explanation does not address yet where the original functions Wrapped a
and a > Wrapped b
come from in the first place, but they do explain what you can do with them once you have them.
Turnstile using State
edit
We now look at how the State
type can help with the turnstile example. Firstly, by comparing the type of coin :: TurnstileState > (TurnstileOutput, TurnstileState)
with newtype State s a = State { runState :: s > (a, s) }
, we can see that, by replacing s
with TurnstileState
and a
by TurnstileOutput
we can define:
coinS, pushS :: State TurnstileState TurnstileOutput
coinS = State coin
pushS = State push
Note
I've added S at the end of these names, just to distinguish them from those on this page that aren't based on the State
monad. It's not something you'd normally do.
We can then use runState
to extract the underlying functions and apply them to a state, for example:
GHCi> :t runState coinS
runState coinS :: TurnstileState > (TurnstileOutput, TurnstileState)
GHCi> runState coinS Locked
(Thank,Unlocked)
Note
There is an interesting comparison here between runState
and partial application of functions.
We normally consider, for example, take :: Int > [a] > [a]
to be a function of two arguments. But we also saw (briefly, here) that we can apply it to only one argument to get a function of the one remaining argument, e.g. take 2 :: [a] > [a]
. We could then do e.g. map (take 2) ["every", "good", "boy"]
to get ["ev", "go", "bo"]
. In fact, Haskell always applies arguments one step at a time, so that take 2 "every"
first applies the 2
to get a new function, to which it then applies "every"
. It could be written (take 2) "every"
.
runState
is defined as a function that takes a single argument. And it returns a function that takes a single argument, to which we can then apply an initial state value. So runState coinS Locked
means (runState coinS) Locked
. But, as with (take 2) "every"
, the brackets are not needed.
In terms of application of arguments, take
and runState
are similar: they take one argument and return a function that takes another. The big difference between them is in their definitions. When take
is defined it declares two parameters and uses both of them in the function definition. runState
, however, is (implicitly) defined as a function of a single parameter. The function it returns is defined separately (by a user of State
), and the type of State
requires it to be a function of one parameter. Each of these implementations reference only their own parameters directly.
Using the Turnstile State
monad
edit
Not yet too exciting, but now coinS
and pushS
are monadic (they are functions — admittedly of zero parameters — that return Monad instances) we can do monadic stuff with them, including using do
notation:
mondayS :: State TurnstileState [TurnstileOutput]
mondayS = do
a1 < coinS
a2 < pushS
a3 < pushS
a4 < coinS
a5 < pushS
return [a1, a2, a3, a4, a5]
GHCi> :t runState mondayS
runState mondayS :: TurnstileState > ([TurnstileOutput], TurnstileState)
GHCi> runState mondayS Locked
([Thank,Open,Tut,Thank,Open],Locked)
Note that we're no longer writing all the code to thread the output state from each step into the next: the State
monad is doing that for us. A lot of the tedious and errorprone work has been removed. How? Remember that do
is simply syntactic sugar for the bind (>>=)
operator so the above is equivalent to:
mondayS =
coinS >>= (\ a1 >
pushS >>= (\ a2 >
pushS >>= (\ a3 >
coinS >>= (\ a4 >
pushS >>= (\ a5 >
return [a1, a2, a3, a4, a5] )))))
This uses the (>>=)
operator we defined for State
above, unwraps each stateprocessing function from its State
wrapper, applies the output state from it as an argument into the next step, and wraps the result back in a State
wrapper. The sequence of (>>=)
operators, along with return
combines all the steps into a single combined stateprocessing function wrapped in a State
wrapper, which we can access and run with runState
.
A monad is sometimes described as providing a value in a context. An IO
monad can provide values from the real world when we ask it to. A Maybe
monad can provide values if it's there, or not otherwise. What about the State
monad? It can provide a value when we execute a step of a stateprocessor. (And the monad "automatically" ensures that state changes are passed from step to step without us having to worrying about it).
In this example, some tedium remains in obtaining the list of outputs from each step and combining them into a list. Can we do better? Yes we can:
mondayS :: State TurnstileState [TurnstileOutput]
mondayS = sequence [coinS, pushS, pushS, coinS, pushS]
We met sequence
in the section on IO Monads. It creates a single action (in this case a state processing function) which, when executed, runs through each of the actions (in this case state processing steps) in turn, executing them and combining the results into a list.
