User:Duplode/State combined
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Stateful computations are very common operations, but the way they are usually implemented in procedural or objectoriented languages cannot be replicated in Haskell. A State
monad is introduced to allow states of any complexity to be represented.
The Problem with Haskell and State[edit]
If you programmed in any language before, chances are you wrote some functions that "kept state". In case you did not encounter the concept before, a state is one or more variables that are required to perform some computation, but are not among the arguments of the relevant function. In fact, objectoriented languages like C++ make extensive usage of state variables in objects in the form of member variables. Procedural languages like C use variables outside the current scope to keep track of state.
In Haskell we can very often keep track of state by passing parameters or by pattern matching of various sorts, but in some cases it is appropriate to find a more general solution. We will consider the common example of generation of pseudorandom numbers in pure functions.
PseudoRandom Numbers[edit]
Generating actually random numbers is a very complicated subject; we will consider pseudorandom numbers. They are called "pseudo" because they are not really random, they only look like it. Starting from an initial state (commonly called the seed), they produce a sequence of numbers that have the appearance of being random.
Every time a pseudorandom number is requested, a global state is updated: that's the part we have problems with in Haskell, since it is a side effect from the point of view of the function requesting the number. Sequences of pseudorandom numbers can be replicated exactly if the initial seed and the algorithm is known.
Implementation in Haskell[edit]
Producing a pseudorandom number in most programming languages is very simple: there is usually a function, such as C or C++'s rand()
, that provides a pseudorandom value (or a random one, depending on the implementation). Haskell has a similar one in the System.Random
module:
> :module System.Random > :type randomIO randomIO :: (Random a) => IO a > randomIO 1557093684
Obviously, save eerie coincidences, the value you will obtain will be different. A disadvantage of randomIO
is that it requires us to utilise the IO
monad, which breaks purity requirements. Usage of the IO
monad is dictated by the process of updating the global generator state, so that the next time we call randomIO
the value will be different.
Implementation with Functional Purity[edit]
In general, we do not want to use the IO
monad if we can help it, because of the loss of guarantees on no side effects and functional purity. Indeed, we can build a local generator (as opposed to the global generator, invisible to us, that randomIO
uses) using mkStdGen
, and use it as seed for the random
function, which in turn returns a tuple with the pseudorandom number that we want and the generator to use the next time:
> :module System.Random > let generator = mkStdGen 0  "0" is our seed > random generator :: (Int, StdGen) (2092838931,1601120196 1655838864)
And in this case, since we are using exactly the same generator, you will obtain the same value 2092838931, always the same no matter how many times you call random
. We have now regained functional purity, but a function supposed to provide pseudorandom numbers that generates always the same value is not very helpful: what we need is a way to automate the extraction of the second member of the tuple (i.e. the new generator) and feed it to a new call to random
; and that is where the State
monad comes into the picture.
Definition of the State Monad[edit]
 Note: in some package systems used for GHC, the
Control.Monad.State
module is in a separate package, usually indicated by MTL (Monad Transformer Library).
The Haskell type State
is defined as a function that consumes state, and produces a result and the state after the result has been extracted. The definition is wrapped inside a newtype
to avoid pattern matching, so that no one can explicitly patternmatch and extract state unless we allow it.
newtype State state result = State { runState :: state > (result, state) }
The name State
is actually a misnomer: it is not the state itself, but rather a state processor.
Instantiating the Monad[edit]
Note also that State
has two type parameters, one for the state and one for the result: all other main types of monads have only one (Maybe
, lists, IO
). This means that, when we instantiate the monad, we are actually leaving the parameter for the state type:
instance Monad (State state_type)
This means that the "real" monad will be State String
, State Int
, or State SomeLargeDataStructure
, not State
itself.
The return
function is implemented as:
return :: result > State state result
return r = State ( \s > (r, s) )
In words, giving a value to return
produces a function, wrapped in the State
constructor: this function takes a state value, and returns it unchanged as the second member of a tuple, together with the specified result value.
Binding is a bit intricate:
(>>=) :: State state result_a > (result_a > State state result_b) > State state result_b
processor >>= processorGenerator = State $ \state >
let (result, state') = runState processor state
in runState (processorGenerator result) state'
The idea is that, given a state processor and a function that can generate another processor given the result of the first one, these two processors are combined to obtain a function that takes the initial state, and returns the second result and state (i.e. after the second function has processed them).
Setting and Accessing the State[edit]
The monad instantiation allows us to manipulate various state processors, but you may at this point wonder where exactly the state comes from in the first place. State state_type
is also an instance of the MonadState
class, which provides two additional functions:
put newState = State $ \_ > ((), newState)
This function will generate a state processor given a state. The processor's input will be disregarded, and the output will be a tuple carrying the state we provided. Since we do not care about the result (we are discarding the input, after all), the first element of the tuple will be null.
The specular operation is to read the state. This is accomplished by get
:
get = State $ \state > (state, state)
The resulting state processor is going to produce the input state
in both positions of the output tuple, as a result and as a state, so that it may be bound to other processors.
