[Haskell-cafe] arr considered harmful

[Ryan Ingram]

I know it's a bit of an 'intentionally provocative' title, but with the
recent discussions on Arrows I thought it timely to bring this up.

Most of the conversion from arrow syntax into arrows uses 'arr' to move
components around. However, arr is totally opaque to the arrow itself, and
prevents describing some very useful objects as arrows.

For example, I would love to be able to use the arrow syntax to define
objects of this type:

data Circuit a b where
    Const :: Bool -> Circuit () Bool
    Wire :: Circuit a a
    Delay :: Circuit a a
    And :: Circuit (Bool,Bool) Bool
    Or :: Circuit (Bool,Bool) Bool
    Not :: Circuit Bool Bool
    Then :: Circuit a b -> Circuit b c -> Circuit a c
    Pair :: Circuit a c -> Circuit b d -> Circuit (a,b) (c,d)
    First :: Circuit a b -> Circuit (a,c) (b,c)
    Swap :: Circuit (a,b) (b,a)
    AssocL :: Circuit ((a,b),c) (a,(b,c))
    AssocR :: Circuit (a,(b,c)) ((a,b),c)
    Loop :: Circuit (a,b) (a,c) -> Circuit b c
etc.

Then we can have code that examines this concrete data representation,
converts it to VHDL, optimizes it, etc.

However, due to the presence of the opaque 'arr', there's no way to make
this type an arrow without adding an 'escape hatch'
    Arr :: (a -> b) -> Circuit a b
which breaks the abstraction: circuit is supposed to represent an actual
boolean circuit; (Arr not) is not a valid circuit because we've lost the
information about the existence of a 'Not' gate.

The arrow syntax translation uses arr to do plumbing of variables.  I think
a promising project would be to figure out exactly what plumbing is needed,
and add those functions to a sort of 'PrimitiveArrow' class.  All of these
plumbing functions are trivially implemented in terms of 'arr', when it
exists, but if it doesn't, it should be possible to use the arrow syntax
regardless.

  -- ryan
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