Exercises 


evalState
and execState
edit
We have seen how runState
accesses the state processing function so that we can do, for example, runState mondayS Locked
. (We also used it in the definition of (>>=)
.)
Other functions which are used in similar ways are evalState
and execState
. Given a State a b
and an initial state, the function evalState
will give back only the result value of the state processing, whereas execState
will give back just the new state.
evalState :: State s a > s > a
evalState p s = fst (runState p s)
execState :: State s a > s > s
execState p s = snd (runState p s)
OK, they're not much. But they're not nothing, and they allow us to do e.g.:
GHCi> evalState mondayS Locked
[Thank,Open,Tut,Thank,Open]
if we only want to see the output sequence, and not the final state.
Setting the State
editWhat if we had an turnstile engineer who wanted to test the locking mechanism with code like this:
testTurnstile :: State TurnstileState Bool
testTurnstile = do
somehow set state to Locked
check1 < pushS
somehow set state to Unlocked
check2 < pushS
somehow set state to Locked again
return (check1 == Tut && check2 == Open)
This handy function comes to the rescue:
put :: s > State s ()
put newState = state $ \_ > ((), newState)
put
is a monadic function that can be bound with (>>=)
operators or fit in do constructs in sequence with other actions. It takes a state parameter (the one we want to introduce) and generates a state processor which ignores whatever state it receives and gives back the new state we introduced as the next state. Since we don't care about the result of this processor (all we want to do is to replace the state), the first element of the tuple will be ()
, the universal placeholder value.^{[3]}
Let's see how it helps the engineer:
testTurnstile :: State TurnstileState Bool
testTurnstile = do
put Locked
check1 < pushS
put Unlocked
check2 < pushS
put Locked
return (check1 == Tut && check2 == Open)
GHCi> runState testTurnstile Locked
(True,Locked)
HHCi> runState testTurnstile Unlocked
(True,Locked)
Accessing the State
editIn the definition of pushS
above we made use of the existing code push
. What if we wanted to write it without such preexisting function? Obviously we could do this:
pushS = state $ \s > case s of
Locked > (Tut , Locked)
Unlocked > (Open, Locked)
but could we write it all using a do construct? Yes, using this:
get :: State s s
get = state $ \s > (s, s)
get
is also monadic and creates a state processor that gives back the state s
it is given both as a result and as the next state. That means the state will remain unchanged, and that a copy of it will be made available for us to use.
We could use get
like this:
pushS = do
s < get
put Locked
case s of
Locked > return Tut
Unlocked > return Open
Exercises 


Monadic Control Structures
editThe second version of mondayS
above shows another benefit of using the monad, in addition to the hiding of the state threading and ability to use do notation and the like: we are also able to use great functions like sequence
. In this section we look at some more of these functions. (You will need to import Control.Monad
, or do GHCi> :m Control.Monad
to ensure all of these are in scope).
First, here's replicateM
:
GHCi> evalState (replicateM 6 pushS) Unlocked
[Open,Tut,Tut,Tut,Tut,Tut]
Which is pretty selfexplanatory.
Before we look at any more, we need a slightly different (arguably better) implementation of the turnstile finitestate machine, using an input type and a single processing function:
data TurnstileInput = Coin  Push
deriving (Eq, Show)
turnS :: TurnstileInput > State TurnstileState TurnstileOutput
turnS = state . turn where
turn Coin _ = (Thank, Unlocked)
turn Push Unlocked = (Open, Locked)
turn Push Locked = (Tut, Locked)
GHCi> runState (turnS Coin) Locked
(Thank,Unlocked)
We can now use mapM
, like this:
GHCi> evalState (mapM turnS [Coin, Push, Push, Coin, Push]) Locked
[Thank,Open,Tut,Thank,Open]
This very nicely illustrates how the finitestate machine is a transducer: it converts an ordered sequence of inputs to an ordered sequence of outputs, maintaining the state as it goes along.
