Control.Auto.Process.Random

This module provides Autos (purely) generating entropy in the form of random or noisy processes. Note that every Auto here is completely deterministic --- given the same initial seed, one would expect the same stream of outputs on every run. Furthermore, if a serializable Auto is serialized and resumed, it will continue along the deterministic path dictated by the original seed given.

All of these Autos come in three flavors: one serializing one that works with any serializable RandomGen instance, one serializing one that works specifically with StdGen from System.Random, and one that takes any RandomGen (including StdGen) and runs it without the ability to serialize and resume deterministically.

The reason why there's a specialized StdGen version for all of these is that StdGen actually doesn't have a Serialize instance, so a rudimentary serialization process is provded with the StdGen versions.

The first class of generators take arbitrary g -> (b, g) functions: "Generate a random b, using the given function, and replace the seed with the resulting seed". Most "random" functions follow this pattern, including random and randomR, and if you are using something from MonadRandom, then you can use the runRand function to turn a Rand g b into a g -> (b, g), as well:

runRand :: RandomGen g => Rand g b -> (g -> (b, g))

These are useful for generating noise...a new random value at every stoep. They are entropy sources.

Alternatively, if you want to give up parallelizability and determinism and have your entire Auto be sequential, you can make your entire Auto run under Rand or RandT as its internal monad, from MonadRandom.

Auto (Rand g) a b
Auto (RandT g m) a b

In this case, if you wanted to pull a random number, you could do:

effect random :: (Random r, RandomGen g) => Auto (Rand g) a r
effect random :: (Random r, RandomGen g) => Auto (RandT g m) a r

Which pulls a random r from "thin air" (from the internal Rand monad).

However, you lose a great deal of determinism from this method, as your Autos are no longer deterministic with a given seed...and resumability becomes dependent on starting everything with the same seed every time you re-load your Auto. Also, Auto's are parallelizable, while Auto (Rand g)s are not.

As a compromise, you can then "seal" away the stateful part with sealState and hoistA:

sealRandom :: Monad m => Auto (RandT g m) a b -> g -> Auto m a b
sealRandom a0 = sealState . hoistA (StateT . runRandT)

sealRandom' :: Auto (Rand g) a b -> g -> Auto' a b
sealRandom' = sealRandom

Where hoistA turns an Auto (RandT g m) into an Auto m.

In this way, you can run any Auto under Rand or RandT as if it was a normal Auto "without" underlying randomness. (These functions aren't given here so that this library doesn't incurr a dependency on MonadRandom). This lets you compose your sequential/non-parallel parts in Rand and use it as a part of an Auto'.

The other generators given are for useful random processes you might run into. The first is a Blip stream that emits at random times with the given frequencyprobability. The second works Interval/ semantics from Control.Auto.Interval, and is a stream that is "on" or "off", chunks at a time, for random lengths. The average length of each on or off period is controlled by the parameter you pass in.

Streams of random values from random generators

rands

Arguments

:: (Serialize g, RandomGen g)

=> (g -> (b, g))

random generating function

-> g

initial generator

-> Auto m a b

Given a seed-consuming generating function of form g -> (b, g) (where g is the seed, and b is the result) and an initial seed, return an Auto that continually generates random values using the given generating funcion.

You'll notice that most of the useful functions from System.Random fit this form:

random  :: RandomGen g =>            g -> (b, g)
randomR :: RandomGen g => (b, b) -> (g -> (b, g))

If you are using something from MonadRandom, then you can use the runRand function to turn a Rand g b into a g -> (b, g):

runRand :: RandomGen g => Rand g b -> (g -> (b, g))

Here is an example using stdRands (for StdGen), but rands works exactly the same way, I promise!

>>> let g = mkStdGen 8675309
>>> let a = stdRands (randomR (1,100)) g :: Auto' a Int
>>> let (res, _) = stepAutoN' 10 a ()
>>> res
[67, 15, 97, 13, 55, 12, 34, 86, 57, 42]

Yeah, if you are using StdGen from System.Random, you'll notice that StdGen has no Serialize instance, so you can't use it with this; you have to either use stdRands or rands_ (if you don't want serialization/resumability).

In the context of these generators, resumability basically means deterministic behavior over re-loads...if "reloading", it'll ignore the seed you pass in, and use the original seed given when originally saved.

stdRands

Arguments

:: (StdGen -> (b, StdGen))

random generating function

-> StdGen

initial generator

-> Auto m a b

Like rands, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of rands for more information.

rands_

Arguments

:: RandomGen g

=> (g -> (b, g))

random generating function

-> g

initial generator

-> Auto m a b

The non-serializing/non-resuming version of rands.

randsM :: (Serialize g, RandomGen g, Monad m) => (g -> m (b, g)) -> g -> Auto m a b

Like rands, except taking a "monadic" random seed function g -> m (b, g), instead of g -> (b, g). Your random generating function has access to the underlying monad.