Getting Values and State[edit]
From the definition of State
, we know that runState
is an accessor to apply to a State a b
value to get the stateprocessing function; this function, given an initial state, will return the extracted value and the new state. Other similar, useful functions are evalState
and execState
, which work in a very similar fashion.
Function evalState
, given a State a b
and an initial state, will return the extracted value only, whereas execState
will return only the new state; it is possibly easiest to remember them as defined as:
evalState stateMonad value = fst ( runState stateMonad value )
execState stateMonad value = snd ( runState stateMonad value )
Example: Rolling Dice[edit]
Suppose 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, which we will now try to simulate with Haskell code. For starters, we will consider the result of throwing two dice: to do that, we resort to the function randomR
, which allows to specify an interval from which the pseudorandom values will be taken; in the case of a die, it is randomR (1,6)
.
In case we are willing to use the IO
monad, the implementation is quite simple, using the IO
version of randomR
:
import Control.Monad
import System.Random
rollDiceIO :: IO (Int, Int)
rollDiceIO = liftM2 (,) (randomRIO (1,6)) (randomRIO (1,6))
The two numbers will be returned as a tuple.
Exercises 


Getting Rid of the IO
Monad[edit]
Suppose that for some reason we do not want to use the IO
monad: we may want the function to stay pure, or we may want a sequence of numbers that is the same in every run, for repeatability.
To do that, we can produce a generator using the mkStdGen
function in the System.Random
library:
> mkStdGen 0 1 1
The argument to mkStdGen
is an Int
that functions as a seed. With that, we can generate a pseudorandom integer number in the interval between 1 and 6 with:
> randomR (1,6) (mkStdGen 0) (6,40014 40692)
We obtained a tuple with the result of the dice throw (6) and the new generator (40014 40692). A simple implementation that produces a tuple of two pseudorandom integers is then:
clumsyRollDice :: (Int, Int)
clumsyRollDice = (n, m)
where
(n, g) = randomR (1,6) (mkStdGen 0)
(m, _) = randomR (1,6) g
When we run the function, we get:
> clumsyRollDice (6, 6)
The implementation of clumsyRollDice
works, but we have to manually write the passing of generator g
from one where
clause to the other. This is pretty easy now, but will become increasingly cumbersome if we want to produce large sets of pseudorandom numbers. It is also errorprone: what if we pass one of the middle generators to the wrong line in the where
clause?
Exercises 


Introducing State
[edit]
We will now try to solve the clumsiness of the previous approach introducing the State StdGen
monad. For convenience, we give it a name with a type synonym:
type GeneratorState = State StdGen
Remember, however, that the type of GeneratorState Int
is really StdGen > (Int, StdGen)
, so it is not really the generator state, but a processor of the generator state. The generator state itself is produced by the mkStdGen
function. Note that GeneratorState
does not specify what type of values we are going to extract, only the type of the state.
We can now produce a function that, given a StdGen
generator, outputs a number between 1 and 6:
rollDie :: GeneratorState Int
rollDie = do generator < get
let (value, newGenerator) = randomR (1,6) generator
put newGenerator
return value
The do
notation is in this case much more readable; let's go through each of the steps:
 First, we take out the pseudorandom generator with
get
: the<
notation extracts the value from theGeneratorState
monad, not the state; since it is the state we want, we useget
, that extracts the state and outputs it as the value (look again at the definition ofget
above, if you have doubts).  Then, we use the
randomR
function to produce an integer between 1 and 6 using the generator we took; we also store the new generator graciously returned byrandomR
.  We then set the state to be the
newGenerator
using theput
function, so that the next call will use a different pseudorandom generator;  Finally, we inject the result into the
GeneratorState
monad usingreturn
.
We can finally use our monadic die:
> evalState rollDie (mkStdGen 0) 6
At this point, a legitimate question is why we have involved monads and built such an intricate framework only to do exactly what fst $ randomR (1,6)
does. The answer is illustrated by the following function:
rollDice :: GeneratorState (Int, Int)
rollDice = liftM2 (,) rollDie rollDie
We obtain a function producing two pseudorandom numbers in a tuple. Note that these are in general different:
> evalState rollDice (mkStdGen 666) (6,1)
That is because, under the hood, the monads are passing state to each other. This used to be very clunky using randomR (1,6)
, because we had to pass state manually; now, the monad is taking care of that for us. Assuming we know how to use the lifting functions, constructing intricate combinations of pseudorandom numbers (tuples, lists, whatever) has suddenly become much easier.
Exercises 


Producing PseudoRandom Values of Different Types: the Random
class[edit]
Until now, absorbed in the die example, we considered only Int
as the type of the produced pseudorandom number. However, already when we defined the GeneratorState
monad, we noticed that it did not specify anything about the type of the returned value. In fact, there is one implicit assumption about it, and that is that we can produce values of such a type with a call to random
.
Values that can be produced by random
and similar function are of types that are instances of the Random
class (capitalised). There are default implementations for Int
, Char
, Integer
, Bool
, Double
and Float
, so you can immediately generate any of those.
