Control.Monad.Random.Lazy

A random monad parameterized by the type g of the generator to carry.

The return function leaves the generator unchanged, while >>= uses the final generator of the first computation as the initial generator of the second.

Construct a random monad computation from a function. (The inverse of runRand.)

Unwrap a random monad computation as a function. (The inverse of liftRand.)

Evaluate a random computation with the given initial generator and return the final value, discarding the final generator.

• evalRand m s = fst (runRand m s)

Evaluate a random computation with the given initial generator and return the final generator, discarding the final value.

• execRand m s = snd (runRand m s)

Map both the return value and final generator of a computation using the given function.

• runRand (mapRand f m) = f . runRand m

withRand f m executes action m on a generator modified by applying f.

• withRand f m = modify f >> m

Evaluate a random computation in the IO monad, splitting the global standard generator to get a new one for the computation.

A random transformer monad parameterized by:

• g - The generator.

• m - The inner monad.

The return function leaves the generator unchanged, while >>= uses the final generator of the first computation as the initial generator of the second.

Construct a random monad computation from an impure function. (The inverse of runRandT.)

Unwrap a random monad computation as an impure function. (The inverse of liftRandT.)

Evaluate a random computation with the given initial generator and return the final value, discarding the final generator.

• evalRandT m g = liftM fst (runRandT m g)

Evaluate a random computation with the given initial generator and return the final generator, discarding the final value.

• execRandT m g = liftM snd (runRandT m g)

Map both the return value and final generator of a computation using the given function.

• runRandT (mapRandT f m) = f . runRandT m

withRandT f m executes action m on a generator modified by applying f.

• withRandT f m = modify f >> m

Evaluate a random computation that is embedded in the IO monad, splitting the global standard generator to get a new one for the computation.

Generates a ByteString of the specified size using a pure pseudo-random number generator. See uniformByteStringM for the monadic version.

Examples

>>> import System.Random>>> import Data.ByteString>>> let pureGen = mkStdGen 137>>> :seti -Wno-deprecations>>> unpack . fst . genByteString 10 $ pureGen[51,123,251,37,49,167,90,109,1,4]

Gets the global pseudo-random number generator. Extracts the contents of globalStdGen

Uses the supplied function to get a value from the current global random generator, and updates the global generator with the new generator returned by the function. For example, rollDice produces a pseudo-random integer between 1 and 6:

>>> rollDice = getStdRandom (randomR (1, 6))>>> replicateM 10 (rollDice :: IO Int)[1,1,1,4,5,6,1,2,2,5]

This is an outdated function and it is recommended to switch to its equivalent applyAtomicGen instead, possibly with the globalStdGen if relying on the global state is acceptable.

>>> import System.Random.Stateful>>> rollDice = applyAtomicGen (uniformR (1, 6)) globalStdGen>>> replicateM 10 (rollDice :: IO Int)[2,1,1,5,4,3,6,6,3,2]

Initialize StdGen using system entropy (i.e. /dev/urandom) when it is available, while falling back on using system time as the seed.

Applies split to the current global pseudo-random generator globalStdGen, updates it with one of the results, and returns the other.

A variant of randomM that uses the global pseudo-random number generator globalStdGen.

>>> import Data.Int>>> randomIO :: IO Int32114794456

This function is equivalent to getStdRandom random and is included in this interface for historical reasons and backwards compatibility. It is recommended to use uniformM instead, possibly with the globalStdGen if relying on the global state is acceptable.

>>> import System.Random.Stateful>>> uniformM globalStdGen :: IO Int32-1768545016

A variant of randomRM that uses the global pseudo-random number generator globalStdGen

>>> randomRIO (2020, 2100) :: IO Int2028

Similar to randomIO, this function is equivalent to getStdRandom randomR and is included in this interface for historical reasons and backwards compatibility. It is recommended to use uniformRM instead, possibly with the globalStdGen if relying on the global state is acceptable.

>>> import System.Random.Stateful>>> uniformRM (2020, 2100) globalStdGen :: IO Int2044

Sets the global pseudo-random number generator. Overwrites the contents of globalStdGen

Generates a ByteString of the specified size using a pure pseudo-random number generator. See uniformByteStringM for the monadic version.

