[Haskell] Re: Typing in haskell and mathematics

[Chung-chieh Shan]

Jeremy Gibbons <Jeremy.Gibbons at comlab.ox.ac.uk> wrote in article <Pine.GSO.4.51.0501310314250.3639 at abacus.comlab> in gmane.comp.lang.haskell.general:
> But in FP, of course, functions are first class citizens. So one may get
> ambiguities on account of it being reasonable to treat a particular
> function either as "program" or as "data" - with conflicting outcomes...

> Suppose one wants to streamline notation, so
> that instead of having to write
>   (f . g) $ x
> one can write
>   f @ g @ x
> Here, "@" means either composition or application according to context. It
> is supposed to be associative, so it isn't the same as normal Haskell $;
> in particular,
>   f @ g
> is valid and represents the composition of f and g.

Indeed, I've been thinking about such a notation, but one that
explicitly specifies whether to treat a function as "program" or "data".
Variables like "f" always denote data to start with; such data can be
thought of as a program that maps unit to a function.  Data can be
turned into a program by unquoting it with "," (usually I write "@" for
unquotation but you use it for composition already).  (Conversely, a
program can be turned into data by enclosing it in a pair of quotation
brackets "[]", a la Joy; unquotation cancels quotation by the reduction
step ,[X] => X.)  Now

    (f . g) $ x

in Haskell notation is written

    ,f @ ,g @ x

We also distinguish between

    ,thrice @ thrice

and

    ,thrice @ ,thrice

An advantage of this notation is that it exposes the symmetry between
call-by-value and call-by-name evaluation: right-to-left evaluation here
corresponds to call-by-value, argument-first evaluation; left-to-right
evaluation here corresponds to call-by-name evaluation.  I could go on
and on about this.

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