Lax arrows.

In order to get an understanding of what a lax arrow is, consider the following code example:

    looping :: IO ()
    looping = fixIO (\char -> putChar char >> return 'A')

One might expect that executing looping will result in a capital A being printed but this is not the case. The resulting 'A' will become “available” not until the action putChar char has been executed.

In order to explain this, let’s think of IO o as being equivalent to World -> Either Exception (o,World). The >>= operator could now be defined as follows:

    io1 >>= io2Gen = \world -> case io1 world of
                                   Left  exc         -> Left exc
                                   Right (o1,world') -> io2Gen o1 world'

This results in the following situation:

• In order to decide whether looping outputs a value or throws an exception, the system has to decide whether putChar char >> return 'A' outputs a value or throws an exception.
• In order to decide whether putChar char >> return 'A' outputs a value or throws an exception, the system has to decide whether putChar char outputs a value or throws an exception (because of the way, >>= is implemented).
• In order to decide whether putChar char outputs a value or throws an exception, the system has to know if there is really a char to output or whether there is none because of an exception. So it has to decide whether putChar char >> return 'A' outputs a value or throws an exception.

So we have a circular dependency resulting in an output value of _|_ for looping.

The LaxArrow type constructor transforms a given arrow into a new arrow which works mostly like the base arrow but is “a bit less strict”. To be more precise, all parts of a lax arrow value which are constructed with arr are internally moved to the beginning. This way, data produced by such parts is available at each point in the arrow when loop is used. For the above example, this would mean that the result 'A' is already known before the putChar action is executed and can therefore be used by this action.

Note that relaxation only works for arrows, not directly for monads. It is unknown whether a relaxation mechanism for monads exists but it is considered unlikely. Of course, you can transform any monad into an arrow by using Kleisli. However, the lax arrow type is not an instance of ArrowChoice nor is it one of ArrowApply, and at least the current implementation does not allow it to be an instance of either class.

Further note that the implementation of lax arrows does not use unsafePerformIO nor unsafeInterleaveIO and is not tied to IO at all.

The lax arrow version of the looping example would be as follows:

    looping :: IO ()
    looping = runKleisli (runLax (loop $ second $ impure (Kleisli putChar) >>> arr (const 'A')))
                         ()

Transforms a value of the base arrow type into a lax arrow value. Pure parts of the argument are not affected by relaxation, only parts of the lax arrow value which are constructed with arr from the LaxArrow instance of Arrow.

lift from the ArrowTransformer class is not used since it is probably supposed to be a homomorphism but impure is not a homomorphism. While impure preserves >>>, it does not preserve arr, first and loop. If it would then we would have no relaxation effect at all.