Examples

>>> import System.Random>>> import Data.ByteString (unpack)>>> let pureGen = mkStdGen 137>>> unpack . fst $ uniformByteString 10 pureGen[51,123,251,37,49,167,90,109,1,4]

Fill in a slice of a mutable byte array with randomly generated bytes. This function does not fail, instead it clamps the offset and number of bytes to generate into a valid range.

Produce a list of the supplied length with elements generated uniformly.

See uniformListM for a stateful counterpart.

Examples

>>> let gen = mkStdGen 2023>>> import Data.Word (Word16)>>> uniformList 5 gen :: ([Word16], StdGen)([56342,15850,25292,14347,13919],StdGen {unStdGen = SMGen 6446154349414395371 1920468677557965761})

Produce a list of the supplied length with elements generated uniformly.

See uniformListM for a stateful counterpart.

Examples

>>> let gen = mkStdGen 2023>>> uniformListR 10 (20, 30) gen :: ([Int], StdGen)([26,30,27,24,30,25,27,21,27,27],StdGen {unStdGen = SMGen 12965503083958398648 1920468677557965761})

Produce an infinite list of pseudo-random values in a specified range. Same as uniforms, integrates nicely with list fusion. There is no way to recover the final generator, therefore either use split before calling uniformRs or use uniformListR instead.

Similar to randomRs, except it relies on UniformRange type class instead of Random.

Examples

>>> let gen = mkStdGen 2023>>> take 5 $ uniformRs (10, 100) gen :: [Int][32,86,21,57,39]

Same as uniformByteArray False, but for ShortByteString.

Returns a ShortByteString of length n filled with pseudo-random bytes.

Examples

>>> import System.Random>>> import Data.ByteString.Short (unpack)>>> let pureGen = mkStdGen 137>>> unpack . fst $ uniformShortByteString 10 pureGen[51,123,251,37,49,167,90,109,1,4]

Shuffle elements of a list in a uniformly random order.

Examples

>>> uniformShuffleList "ELVIS" $ mkStdGen 252("LIVES",StdGen {unStdGen = SMGen 17676540583805057877 5302934877338729551})

Produce an infinite list of pseudo-random values. Integrates nicely with list fusion. Naturally, there is no way to recover the final generator, therefore either use split before calling uniforms or use uniformList instead.

Similar to randoms, except it relies on Uniform type class instead of Random

Examples

>>> let gen = mkStdGen 2023>>> import Data.Word (Word16)>>> take 5 $ uniforms gen :: [Word16][56342,15850,25292,14347,13919]

Constructs a StdGen deterministically from an Int seed. See mkStdGen64 for a Word64 variant that is architecture agnostic.

Constructs a StdGen deterministically from a Word64 seed.

The difference between mkStdGen is that mkStdGen64 will work the same on 64-bit and 32-bit architectures, while the former can only use 32-bit of information for initializing the psuedo-random number generator on 32-bit operating systems

Efficiently generates a sequence of pseudo-random bytes in a platform independent manner.

Construct a Seed from a ByteArray of expected length. Whenever ByteArray does not match the SeedSize specified by the pseudo-random generator, this function will fail.

Just like mkSeed, but uses ByteString as argument. Results in a memcopy of the seed.

Convert a Seed to a list of 64bit words.

Construct a seed from a list of 64-bit words. At most SeedSize many bytes will be used.

This is a function that shows the name of the generator type, which is useful for error reporting.

Get the expected size of the Seed in number bytes

Just like seedSize, except it accepts a proxy as an argument.

Unwrap the Seed and get the underlying ByteArray

Just like unSeed, but produced a ByteString. Results in a memcopy of the seed.

Helper function that allows for operating directly on the Seed, while supplying a function that uses the pseudo-random number generator that is constructed from that Seed.

Example

>>> :set -XTypeApplications>>> import System.Random>>> withSeed (nonEmptyToSeed (pure 2024) :: Seed StdGen) (uniform @Int)(1039666877624726199,Seed [0xe9, 0x07, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00])

Read the seed from a file and use it for constructing a pseudo-random number generator. After supplied action has been applied to the constructed generator, the resulting generator will be converted back to a seed and written to the same file.

