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.

A signal generator is the only source of stateful signals. It can be thought of as a function of type Nat -> a, where the result is an arbitrary data structure that can potentially contain new signals, and the argument is the creation time of these new signals. It exposes the MonadFix interface, which makes it possible to define signals in terms of each other. The denotation of signal generators happens to be the same as that of signals, but this partly accidental (it does not hold in the other variants), so we will use a separate notation for generators:

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

Just like signals, generators behave as functions of time:

 g t_start = [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.

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. 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. In general:

 replicateM n =<< start <|<<x0 x1 x2 x3 ...>> ...|> == take n [x0,x1,x2,x3,...]

Example:

 do
     smp <- start (stateful 3 (+2))
     res <- replicateM 5 smp
     print res

Output:

 [3,5,7,9,11]

A signal that can be directly fed through the sink function returned. This can be used to attach the network to the outer world. The signal always yields the value last written to the sink. In other words, if the sink is written less frequently than the network sampled, the output remains the same during several samples. If values are pushed in the sink more frequently, only the last one before sampling is visible on the output.

Example:

 do
     (sig,snk) <- external 4
     smp <- start (return sig)
     r1 <- smp
     r2 <- smp
     snk 7
     r3 <- smp
     snk 9
     snk 2
     r4 <- smp
     print [r1,r2,r3,r4]

Output:

 [4,4,7,2]

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.

Example:

 do
     (gen,snk) <- externalMulti
     smp <- start gen
     r1 <- smp
     snk 7
     r2 <- smp
     r3 <- smp
     snk 9
     snk 2
     r4 <- smp
     print [r1,r2,r3,r4]

Output:

 [[],[7],[],[2,9]]

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 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.

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
     print res

Output:

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

A formal conversion from signals to signal generators, which effectively allows for retrieving the current value of a previously created signal within a generator. This includes both signals defined in an external scope as well as those created earlier in the same generator. In the model, it corresponds to the identity function.

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 t_sample = g t_sample t_sample

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.

Refer to the longer example at the bottom to see how it can be used.

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.

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 (||) s
     dstep <- delay False step
     memo (liftA2 (/=) step dstep)

Example:

 do
     smp <- start $ do
         cnt <- stateful 0 (+1)
         tick <- until ((>=3) <$> cnt)
         return $ liftA2 (,) cnt tick
     res <- replicateM 6 smp
     print res

Output:

 [(0,False),(1,False),(2,False),(3,True),(4,False),(5,False)]

A pure stateful signal. The initial state is the first output, and every subsequent state is derived from the preceding one by applying a pure transformation.

Example:

 do
     smp <- start (stateful "x" ('x':))
     res <- replicateM 5 smp
     print res

Output:

 ["x","xx","xxx","xxxx","xxxxx"]

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, and subsequent states are determined by combining the preceding state with the current output of the input signal using the function supplied.

Example:

 do
     smp <- start $ do
         cnt <- stateful 1 (+1)
         transfer 10 (+) cnt
     res <- replicateM 5 smp
     print res

Output:

 [11,13,16,20,25]

A variation of transfer with two input signals.

A variation of transfer with three input signals.

A variation of transfer with four input signals.

An IO action executed in the SignalGen monad. Can be used as liftIO.

A signal that executes a given IO action once at every sampling.

In essence, this combinator provides cooperative multitasking capabilities, and its primary purpose is to assist library writers in wrapping effectful APIs as conceptually pure signals. If there are several effectful signals in the system, their order of execution is undefined and should not be relied on.

Example:

 do
     smp <- start $ do
         ref <- execute $ newIORef 0
         effectful $ do
             x <- readIORef ref
             putStrLn $ "Count: " ++ show x
             writeIORef ref $! x+1
             return ()
     replicateM_ 5 smp

Output:

 Count: 0
 Count: 1
 Count: 2
 Count: 3
 Count: 4

Another example (requires mersenne-random):

 do
     smp <- start $ effectful randomIO :: IO (IO Double)
     res <- replicateM 5 smp
     print res

Output:

 [0.12067753390401374,0.8658877349182655,0.7159264443196786,0.1756941896012891,0.9513646060896676]

A signal that executes a parametric IO action once at every sampling. The parameter is supplied by another signal at every sampling step.

Like effectful1, but with two parameter signals.

Like effectful1, but with three parameter signals.

Like effectful1, but with four parameter signals.