Documentation

Suppression of initial (at local time 0) event.

Event source that never occurs.

Event source with a single occurrence at time 0. The value of the event is given by the function argument.

Suppress all but the first event.

Suppress all but the first n events.

Initialization operator (cf. Lustre/Lucid Synchrone).

The output at time zero is the first argument, and from that point on it behaves like the signal function passed as second argument.

Input initialization operator.

The input at time zero is the first argument, and from that point on it behaves like the signal function passed as second argument.

Any instance of ArrowApply can be made into an instance of ArrowChoice by defining left = leftApp.

Postcomposition with a pure function (right-to-left variant).

Precomposition with a pure function (right-to-left variant).

Postcomposition with a pure function.

Precomposition with a pure function.

The identity arrow, which plays the role of return in arrow notation.

The basic arrow class.

Instances should satisfy the following laws:

• arr id = id

• arr (f >>> g) = arr f >>> arr g

• first (arr f) = arr (first f)

• first (f >>> g) = first f >>> first g

• first f >>> arr fst = arr fst >>> f

• first f >>> arr (id *** g) = arr (id *** g) >>> first f

• first (first f) >>> arr assoc = arr assoc >>> first f

where

assoc ((a,b),c) = (a,(b,c))

The other combinators have sensible default definitions, which may be overridden for efficiency.

Kleisli arrows of a monad.

A monoid on arrows.

Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation.

Instances should satisfy the following laws:

• left (arr f) = arr (left f)

• left (f >>> g) = left f >>> left g

• f >>> arr Left = arr Left >>> left f

• left f >>> arr (id +++ g) = arr (id +++ g) >>> left f

• left (left f) >>> arr assocsum = arr assocsum >>> left f

where

assocsum (Left (Left x)) = Left x
assocsum (Left (Right y)) = Right (Left y)
assocsum (Right z) = Right (Right z)

The other combinators have sensible default definitions, which may be overridden for efficiency.

Some arrows allow application of arrow inputs to other inputs. Instances should satisfy the following laws:

• first (arr (\x -> arr (\y -> (x,y)))) >>> app = id

• first (arr (g >>>)) >>> app = second g >>> app

• first (arr (>>> h)) >>> app = app >>> h

Such arrows are equivalent to monads (see ArrowMonad).

The ArrowApply class is equivalent to Monad: any monad gives rise to a Kleisli arrow, and any instance of ArrowApply defines a monad.

The loop operator expresses computations in which an output value is fed back as input, although the computation occurs only once. It underlies the rec value recursion construct in arrow notation. loop should satisfy the following laws:

extension

loop (arr f) = arr (\ b -> fst (fix (\ (c,d) -> f (b,d))))

left tightening

loop (first h >>> f) = h >>> loop f

right tightening

loop (f >>> first h) = loop f >>> h

sliding

loop (f >>> arr (id *** k)) = loop (arr (id *** k) >>> f)

vanishing

loop (loop f) = loop (arr unassoc >>> f >>> arr assoc)

superposing

second (loop f) = loop (arr assoc >>> second f >>> arr unassoc)

where

assoc ((a,b),c) = (a,(b,c))
unassoc (a,(b,c)) = ((a,b),c)

Left-to-right composition

Right-to-left composition

Outputs every input sample, with a given message prefix, when a condition is met, and waits for some input / enter to continue.

Outputs every input sample, with a given message prefix, using an auxiliary printing function, when a condition is met.

Outputs every input sample, with a given message prefix, using an auxiliary printing function.

Generate outputs using a step-wise generation function and an initial value. Version of unfold in which the output and the new accumulator are the same. Should be equal to f a -> unfold (f >>> dup) a.

Generate outputs using a step-wise generation function and an initial value.

Applies a transfer function to the input and an accumulator, returning the updated accumulator and output.

Applies a function to the input and an accumulator, outputting the updated accumulator. Equal to f s0 -> feedback s0 $ arr (uncurry f >>> dup).

Accumulate the inputs, starting from an initial monoid value.

Accumulate the inputs, starting from mempty.

Sums the inputs, starting from an initial vector.

Sums the inputs, starting from zero.

Count the number of simulation steps. Produces 1, 2, 3,...

Buffers and returns the elements in FIFO order, returning Nothing whenever the buffer is empty.

Preprends a fixed output to an MSF, shifting the output.

Preprends a fixed output to an MSF. The first input is completely ignored.

Delay a signal by one sample.

Produces an additional side effect and passes the input unchanged.

Applies a function to produce an additional side effect and passes the input unchanged.

Apply an MSF to every input. Freezes temporarily if the input is Nothing, and continues as soon as a Just is received.

A stream is an MSF that produces outputs, while ignoring the input. It can obtain the values from a monadic context.

A sink is an MSF that consumes inputs, while producing no output. It can consume the values with side effects.

Apply trans-monadic actions (in an arbitrary way).

This is just a convenience function when you have a function to move across monads, because the signature of morphGS is a bit complex.

Lift inner monadic actions in monad stacks.

Lift the second MSF into the monad of the first.

Lift the first MSF into the monad of the second.

Lift innermost monadic actions in monad stack (generalisation of liftIO).

Monadic lifting from one monad into another

Apply a monadic transformation to every element of the input stream.

Generalisation of arr from Arrow to monadic functions.

Lifts a monadic computation into a Stream.

Apply a monadic stream function to a list.

Because the result is in a monad, it may be necessary to traverse the whole list to evaluate the value in the results to WHNF. For example, if the monad is the maybe monad, this may not produce anything if the MSF produces Nothing at any point, so the output stream cannot consumed progressively.

To explore the output progressively, use liftMSF and '(>>>)'', together with some action that consumes/actuates on the output.

This is called runSF in Liu, Cheng, Hudak, "Causal Commutative Arrows and Their Optimization"

Well-formed looped connection of an output component as a future input.

Generic lifting of a morphism to the level of MSFs.

Natural transformation to the level of MSFs.

Mathematical background: The type a -> m (b, c) is a functor in c, and MSF m a b is its greatest fixpoint, i.e. it is isomorphic to the type a -> m (b, MSF m a b), by definition. The types m, a and b are parameters of the functor. Taking a fixpoint is functorial itself, meaning that a morphism (a natural transformation) of two such functors gives a morphism (an ordinary function) of their fixpoints.

This is in a sense the most general "abstract" lifting function, i.e. the most general one that only changes input, output and side effect types, and doesn't influence control flow. Other handling functions like exception handling or ListT broadcasting necessarily change control flow.

Stepwise, side-effectful MSFs without implicit knowledge of time.

MSFs should be applied to streams or executed indefinitely or until they terminate. See reactimate and reactimateB for details. In general, calling the value constructor MSF or the function unMSF is discouraged.