Same as withSeed, except it is useful with monadic computation and frozen generators.

See withSeedMutableGen for a helper that also handles seeds for mutable pseduo-random number generators.

The class of types for which random values can be generated. Most instances of Random will produce values that are uniformly distributed on the full range, but for those types without a well-defined "full range" some sensible default subrange will be selected.

Random exists primarily for backwards compatibility with version 1.1 of this library. In new code, use the better specified Uniform and UniformRange instead.

A type class for data with a finite number of inhabitants. This type class is used in the default implementation of System.Random.Stateful.Uniform.

Users are not supposed to write instances of Finite manually. There is a default implementation in terms of Generic instead.

>>> :seti -XDeriveGeneric -XDeriveAnyClass>>> import GHC.Generics (Generic)>>> data MyBool = MyTrue | MyFalse deriving (Generic, Finite)>>> data Action = Code MyBool | Eat (Maybe Bool) | Sleep deriving (Generic, Finite)

RandomGen is an interface to pure pseudo-random number generators.

StdGen is the standard RandomGen instance provided by this library.

This is a binary form of pseudo-random number generator's state. It is designed to be safe and easy to use for input/output operations like restoring from file, transmitting over the network, etc.

Constructor is not exported, becasue it is important for implementation to enforce the invariant of the underlying byte array being of the exact same length as the generator has specified in SeedSize. Use mkSeed and unSeed to get access to the raw bytes in a safe manner.

Pseudo-random generators that can be split into two separate and independent psuedo-random generators should provide an instance for this type class.

Historically this functionality was included in the RandomGen type class in the split function, however, few pseudo-random generators possess this property of splittability. This lead the old split function being usually implemented in terms of error.

The standard pseudo-random number generator.

The class of types for which a uniformly distributed value can be drawn from all possible values of the type.

The class of types for which a uniformly distributed value can be drawn from a range.

Interface for converting a pure pseudo-random number generator to and from non-empty sequence of bytes. Seeds are stored in Little-Endian order regardless of the platform it is being used on, which provides cross-platform compatibility, while providing optimal performance for the most common platform type.

Conversion to and from a Seed serves as a building block for implementing serialization for any pure or frozen pseudo-random number generator.

It is not trivial to implement platform independence. For this reason this type class has two alternative ways of creating an instance for this class. The easiest way for constructing a platform indepent seed is by converting the inner state of a generator to and from a list of 64 bit words using toSeed64 and fromSeed64 respectively. In that case cross-platform support will be handled automaticaly.

>>> :set -XDataKinds -XTypeFamilies>>> import Data.Word (Word8, Word32)>>> import Data.Bits ((.|.), shiftR, shiftL)>>> import Data.List.NonEmpty (NonEmpty ((:|)))>>> data FiveByteGen = FiveByteGen Word8 Word32 deriving Show>>> :{instance SeedGen FiveByteGen where  type SeedSize FiveByteGen = 5  fromSeed64 (w64 :| _) =    FiveByteGen (fromIntegral (w64 `shiftR` 32)) (fromIntegral w64)  toSeed64 (FiveByteGen x1 x4) =    let w64 = (fromIntegral x1 `shiftL` 32) .|. fromIntegral x4     in (w64 :| []):}

>>> FiveByteGen 0x80 0x01020304FiveByteGen 128 16909060>>> fromSeed (toSeed (FiveByteGen 0x80 0x01020304))FiveByteGen 128 16909060>>> toSeed (FiveByteGen 0x80 0x01020304)Seed [0x04, 0x03, 0x02, 0x01, 0x80]>>> toSeed64 (FiveByteGen 0x80 0x01020304)549772722948 :| []

However, when performance is of utmost importance or default handling of cross platform independence is not sufficient, then an adventurous developer can try implementing conversion into bytes directly with toSeed and fromSeed.