If you are using something from MonadRandom, then you can use the runRandT function to turn a RandT g m b into a g -> m (b, g):

runRandT :: (Monad m, RandomGen g)
           => RandT g m b -> (g -> m (b, g))

stdRandsM :: Monad m => (StdGen -> m (b, StdGen)) -> StdGen -> Auto m a b

Like randsM, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of randsM for more information.

randsM_ :: (RandomGen g, Monad m) => (g -> m (b, g)) -> g -> Auto m a b

The non-serializing/non-resuming version of randsM.

Lifting/wrapping random functions

arrRand :: (Serialize g, RandomGen g) => (a -> g -> (b, g)) -> g -> Auto m a b

Takes a "random function", or "random arrow" --- a function taking an input value and a starting seed/entropy generator and returning a result and an ending seed/entropy generator --- and turns it into an Auto that feeds its input into such a function and outputs the result, with a new seed every time.

>>> let f x = randomR (0 :: Int, x)
>>> streamAuto' (arrRandStd f (mkStdGen 782065)) [1..10]
-- [1,2,3,4,5,6,7,8,9,10] <- upper bounds
   [1,2,0,1,5,3,7,6,8,10] -- random number from 0 to upper bound

If you are using something from MonadRandom, then you can use the (runRand .) function to turn a a -> Rand g b into a a -> g -> (b, g):

(runRand .) :: RandomGen g => (a -> Rand g b) -> (a -> g -> (b, g))

(This is basically mkState, specialized.)

arrRandM :: (Monad m, Serialize g, RandomGen g) => (a -> g -> m (b, g)) -> g -> Auto m a b

Like arrRand, except the result is the result of a monadic action. Your random arrow function has access to the underlying monad.

If you are using something from MonadRandom, then you can use the (runRandT .) function to turn a a -> RandT m g b into a a -> g -> m (b, g):

(runRandT .) :: RandomGen g => (a -> RandT g b) -> (a -> g -> m (b, g))

arrRandStd :: (a -> StdGen -> (b, StdGen)) -> StdGen -> Auto m a b

Like arrRand, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of arrRand for more information.

arrRandStdM :: (a -> StdGen -> m (b, StdGen)) -> StdGen -> Auto m a b

Like arrRandM, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of arrRandM for more information.

arrRand_ :: RandomGen g => (a -> g -> (b, g)) -> g -> Auto m a b

The non-serializing/non-resuming version of arrRand.

arrRandM_ :: RandomGen g => (a -> g -> m (b, g)) -> g -> Auto m a b

The non-serializing/non-resuming version of arrRandM.

Random processes

Bernoulli (on/off) processes

bernoulli

Arguments

:: (Serialize g, RandomGen g)

=> Double

probability of success per step

-> g

initial seed

-> Auto m a (Blip a)

Simulates a Bernoulli Process: a process of sequential independent trials each with a success of probability p.

Implemented here is an Auto producing a blip stream that emits whenever the bernoulli process succeeds with the value of the received input of the Auto, with its probability of succuss per each trial as the Double parameter.

It is expected that, for probability p, the stream will emit a value on average once every 1/p ticks.

stdBernoulli

Arguments

:: Double

probability of any step emitting

-> StdGen

initial seed

-> Auto m a (Blip a)

Like bernoulli, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of bernoulli for more information.

bernoulli_

Arguments

:: RandomGen g

=> Double

probability of any step emitting

-> g

initial seed

-> Auto m a (Blip a)

The non-serializing/non-resuming version of bernoulli.

Random-length intervals

randIntervals :: (Serialize g, RandomGen g) => Double -> g -> Interval m a a

An Interval that is "on" and "off" for contiguous but random intervals of time...when "on", allows values to pass as "on" (Just), but when "off", suppresses all incoming values (outputing Nothing).

You provide a Double, an l parameter, representing the averageexpected length of each onoff interval.

The distribution of interval lengths follows a Geometric Distribution. This distribution is, as we call it in maths, "memoryless", which means that the "time left" that the Auto will be "on" or "off" at any given time is going to be, on average, the given l parameter.

Internally, the "toggling" events follow a bernoulli process with a p parameter of 1 / l.

stdRandIntervals :: Double -> StdGen -> Interval m a a

Like randIntervals, but specialized for StdGen from System.Random, so that you can serialize and resume. This is needed because StdGen doesn't have a Serialize instance.

See the documentation of randIntervals for more information.

randIntervals_ :: RandomGen g => Double -> g -> Interval m a a

The non-serializing/non-resuming version of randIntervals.