Now we'll look at filterM
:
getsThroughS :: TurnstileInput > State TurnstileState Bool
getsThroughS input = do
output < turnS input
return $ output == Open
GHCi> evalState (filterM getsThroughS [Push, Coin, Coin, Push, Push, Coin, Push]) Locked
[Push,Push]
We can see two people made it through (not surprisingly, when they pushed the arm). If we switch the order of the first two inputs more people get through:
GHCi> evalState (filterM getsThroughS [Coin, Push, Coin, Push, Push, Coin, Push]) Locked
[Push,Push,Push]
Here's a different way of counting the number of openings using foldM
:
countOpens :: [TurnstileInput] > State TurnstileState Int
countOpens = foldM incIfOpens 0 where
incIfOpens :: Int > TurnstileInput > State TurnstileState Int
incIfOpens n i = do
g < getsThroughS i
if g then return (n+1) else return n
GHCi> evalState (countOpens [Coin, Push, Coin, Push, Push, Coin, Push]) Locked
3
Note that sequence
, mapM
and filterM
always execute all of the actions in the input list, but foldM
could skip some.
Exercises 


PseudoRandom Numbers
editSuppose we are coding a game in which at some point we need an element of chance. In reallife games that is often obtained by means of dice or similar. For a computer program we need something to emulate such an object, and most programming languages provide some concept of random numbers that can be used for this purpose^{[4]}.
Generating actual random numbers is hard. Computer programs almost always use pseudorandom numbers instead. They are "pseudo" because they are not actually random, and that they are known in advance. Indeed, they are generated by algorithms (the pseudorandom number generators) which take an initial state (commonly called the seed) and produce from it a sequence of numbers that have the appearance of being random.^{[5]} Every time a pseudorandom number is requested, state somewhere must be updated, so that the generator can be ready for producing a fresh, different random number the next time. Sequences of pseudorandom numbers can be replicated exactly if the initial seed and the generating algorithm are known.
Haskell Global PseudoRandom Number Generator
editProducing a pseudorandom number in most programming languages is very simple: there is a function somewhere in the libraries that provides a pseudorandom value (and also updates an internal mutable state so that it produces a different value next time, although some implementations perhaps produce a truly random value). Haskell has a similar one in the System.Random
module from the random
package:
GHCi> :m System.Random
GHCi> :t randomIO
randomIO :: Random a => IO a
What is Random
? It's the class of types that can have pseudorandom values generated by the System.Random
module functions. Int
, Integer
, Bool
and others are all instances of Random
. You can "request" any of these by specifying the result type:
GHCi> randomIO :: IO Int
1557093684
GHCi> randomIO :: IO Int
1342278538
GHCi> randomIO :: IO Bool
True
More interestingly, randomIO
is an IO
action. It couldn't be otherwise, as it makes use of mutable state, which is kept out of reach from our Haskell programs. Thanks to this hidden dependency, the pseudorandom values it gives back can be different every time.
However, we're here to study the State
monad, so let's look at functions that take and return an explicit representation of the random number generator state.
Haskell PseudoRandom Number Generator with Explicit State
editHere's a slightly different function in the System.Random
module:
GHCi> :t random
random :: (Random a, RandomGen g) => g > (a, g)
Now there's no IO
, and we should recognise the g > (a, g)
pattern as something we could put inside a State
wrapper.
What is RandomGen
? It is another class defined in the System.Random
module. The module also provides a single instance StdGen
. There are a couple of ways to create values of this type. The one we will use first is mkStdGen :: Int > StdGen
which creates a StdGen
value from a given seed:
GHCi> mkStdGen 666
667 1
GHCi> mkStdGen 666
667 1
Note that, given the same seed, you get the same StdGen. What is StdGen
? The documentation calls it "the standard pseudorandom number generator", but it might be better to call it the state of the standard pseudorandom number generator. We can see that here:
GHCi> let s = mkStdGen 666
GHCi> s
667 1
GHCi> random s :: (Int, StdGen)
(6438947685955577547,392509921 2103410263)
The first function (mkStdGen 666
) returns an initial state, based on a given seed of 666. The second function (random s
) takes the initial StdGen
state and returns a pair: a random value (we've requested an Int
) and a new StdGen
state. How is the state represented internally? The System.Random
module does it somehow, and we don't really care how. (We can see StdGen
implements show
, which displays two funny numbers. We could go and look at the source code if we really wanted to see how it works, but some clever person might go and change it one day anyway). How does random
calculate a new state? We also don't care; we can just be happy that it does.
Example: Rolling Dice
editWe are going to build a dicethrowing example. And for this, we'll use a slightly different function:
GHCi> let s = mkStdGen 666
GHCi> randomR (1,6) s
(6,26689338 40692)
randomR
takes a range (in this case 1 to 6) and returns a pseudorandom number in the range (we were lucky: we got a 6!).
