elerea-2.3.0: A minimalistic FRP library

FRP.Elerea.Param

Description

This module provides leak-free and referentially transparent higher-order discrete signals. Unlike in FRP.Elerea.Simple, the sampling action has an extra argument that will be globally distributed to every node and can be used to update the state. For instance, it can hold the time step between the two samplings, but it could also encode all the external input to the system.

Synopsis

# The signal abstraction

data Signal a Source

A signal represents a value changing over time. It can be thought of as a function of type `Nat -> a`, where the argument is the sampling time, and the `Monad` instance agrees with the intuition (bind corresponds to extracting the current sample). Signals and the values they carry are denoted the following way in the documentation:

``` s = <<s0 s1 s2 ...>>
```

This says that `s` is a signal that reads `s0` during the first sampling, `s1` during the second and so on. You can also think of `s` as the following function:

``` s t_sample = [s0,s1,s2,...] !! t_sample
```

Signals are constrained to be sampled sequentially, there is no random access. The only way to observe their output is through `start`.

Instances

 Monad Signal Functor Signal Applicative Signal Bounded t => Bounded (Signal t) Enum t => Enum (Signal t) Eq (Signal a) Equality test is impossible. Floating t => Floating (Signal t) Fractional t => Fractional (Signal t) Integral t => Integral (Signal t) Num t => Num (Signal t) Ord t => Ord (Signal t) Real t => Real (Signal t) Show (Signal a) The `Show` instance is only defined for the sake of `Num`...

data SignalGen p a Source

A signal generator is the only source of stateful signals. It can be thought of as a function of type `Nat -> Signal p -> a`, where the result is an arbitrary data structure that can potentially contain new signals, the first argument is the creation time of these new signals, and the second is a globally accessible input signal. It exposes the `MonadFix` interface, which makes it possible to define signals in terms of each other. Unlike the simple variant, the denotation of signal generators differs from that of signals. We will use the following notation for generators:

``` g = <|g0 g1 g2 ...|>
```

Just like signals, generators behave as functions of time, but they can also refer to the input signal:

``` g t_start s_input = [g0,g1,g2,...] !! t_start
```

The conceptual difference between the two notions is that signals are passed a sampling time, while generators expect a start time that will be the creation time of all the freshly generated signals in the resulting structure.

Instances

 Monad (SignalGen p) Functor (SignalGen p) MonadFix (SignalGen p) Applicative (SignalGen p)

# Embedding into I/O

Arguments

 :: SignalGen p (Signal a) the generator of the top-level signal -> IO (p -> IO a) the computation to sample the signal

Embedding a signal into an `IO` environment. Repeated calls to the computation returned cause the whole network to be updated, and the current sample of the top-level signal is produced as a result. The computation accepts a global parameter that will be distributed to all signals. For instance, this can be the time step, if we want to model continuous-time signals. This is the only way to extract a signal generator outside the network, and it is equivalent to passing zero to the function representing the generator.

Example:

``` do
smp <- start (stateful 10 (+))
res <- forM [5,3,2,9,4] smp
print res
```

Output:

``` [10,15,18,20,29]
```

Arguments

 :: a initial value -> IO (Signal a, a -> IO ()) the signal and an IO function to feed it

A signal that can be directly fed through the sink function returned. This can be used to attach the network to the outer world. Note that this is optional, as all the input of the network can be fed in through the global parameter, although that is not really convenient for many signals.

Arguments

 :: IO (SignalGen p (Signal [a]), a -> IO ()) a generator for the event signal and the associated sink

An event-like signal that can be fed through the sink function returned. The signal carries a list of values fed in since the last sampling, i.e. it is constantly [] if the sink is never invoked. The order of elements is reversed, so the last value passed to the sink is the head of the list. Note that unlike `external` this function only returns a generator to be used within the expression constructing the top-level stream, and this generator can only be used once.

A printing action within the `SignalGen` monad.