Properties that must hold:

> fromSeed (toSeed gen) == gen

> fromSeed64 (toSeed64 gen) == gen

Note, that there is no requirement for every Seed to roundtrip, eg. this proprty does not even hold for StdGen:

>>> let seed = nonEmptyToSeed (0xab :| [0xff00]) :: Seed StdGen>>> seed == toSeed (fromSeed seed)False

Number of bytes that is required for storing the full state of a pseudo-random number generator. It should be big enough to satisfy the roundtrip property:

> fromSeed (toSeed gen) == gen

Same as >>=, but with the arguments interchanged.

as >>= f == f =<< as

In many situations, the liftM operations can be replaced by uses of ap, which promotes function application.

return f `ap` x1 `ap` ... `ap` xn

is equivalent to

liftM<n> f x1 x2 ... xn

Examples

>>> pure (\x y z -> x + y * z) `ap` Just 1 `ap` Just 5 `ap` Just 10Just 51

The join function is the conventional monad join operator. It is used to remove one level of monadic structure, projecting its bound argument into the outer level.

'join bss' can be understood as the do expression

do bs <- bss
   bs

Examples

>>> join [[1, 2, 3], [4, 5, 6], [7, 8, 9]][1,2,3,4,5,6,7,8,9]

>>> join (Just (Just 3))Just 3

A common use of join is to run an IO computation returned from an GHC.Conc.STM transaction, since GHC.Conc.STM transactions can't perform IO directly. Recall that

GHC.Internal.Conc.atomically :: STM a -> IO a

is used to run GHC.Conc.STM transactions atomically. So, by specializing the types of GHC.Internal.Conc.atomically and join to

GHC.Internal.Conc.atomically :: STM (IO b) -> IO (IO b)
join       :: IO (IO b)  -> IO b

we can compose them as

join . GHC.Internal.Conc.atomically :: STM (IO b) -> IO b

to run an GHC.Conc.STM transaction and the IO action it returns.

Promote a function to a monad. This is equivalent to fmap but specialised to Monads.

Promote a function to a monad, scanning the monadic arguments from left to right.

Examples

>>> liftM2 (+) [0,1] [0,2][0,2,1,3]

>>> liftM2 (+) (Just 1) NothingNothing

>>> liftM2 (+) (+ 3) (* 2) 518

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

Conditional execution of Applicative expressions. For example,

Examples

when debug (putStrLn "Debugging")

will output the string Debugging if the Boolean value debug is True, and otherwise do nothing.

>>> putStr "pi:" >> when False (print 3.14159)pi:

Strict version of <$>.

Right-to-left composition of Kleisli arrows. (>=>), with the arguments flipped.

Note how this operator resembles function composition (.):

(.)   ::            (b ->   c) -> (a ->   b) -> a ->   c
(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c

Left-to-right composition of Kleisli arrows.

'(bs >=> cs) a' can be understood as the do expression

do b <- bs a
   cs b

or in terms of (>>=) as

bs a >>= cs

This generalizes the list-based Data.List.filter function.

runIdentity (filterM (Identity . p) xs) == filter p xs

Examples

>>> filterM (\x -> do      putStrLn ("Keep: " ++ show x ++ "?")      answer <- getLine      pure (answer == "y"))    [1, 2, 3]Keep: 1?yKeep: 2?nKeep: 3?y[1,3]

>>> filterM (\x -> do      putStr (show x)      x' <- readLn      pure (x == x'))    [1, 2, 3]122233[2,3]

The foldM function is analogous to foldl, except that its result is encapsulated in a monad. Note that foldM works from left-to-right over the list arguments. This could be an issue where (>>) and the `folded function' are not commutative.

foldM f a1 [x1, x2, ..., xm]

==

do
  a2 <- f a1 x1
  a3 <- f a2 x2
  ...
  f am xm

If right-to-left evaluation is required, the input list should be reversed.

Note: foldM is the same as foldlM

Like foldM, but discards the result.

Repeat an action indefinitely.

Examples

A common use of forever is to process input from network sockets, System.IO.Handles, and channels (e.g. MVar and Chan).

For example, here is how we might implement an echo server, using forever both to listen for client connections on a network socket and to echo client input on client connection handles:

echoServer :: Socket -> IO ()
echoServer socket = forever $ do
  client <- accept socket
  forkFinally (echo client) (\_ -> hClose client)
  where
    echo :: Handle -> IO ()
    echo client = forever $
      hGetLine client >>= hPutStrLn client

Note that "forever" isn't necessarily non-terminating. If the action is in a MonadPlus and short-circuits after some number of iterations. then forever actually returns mzero, effectively short-circuiting its caller.