Suppose we want a function that rolls two dice and returns a pair representing the result of each throw. Here's one way:
import System.Random put this towards the top of the file
rollPair :: StdGen > ((Int, Int), StdGen)
rollPair s0 =
let (r1, s1) = randomR (1,6) s0
(r2, s2) = randomR (1,6) s1
in ((r1, r2), s2)
GHCi> rollPair (mkStdGen 666)
((6,1),647839921 1655838864)
Doesn't this remind us of the tedious and errorprone approach we first tried in the turnstile example? Not convinced it's tedious? Try the first exercise:
Exercises 


Dice with State
edit
So, a better way, using State
:
rollDieS :: State StdGen Int
rollDieS = state $ randomR (1,6)
GHCi> runState rollDieS (mkStdGen 666)
(6,26689338 40692)
This is very similar to the original versions of coinS
and pushS
: there was already a function of form s > (a, s)
, and we just wrapped in in a State
wrapper. Now we have monadic power! We can write:
rollPairS :: State StdGen (Int, Int)
rollPairS = do
r1 < rollDieS
r2 < rollDieS
return (r1, r2)
GHCi> runState rollPairS (mkStdGen 666)
((6,1),647839921 1655838864)
And we avoid all the tedious threading of state from one step to the next.
Exercises 


State
is also a Functor
and an Applicative
edit
Here's another dice throwing function:
rollDieDoubledS :: State StdGen Int
rollDieDoubledS = do
r < rollDieS
return (r * 2)
Its behaviour should be clear. But it seems a bit verbose for such a simple function. Can we do better?
As we noted previously (and saw above), State
(and all other monads) are also instances of Functor
and Applicative
. And in the prologue we did:
...
let mx = readMaybe s :: Maybe Double
case fmap (2*) mx of
...
This leveraged the fact that Maybe
is a Functor
. The fmap (2*) mx
converts a Just x
to Just (2*x)
(or Nothing
to Nothing
). If we think of x
as a value wrapped in a context, we can see that the fmap
has kept the same context (it's still a Just
, or still Nothing
), but applied a conversion to the wrapped value. We can do the same with the State
functor:
rollDieDoubledS = fmap (*2) rollDieS
The meaning of State
is different to the meaning of Maybe
(it's the output of a stateprocessing step, not a possiblyexisting value), but we've applied the same conversion to the wrapped value. Now, when we unwrap the value from rollDieDoubledS
we get double what we would have got had we unwrapped rollDieS
.
Suppose we also wanted rollTwoSummedS :: State StdGen Int
? In the prologue section did sz < (++) <$> getLine <*> getLine
, and again we can do something similar:
rollTwoSummedS :: State StdGen Int
rollTwoSummedS = (+) <$> rollDieS <*> rollDieS
This code depends on State
being also an Applicative
, but not on it being a Monad
. It will ensure each of the rollDieS
actions is executed in order, and chain the state correctly between them. It will then repackage the combination as a stateprocessing function wrapped in a State
wrapper. The combined function will return the sum of the two successive throws (and also, but quietly, ensure the state is added as an input parameter and an output value).
The Control.Applicative
module provides a function liftA2
:
liftA2 f u v = f <$> u <*> v
Using this, it rollTwoSummedS
could be defined as:
import Control.Applicative this needs to be at the top of your file
rollTwoSummedS = liftA2 (+) rollDieS rollDieS
Exercises 


More is said later on the relationship between Functor
, Applicative
and Monad
, and choosing which one to use.
Pseudorandom values of different types
editWe saw that randomIO :: Random a => IO a
can return a value of a type other than Int
. So can its IO
free equivalent random :: (Random a, RandomGen g) => g > (a, g)
.