# Basic building blocks

Arguments

 :: a initial output -> Signal a the signal to delay -> SignalGen p (Signal a)

The `delay` combinator is the elementary building block for adding state to the signal network by constructing delayed versions of a signal that emit a given value at creation time and the previous output of the signal afterwards (`--` is undefined):

``` delay x0 s = <| <<x0 s0 s1 s2 s3 ...>>
<<-- x0 s1 s2 s3 ...>>
<<-- -- x0 s2 s3 ...>>
<<-- -- -- x0 s3 ...>>
...
|>
```

It can be thought of as the following function (which should also make it clear why the return value is `SignalGen`):

``` delay x0 s t_start s_input t_sample
| t_start == t_sample = x0
| t_start < t_sample  = s (t_sample-1)
| otherwise           = error \"Premature sample!\"
```

The way signal generators are extracted by `generator` ensures that the error can never happen. It is also clear that the behaviour of `delay` is not affected in any way by the global input.

Example (requires the `DoRec` extension):

``` do
smp <- start \$ do
rec let fib'' = liftA2 (+) fib' fib
fib' <- delay 1 fib''
fib <- delay 1 fib'
return fib
res <- replicateM 7 (smp undefined)
print res
```

Output:

``` [1,1,2,3,5,8,13]
```

Arguments

 :: Signal (SignalGen p a) a stream of generators to potentially run -> SignalGen p (Signal a)

A reactive signal that takes the value to output from a signal generator carried by its input with the sampling time provided as the start time for the generated structure. It is possible to create new signals in the monad, which is the key to defining dynamic data-flow networks.

``` generator << <|x00 x01 x02 ...|>
<|x10 x11 x12 ...|>
<|x20 x21 x22 ...|>
...
>> = <| <<x00 x11 x22 ...>>
<<x00 x11 x22 ...>>
<<x00 x11 x22 ...>>
...
|>
```

It can be thought of as the following function:

``` generator g t_start s_input t_sample = g t_sample t_sample s_input
```

It has to live in the `SignalGen` monad, because it needs to maintain an internal state to be able to cache the current sample for efficiency reasons. However, this state is not carried between samples, therefore start time doesn't matter and can be ignored. Also, even though it does not make use of the global input itself, part of its job is to distribute it among the newly generated signals.

Refer to the longer example at the bottom of FRP.Elerea.Simple to see how it can be used.

Arguments

 :: Signal a signal to memoise -> SignalGen p (Signal a)

Memoising combinator. It can be used to cache results of applicative combinators in case they are used in several places. It is observationally equivalent to `return` in the `SignalGen` monad.

``` memo s = <|s s s s ...|>
```

For instance, if `s = f <\$> s'`, then `f` will be recalculated once for each sampling of `s`. This can be avoided by writing ```s <- memo (f <\$> s')``` instead. However, `memo` incurs a small overhead, therefore it should not be used blindly.

All the functions defined in this module return memoised signals. Just like `delay`, it is independent of the global input.

Arguments

 :: Signal Bool the boolean input signal -> SignalGen p (Signal Bool) a one-shot signal true only the first time the input is true

A signal that is true exactly once: the first time the input signal is true. Afterwards, it is constantly false, and it holds no reference to the input signal. For instance (assuming the rest of the input is constantly `False`):

``` until <<False False True True False True ...>> =
<| <<False False True  False False False False False False False ...>>
<< ---  False True  False False False False False False False ...>>
<< ---   ---  True  False False False False False False False ...>>
<< ---   ---   ---  True  False False False False False False ...>>
<< ---   ---   ---   ---  False True  False False False False ...>>
<< ---   ---   ---   ---   ---  True  False False False False ...>>
<< ---   ---   ---   ---   ---   ---  False False False False ...>>
...
|>
```

It is observationally equivalent to the following expression (which would hold onto `s` forever):

``` until s = do
step <- transfer False (const (||)) s
dstep <- delay False step
memo (liftA2 (/=) step dstep)
```