Conditional failure of Alternative computations. Defined by

guard True  = pure ()
guard False = empty

Examples

Common uses of guard include conditionally signalling an error in an error monad and conditionally rejecting the current choice in an Alternative-based parser.

As an example of signalling an error in the error monad Maybe, consider a safe division function safeDiv x y that returns Nothing when the denominator y is zero and Just (x `div` y) otherwise. For example:

>>> safeDiv 4 0Nothing

>>> safeDiv 4 2Just 2

A definition of safeDiv using guards, but not guard:

safeDiv :: Int -> Int -> Maybe Int
safeDiv x y | y /= 0    = Just (x `div` y)
            | otherwise = Nothing

A definition of safeDiv using guard and Monad do-notation:

safeDiv :: Int -> Int -> Maybe Int
safeDiv x y = do
  guard (y /= 0)
  return (x `div` y)

The mapAndUnzipM function maps its first argument over a list, returning the result as a pair of lists. This function is mainly used with complicated data structures or a state monad.

Direct MonadPlus equivalent of Data.List.filter.

Examples

The Data.List.filter function is just mfilter specialized to the list monad:

Data.List.filter = ( mfilter :: (a -> Bool) -> [a] -> [a] )

An example using mfilter with the Maybe monad:

>>> mfilter odd (Just 1)Just 1>>> mfilter odd (Just 2)Nothing

replicateM n act performs the action act n times, and then returns the list of results.

replicateM n (pure x) == replicate n x

Examples

>>> replicateM 3 getLinehiheyahiya["hi","heya","hiya"]

>>> import Control.Monad.State>>> runState (replicateM 3 $ state $ \s -> (s, s + 1)) 1([1,2,3],4)

Like replicateM, but discards the result.

Examples

>>> replicateM_ 3 (putStr "a")aaa

The reverse of when.

Examples

>>> do x <- getLine       unless (x == "hi") (putStrLn "hi!")comingupwithexamplesisdifficulthi!

>>> unless (pi > exp 1) NothingJust ()

The zipWithM function generalizes zipWith to arbitrary applicative functors.

zipWithM_ is the extension of zipWithM which ignores the final result.

forM_ is mapM_ with its arguments flipped. For a version that doesn't ignore the results see Data.Traversable.forM.

forM_ is just like for_, but specialised to monadic actions.

Map each element of a structure to a monadic action, evaluate these actions from left to right, and ignore the results. For a version that doesn't ignore the results see Data.Traversable.mapM.

mapM_ is just like traverse_, but specialised to monadic actions.

The sum of a collection of actions using (<|>), generalizing concat.

msum is just like asum, but specialised to MonadPlus.

Examples

Basic usage, using the MonadPlus instance for Maybe:

>>> msum [Just "Hello", Nothing, Just "World"]Just "Hello"

Evaluate each monadic action in the structure from left to right, and ignore the results. For a version that doesn't ignore the results see Data.Traversable.sequence.

sequence_ is just like sequenceA_, but specialised to monadic actions.

void value discards or ignores the result of evaluation, such as the return value of an System.IO.IO action.

Examples

Replace the contents of a Maybe Int with unit:

>>> void NothingNothing

>>> void (Just 3)Just ()

Replace the contents of an Either Int Int with unit, resulting in an Either Int ():

>>> void (Left 8675309)Left 8675309

>>> void (Right 8675309)Right ()

Replace every element of a list with unit:

>>> void [1,2,3][(),(),()]

Replace the second element of a pair with unit:

>>> void (1,2)(1,())

Discard the result of an System.IO.IO action:

>>> mapM print [1,2]12[(),()]

>>> void $ mapM print [1,2]12

forM is mapM with its arguments flipped. For a version that ignores the results see Data.Foldable.forM_.