Since we noticed already that the GeneratorState
is "agnostic" in regard to the type of the pseudorandom value it produces, we can write down a similarly "agnostic" function, analogous to rollDie
, that provides a pseudorandom value of unspecified type (as long as it is an instance of Random
):
getRandom :: Random a => GeneratorState a
getRandom = do generator < get
let (value, newGenerator) = random generator
put newGenerator
return value
Compared to rollDie
, this function does not specify the Int
type in its signature and uses random
instead of randomR
; otherwise, it is just the same. What is notable is that getRandom
can be used for any instance of Random
:
> evalState getRandom (mkStdGen 0) :: Bool True > evalState getRandom (mkStdGen 0) :: Char '\64685' > evalState getRandom (mkStdGen 0) :: Double 0.9872770354820595 > evalState getRandom (mkStdGen 0) :: Integer 2092838931
Indeed, it becomes quite easy to conjure all these at once:
allTypes :: GeneratorState (Int, Float, Char, Integer, Double, Bool, Int)
allTypes = liftM (,,,,,,) getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
Here we are forced to used the ap
function, defined in Control.Monad
, since there exists no liftM7
. As you can see, its effect is to concatenate multiple monads into a lifting operation of the 7elementtuple operator, (,,,,,,)
. To understand what ap
does, look at its signature:
>:type ap ap :: (Monad m) => m (a > b) > m a > m b
remember then that type a
in Haskell can be a function as well as a value, and compare to:
>:type liftM (,,,,,,) getRandom liftM (,,,,,) getRandom :: (Random a1) => State StdGen (b > c > d > e > f > (a1, b, c, d, e, f))
The monad m
is obviously State StdGen
(which we "nicknamed" GeneratorState
), while ap
's first argument is function b > c > d > e > f > (a1, b, c, d, e, f)
. Applying ap
over and over (in this case 6 times), we finally get to the point where b
is an actual value (in our case, a 7element tuple), not another function.
So much for understanding the implementation. Function 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.
> evalState allTypes (mkStdGen 0) (2092838931,9.953678e4,'\825586',868192881,0.4188001483955421,False,316817438)
Exercises 


AM source text[edit]
The State monad[edit]
The State monad actually makes a lot more sense when viewed as a computation, rather than a container. Computations in State represents computations that depend on and modify some internal state. For example, say you were writing a program to model the three body problem. The internal state would be the positions, masses and velocities of all three bodies. Then a function, to, say, get the acceleration of a specific body would need to reference this state as part of its calculations.
The other important aspect of computations in State is that they can modify the internal state. Again, in the threebody problem, you could write a function that, given an acceleration for a specific body, updates its position.
The State monad is quite different from the Maybe and the list monads, in that it doesn't represent the result of a computation, but rather a certain property of the computation itself.
What we do is model computations that depend on some internal state as functions which take a state parameter. For example, if you had a function f :: String > Int > Bool
, and we want to modify it to make it depend on some internal state of type s
, then the function becomes f :: String > Int > s > Bool
. To allow the function to change the internal state, the function returns a pair of (return value, new state). So our function becomes f :: String > Int > s > (Bool, s)
It should be clear that this method is a bit cumbersome. However, the types aren't the worst of it: what would happen if we wanted to run two stateful computations, call them f
and g
, one after another, passing the result of f
into g
? The second would need to be passed the new state from running the first computation, so we end up 'threading the state':
fThenG :: (s > (a, s)) > (a > s > (b, s)) > s > (b, s) fThenG f g s = let (v, s' ) = f s  run f with our initial state s. (v', s'') = g v s'  run g with the new state s' and the result of f, v. in (v', s'')  return the latest state and the result of g
All this 'plumbing' can be nicely hidden by using the State monad. The type constructor State
takes two type parameters: the type of its environment (internal state), and the type of its output. (Even though the new state comes last in the result pair, the state type must come first in the type parameters, since the 'real' monad is bound to some particular type of state but lets the result type vary.) So State s a
indicates a stateful computation which depends on, and can modify, some internal state of type s
, and has a result of type a
. How is it defined? Well, simply as a function that takes some state and returns a pair of (value, new state):
newtype State s a = State (s > (a, s))
The above example of fThenG
is, in fact, the definition of >>=
for the State monad, which you probably remember from the first monads chapter.
The meaning of return[edit]
We mentioned earlier that return x
was the computation that 'did nothing' and just returned x
. This idea only really starts to take on any meaning in monads with sideeffects, like State. That is, computations in State have the opportunity to change the outcome of later computations by modifying the internal state. It's a similar situation with IO (because, of course, IO is just a special case of State).
return x
doesn't do this. A computation produced by return
generally won't have any sideeffects. The monad law return x >>= f == f x basically guarantees this, for most uses of the term 'sideeffect'.
Further reading[edit]
 A tour of the Haskell Monad functions by HenkJan van Tuyl
 All about monads by Jeff Newbern explains well the concept of monads as computations, using good examples. It also has a section outlining all the major monads, explains each one in terms of this computational view, and gives a full example.
 Monads by Eugene Kirpichov attempts to give a broader and more intuitive understanding of monads by giving nontrivial examples of them