Because State StdGen
is "agnostic" in regard to the type of the pseudorandom value it produces, we can write a similarly "agnostic" function that provides a pseudorandom value of unspecified type (as long as it is an instance of Random
):
getRandomS :: Random a => State StdGen a
getRandomS = state random
Compared to rollDieS
, this function does not specify the Int
type in its signature and uses random
instead of randomR
; otherwise, it is just the same. getRandomS
can be used for any instance of Random
:
GHCi> evalState getRandomS (mkStdGen 0) :: Bool
True
GHCi> evalState getRandomS (mkStdGen 0) :: Char
'\64685'
GHCi> evalState getRandomS (mkStdGen 0) :: Double
0.9872770354820595
GHCi> evalState getRandomS (mkStdGen 0) :: Integer
2092838931
Indeed, it becomes quite easy to conjure all these at once:
someTypes :: State StdGen (Int, Float, Char)
someTypes = liftA3 (,,) getRandomS getRandomS getRandomS
allTypes :: State StdGen (Int, Float, Char, Integer, Double, Bool, Int)
allTypes = (,,,,,,) <$> getRandomS
<*> getRandomS
<*> getRandomS
<*> getRandomS
<*> getRandomS
<*> getRandomS
<*> getRandomS
For writing allTypes
, there is no liftA7
,^{[6]} and so we resort to plain old (<*>)
instead. Using it, we can apply the tuple constructor to each of the seven random values in the State StdGen
monadic context.
allTypes
provides pseudorandom values for all default instances of Random
; an additional Int
is inserted at the end to prove that the generator is not the same, as the two Int
s will be different.
GHCi> evalState allTypes (mkStdGen 0)
GHCi>(2092838931,9.953678e4,'\825586',868192881,0.4188001483955421,False,316817438)
(Probably) don't put
or get
edit
In the turnstile example, we used put
to set the state and get
to access it. Can we do the same here? Well, yes we can, but there's probably no need.
In that example, we had to code functions (like pushS
) that used the current state to determine outputs and new states, or (like testTurnstile
) that set required states as part of a processing sequence. With our random number examples, all generation, inspection and update of the StdGen
state is done internally within the System.Random
module, without us having to know how.
Now, in our first implementation of rollPair
we were aware of the StdGen
state: we took it as a parameter, threaded it through the successive steps and returned the final state. If we really wanted to use the value (perhaps we wanted to put it in a debugging message using trace
) then we did have the opportunity. And, with our State
monad we still do. The following shows one usage:
rollDieS :: State StdGen Int
rollDieS = do s0 < get
let (value, s1) = randomR (1,6) s0
put s1
return value
This does spell out all of the steps the State
monad takes for us, but would be a rather perverse implementation since the whole point of the monad is so we don't have to spell them out.
Exercises 


Better Random Numbers
editOther than our initial use of randomIO
, all of the above examples have used mkStdGen
, and all with the same seed value 666. This would make for a pretty boring game, where exactly the same dice were rolled each time. (Though this might be useful, e.g. when testing your program.) How can we get better random numbers? Like this:
getRandomPair :: IO (Int, Int)
getRandomPair = do
s < newStdGen
return $ evalState rollPairS s
newStdGen
is (effectively) defined as newStdGen :: IO StdGen
. It is an IO action that spawns a new random state from the same global random state used by randomIO
. It also updates that global state, so that further uses of newStdGen
give a different value.
So, aren't we back a square one, being dependent on IO
? No we're not. We have gained all the power of the State
monad to build up chains of dicerolling steps which we can assemble into bigger and bigger statetransformation functions. We can do all of that without IO
. In the turnstile example, we didn't need IO
at all (although we probably would if we wanted to put our code into some kind of application), and for some uses of random numbers, having the same numbers each time might be beneficial. We only needed IO
to get "really random" numbers, and we may well need newStdGen
only once in a program. Chances are that it would be alongside other IO
actions, for example:
main :: IO ()
main = do
s < newStdGen
let (r1, r2) = evalState rollPairS s
putStrLn $ "You rolled twice and got " ++ show r1 ++ " and " ++ show r2 ++ "."
Handling Combined States
editSuppose we wanted to create a random turnstile, where each visitor would be given a random turnstile input: either they insert a coin (but are not allowed through); or they get to push the arm (and go through if it opens, but are otherwise sent away).
Here's one useful bit of code:
randomInputS :: State StdGen TurnstileInput
randomInputS = do
b < getRandomS
return $ if b then Coin else Push
This allows us to generate random turnstileInput
values^{[7]}. However, our random turnstile machine needs to track both the state of a random number generator and the state of the turnstile. We want to write a function like this:
randomTurnS :: State (StdGen, TurnstileState) TurnstileOutput
And this function needs to call both randomInputS
(which is in the State StdGen
monad) and turnS
(which is in the State TurnstileState
monad).