A type f is a Functor if it provides a function fmap which, given any types a and b lets you apply any function from (a -> b) to turn an f a into an f b, preserving the structure of f. Furthermore f needs to adhere to the following:

Identity

fmap id == id

Composition

fmap (f . g) == fmap f . fmap g

Note, that the second law follows from the free theorem of the type fmap and the first law, so you need only check that the former condition holds. See these articles by School of Haskell or David Luposchainsky for an explanation.

The Monad class defines the basic operations over a monad, a concept from a branch of mathematics known as category theory. From the perspective of a Haskell programmer, however, it is best to think of a monad as an abstract datatype of actions. Haskell's do expressions provide a convenient syntax for writing monadic expressions.

Instances of Monad should satisfy the following:

Left identity

return a >>= k = k a

Right identity

m >>= return = m

Associativity

m >>= (\x -> k x >>= h) = (m >>= k) >>= h

Furthermore, the Monad and Applicative operations should relate as follows:

• pure = return

• m1 <*> m2 = m1 >>= (\x1 -> m2 >>= (\x2 -> return (x1 x2)))

The above laws imply:

• fmap f xs  =  xs >>= return . f

• (>>) = (*>)

and that pure and (<*>) satisfy the applicative functor laws.

The instances of Monad for GHC.List.List, Maybe and System.IO.IO defined in the Prelude satisfy these laws.

Monads that also support choice and failure.

When a value is bound in do-notation, the pattern on the left hand side of <- might not match. In this case, this class provides a function to recover.

A Monad without a MonadFail instance may only be used in conjunction with pattern that always match, such as newtypes, tuples, data types with only a single data constructor, and irrefutable patterns (~pat).

Instances of MonadFail should satisfy the following law: fail s should be a left zero for >>=,

fail s >>= f  =  fail s

If your Monad is also MonadPlus, a popular definition is

fail _ = mzero

fail s should be an action that runs in the monad itself, not an exception (except in instances of MonadIO). In particular, fail should not be implemented in terms of error.

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see Data.Foldable.mapM_.

Examples

mapM is literally a traverse with a type signature restricted to Monad. Its implementation may be more efficient due to additional power of Monad.

Evaluate each monadic action in the structure from left to right, and collect the results. For a version that ignores the results see Data.Foldable.sequence_.

Examples

Basic usage:

The first two examples are instances where the input and and output of sequence are isomorphic.

>>> sequence $ Right [1,2,3,4][Right 1,Right 2,Right 3,Right 4]

>>> sequence $ [Right 1,Right 2,Right 3,Right 4]Right [1,2,3,4]

The following examples demonstrate short circuit behavior for sequence.

>>> sequence $ Left [1,2,3,4]Left [1,2,3,4]

>>> sequence $ [Left 0, Right 1,Right 2,Right 3,Right 4]Left 0

fix f is the least fixed point of the function f, i.e. the least defined x such that f x = x.

When f is strict, this means that because, by the definition of strictness, f ⊥ = ⊥ and such the least defined fixed point of any strict function is ⊥.

Examples

We can write the factorial function using direct recursion as

>>> let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5120

This uses the fact that Haskell’s let introduces recursive bindings. We can rewrite this definition using fix,

Instead of making a recursive call, we introduce a dummy parameter rec; when used within fix, this parameter then refers to fix’s argument, hence the recursion is reintroduced.

>>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5120

Using fix, we can implement versions of repeat as fix . (:) and cycle as fix . (++)

>>> take 10 $ fix (0:)[0,0,0,0,0,0,0,0,0,0]

>>> map (fix (\rec n -> if n < 2 then n else rec (n - 1) + rec (n - 2))) [1..10][1,1,2,3,5,8,13,21,34,55]

Implementation Details

The current implementation of fix uses structural sharing

fix f = let x = f x in x

A more straightforward but non-sharing version would look like

fix f = f (fix f)

Monads having fixed points with a 'knot-tying' semantics. Instances of MonadFix should satisfy the following laws:

Purity

mfix (return . h) = return (fix h)

Left shrinking (or Tightening)

mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)

Sliding

mfix (liftM h . f) = liftM h (mfix (f . h))

, for strict

h

.

Nesting

mfix (\x -> mfix (\y -> f x y)) = mfix (\x -> f x x)

This class is used in the translation of the recursive do notation supported by GHC and Hugs.