Example:

``` do
smp <- start \$ do
accum <- stateful 0 (+)
tick <- until ((>=10) <\$> accum)
return \$ liftA2 (,) accum tick
res <- forM [4,1,3,5,2,8,6] smp
print res
```

Output:

``` [(0,False),(4,False),(5,False),(8,False),(13,True),(15,False),(23,False)]
```

input :: SignalGen p (Signal p)Source

The common input signal that is fed through the function returned by `start`, unless we are in an `embed`ded generator. It is equivalent to the following function:

``` input t_start s_input = s_input
```

Example:

``` do
smp <- start \$ do
sig <- input
return (sig*2)
res <- forM [4,1,3,5,2,8,6] smp
print res
```

Output:

``` [8,2,6,10,4,16,12]
```

embed :: Signal p' -> SignalGen p' a -> SignalGen p aSource

Embed a generator with an overridden input signal. It is equivalent to the following function:

``` embed s g t_start s_input = g t_start s
```

Example:

``` do
smp <- start \$ do
sig <- input
embed (sig*2) \$ do
sig <- input
return (sig+1)
res <- forM [4,1,3,5,2,8,6] smp
print res
```

Output:

``` [9,3,7,11,5,17,13]
```

# Derived combinators

Arguments

 :: a initial state -> (p -> a -> a) state transformation -> SignalGen p (Signal a)

A direct stateful transformation of the input. The initial state is the first output, and every following output is calculated from the previous one and the value of the global parameter (which might have been overridden by `embed`).

Example:

``` do
smp <- start (stateful "" (:))
res <- forM "olleh~" smp
print res
```

Output:

``` ["","o","lo","llo","ello","hello"]
```

Arguments

 :: a initial internal state -> (p -> t -> a -> a) state updater function -> Signal t input signal -> SignalGen p (Signal a)

A stateful transfer function. The current input affects the current output, i.e. the initial state given in the first argument is considered to appear before the first output, and can never be observed. Every output is derived from the current value of the input signal, the global parameter (which might have been overridden by `embed`) and the previous output. It is equivalent to the following expression:

Example (assuming a delta time is passed to the sampling function in each step):

``` integral x0 s = transfer x0 (\dt v x -> x+dt*v)
```

Example for using the above:

``` do
smp <- start (integral 3 (pure 2))
res <- replicateM 7 (smp 0.1)
print res
```

Output:

``` [3.2,3.4,3.6,3.8,4.0,4.2,4.4]
```

Arguments

 :: a initial internal state -> (p -> t1 -> t2 -> a -> a) state updater function -> Signal t1 input signal 1 -> Signal t2 input signal 2 -> SignalGen p (Signal a)

A variation of `transfer` with two input signals.

Arguments

 :: a initial internal state -> (p -> t1 -> t2 -> t3 -> a -> a) state updater function -> Signal t1 input signal 1 -> Signal t2 input signal 2 -> Signal t3 input signal 3 -> SignalGen p (Signal a)

A variation of `transfer` with three input signals.

Arguments

 :: a initial internal state -> (p -> t1 -> t2 -> t3 -> t4 -> a -> a) state updater function -> Signal t1 input signal 1 -> Signal t2 input signal 2 -> Signal t3 input signal 3 -> Signal t4 input signal 4 -> SignalGen p (Signal a)

A variation of `transfer` with four input signals.

# Random sources

noise :: MTRandom a => SignalGen p (Signal a)Source

A random signal.

Example:

``` do
smp <- start noise :: IO (IO Double)
res <- replicateM 5 smp
print res
```

Output:

``` [0.12067753390401374,0.8658877349182655,0.7159264443196786,0.1756941896012891,0.9513646060896676]
```

getRandom :: MTRandom a => SignalGen p aSource

A random source within the `SignalGen` monad.