Exercises 


Much of the code in randomTurnS
deals with managing the state: accessing the combined state, unpacking subcomponents, forwarding them to the individual State monads, recombining them and putting the combined state back. The state management code is not too bad in this case, but could easily become cumbersome in a more complex function. And it is something we wanted the State
monad to hide from us.
StateProcessing a Subcomponent
editIdeally we'd want some utility function(s) that allow us to invoke a State StdGen
monad function (or State TurnstileState
monad function) from within a State (StdGen, TurnstileState)
monad function. These function(s) should take care of the state management for us, ensuring that the right subcomponent of the combined state is updated.
Here's one such a function that works for any combined state represented as a pair, and performs the state update on the fst of the pair:
processingFst :: State a o > State (a,b) o
processingFst m = do
(s1,s2) < get
let (o,s1') = runState m s1
put (s1',s2)
return o
Note the type:
GHCi> :t processingFst randomInputS
processingFst randomInputS :: State (StdGen, b) TurnstileInput
processingFst
"converts" a State
monad (in this case with state type StdGen
) to another State
monad (in this case with state type (StdGen, b)
, where b
can be any type, even a TurstileState
).
Exercises 


Note how randomTurnS
is no longer directly involved in the details of the state management, and its business logic is much more apparent.
Generic Subcomponent Processing
editWe can see that processingFst
and processingSnd
are very similar. They both extract a subcomponent of a combined state, runState
on that subcomponent, then update the combined state with the new value of the subcomponent.
Let's combine them into a single generic subcomponent processing function. To do this, we could pass in separate parameters, one of type (cmb > sub)
(a function that extracts a subcomponent from a combined state value), and another of type (cmb > sub > cmb)
(a function that, given a combined value and a new value for a subcomponent, returns the revised combined value with the updated subcomponent). However, it's a bit neater to package these two functions together in a type which we'll call Lens
:
data Lens cmb sub = Lens
{ view :: cmb > sub,
set :: cmb > sub > cmb
}
We can provide specific lenses onto the fst and snd elements in a pair:
fstL :: Lens (a,b) a
fstL = Lens fst (\(_,y) x > (x,y))
sndL :: Lens (a,b) b
sndL = Lens snd (\(x,_) y > (x,y))
So now:
GHCi> view fstL ("fred", 5)
"fred"
GHCi> set fstL ("fred", 5) "sue"
("sue",5)
Note
Lenses that are more sophisticated and powerful are described later, but it's also harder to understand how they work. Our simple lenses are sufficient for now, but you might want to update the random turnstile code to use "proper lenses" later on.
We can now replace processingFst
and processingSnd
with our generic function.
Exercises 


Our final random turnstile code is neater, with three separate logical functions segregated:
 state management (now in a single
processing
utility function, which can be reused elsewhere);  subcomponent accessing and update (using
Lens
, which can also be reused elsewhere^{[8]}.); and  the "business logic" of the turnstile, which is now very apparent.
In our first implementation, all three of these were muddled together.
Let's give it a go:
GHCi> g < newStdGen
GHCi> evalState (replicateM 10 randomTurnS) (g, Locked)
[Thank,Open,Tut,Thank,Thank,Open,Tut,Tut,Tut,Thank]
I'm not sure we'll sell many of them, though.
Notes
 ↑ Hence our finitestate machine is a transducer.
 ↑ This comparison of
Applicative
andMonad
explains why you can't use justsequence
forhastyPersonS
. In summary, it's because the actions taken (and the number of values in the result list) depend on the outcome of the first action (the initial attempt to push the arm), whereas for the first two always execute two actions and return the corresponding two results.  ↑ The technical term for both
()
and its type is unit.  ↑ Random numbers can also be used for many other things, for example simulation, statistical analysis and cryptography
 ↑ A common source of seeds is the current date and time as given by the internal clock of the computer. Assuming the clock is functioning correctly, it can provide unique seeds suitable for most daytoday needs (as opposed to applications which demand highquality randomness, as in cryptography or statistics).
 ↑ Beyond
liftA3
, the standard libraries only provide the monadonlyliftM4
andliftM5
inControl.Monad
.  ↑ Alternatively, we could make
TurnstileInput
an instance ofUniform
, but this code seems easier.  ↑ We could use
Control.Lens
for this as described in the later chapter. This module also provides more lenses, automatic creation of lenses for custom data types, easy combining of lenses for deeplynested subcomponents, etc. Also,Control.Lens.Combinators
includes azoom
function that is more generic thanprocessing
.