[Haskell-cafe] A Proposed Law for Foldable?

Thu Feb 12 21:39:02 UTC 2015

The argument could be rephrased in terms of something other than toList,
toList was more convenient short-hand than intending to rule out entire
classes of recursion patterns.

If we had a way to talk about roles you could likely recover this law for
representational arguments, but dragging in a bunch of talk about (forall a
b. Coercion a b -> Coercion (f a) (f b)) in order to claim that f is
representational is a great way to make the law completely inaccessible and
GHC specific, it also renders it useless for talking about things like Set,
which aren't representational in their argument.

And as been worked through to death in the past, Functor cannot be a
superclass of Foldable. ;)

-Edward

On Thu, Feb 12, 2015 at 4:31 PM, David Feuer <david.feuer at gmail.com> wrote:

> What I kind of like about this proposal is that it makes it illegal
> for potentially-infinite snoc-lists to be Foldable, since toList will
> produce _|_ on an infinite snoc-list. On the other hand, this is
> reason enough to make a lot of people hate it. The GADT problem Edward
> Kmett points out seems like a more serious issue. The solution is to
> make  Functor f => Foldable f, which will probably eliminate another
> four fifths of the supporters of your idea.
>
> On Thu, Feb 12, 2015 at 3:49 PM, Gershom B <gershomb at gmail.com> wrote:
> > I agree the third issue raised is a pretty tricky one. GADTs can
> effectively
> > pack in a "secret a" that is nonetheless accessible, or a "secret a -> a"
> > for that matter...
> >
> > A quick and dirty repair is to just say that the law only applies to data
> > types that do not quantify over dictionaries (and passes no judgement on
> > data types which do). In such a case I think it is still useful, but
> > unfortunately specialized.
> >
> > I absolutely think a more general law could be possible, but it could be
> > rather tricky to state... Perhaps it could be stated by, instead of
> > quantifying over _all_ a, quantifying over a specific "generic a" which
> > promises it has no dictionaries with a positive occurrence of `a`. This
> > maintains the spirit properly, but makes the statement more complex.
> >
> > With regards to Atze's question, the fact that such a law could rule out
> > giving an arbitrary type an instance where "foldMap = mempty" is exactly
> the
> > sort of thing we would like to see.
> >
> > --gershom
> >
> >
> > On Thu, Feb 12, 2015 at 3:36 PM, Edward Kmett <ekmett at gmail.com> wrote:
> >>
> >> There are 3 cases ruled out by this law. Two of them I'd have no trouble
> >> seeing go, the third one I think damages it beyond repair.
> >>
> >> First,
> >>
> >> `foldMap = mempty`
> >>
> >> is currently an admissable definition of foldMap for anything that is
> not
> >> Traversable.
> >>
> >> The law effectively talks backwards and ensures that you have to give
> back
> >> info on every 'a' in the container, so this is ruled out for any
> container
> >> that actually 'contains' an a.
> >>
> >> I'm pretty much okay with that case being ruled out.
> >>
> >> Second,
> >>
> >> There are instances such as the Foldable instance for `Machine` in the
> >> machines package. Here it starves the machine for input and takes the
> output
> >> and folds over it.
> >>
> >> However, these are not 'all of the 'a's it is possible to generate with
> >> such a machine, as you can construct a function (Machine ((->) b) a ->
> Maybe
> >> a) that feeds the machine a 'b' and then gets out an 'a' that would not
> >> occur in toList.
> >>
> >> One could argue that this Foldable violates the spirit of Foldable.
> >>
> >> I'm somewhat less okay with that case being ruled out as folks have
> found
> >> it useful, but I could accept it.
> >>
> >> Third,
> >>
> >> In the presence of GADTs, the fact that Foldable only accepts 'f' in
> >> negative position means that 'f' might be a GADT, telling us more about
> `a`,
> >> despite your function being parametric.
> >>
> >> e.g. it could carry around a Num constraint on its argument. Extracting
> >> this dictionary from the GADT would enable sum :: Num a => f a -> a to
> be
> >> used in your function (forall a. f a -> Maybe a), preventing
> parametricity
> >> from providing the insurance you seek.
> >>
> >> This means that your law would rule out any `Foldable` that exploits
> >> GADT-like properties.
> >>
> >> A version of `Set` where the data type carries around the `Ord` instance
> >> internally, could for instance instantiate `elem` in log time. That
> example
> >> becomes only marginally safe under your law because of `min` and `max`
> being
> >> in Ord and producing "new" a's, but it also rules out similar O(1)
> >> optimizations for sum or product in other potential containers, which
> could
> >> carry Num.
> >>
> >> These I'm much more reluctant to let go.
> >>
> >> You might be able to repair your law by also quantifying over `f` with a
> >> Foldable constraint or some such, but that re-admits the former 2 laws
> and
> >> seems to make it vacuous.
> >>
> >> -Edward
> >>
> >>
> >> On Thu, Feb 12, 2015 at 2:59 PM, Atze van der Ploeg <atzeus at gmail.com>
> >> wrote:
> >>>
> >>> Hi Gershom!
> >>>
> >>> Do you have an example where this law allows us to conclude something
> >>> interesting we otherwise would not have been able to conclude?
> >>>
> >>> Cheers,
> >>>
> >>> Atze
> >>>
> >>> On Feb 12, 2015 8:47 PM, "Gershom B" <gershomb at gmail.com> wrote:
> >>>>
> >>>> For a long time, many people, including me, have said that "Foldable
> has
> >>>> no laws" (or Foldable only has free laws) -- this is true, as it
> stands,
> >>>> with the exception that Foldable has a non-free law in interaction
> with
> >>>> Traversable (namely that it act as a proper specialization of
> Traversable
> >>>> methods). However, I believe that there is a good law we can give for
> >>>> Foldable.
> >>>>
> >>>> I earlier explored this in a paper presented at IFL 2014 but
> >>>> (rightfully) rejected from the IFL post-proceedings.
> >>>> (http://gbaz.github.io/slides/buildable2014.pdf). That paper got
> part of the
> >>>> way there, but I believe now have a better approach on the question
> of a
> >>>> Foldable law -- as sketched below.
> >>>>
> >>>> I think I now (unlike in the paper) can state a succinct law for
> >>>> Foldable that has desired properties: 1) It is not "free" -- it can be
> >>>> violated, and thus stating it adds semantic content. 2) We typically
> expect
> >>>> it to be true. 3) There are no places where I can see an argument for
> >>>> violating it.
> >>>>
> >>>> If it pans out, I intend to pursue this and write it up more formally,
> >>>> but given the current FTP discussion I thought it was worth
> documenting this
> >>>> earlier rather than later. For simplicity, I will state this property
> in
> >>>> terms of `toList` although that does not properly capture the infinite
> >>>> cases. Apologies for what may be nonstandard notation.
> >>>>
> >>>> Here is the law I think we should discuss requiring:
> >>>>
> >>>> * * *
> >>>> Given Foldable f, then
> >>>> forall (g :: forall a. f a -> Maybe a), (x :: f a). case g x of Just a
> >>>> --> a `elem` toList x
> >>>> * * *
> >>>>
> >>>> Since we do not require `a` to be of type `Eq`, note that the `elem`
> >>>> function given here is not internal to Haskell, but in the metalogic.
> >>>>
> >>>> Furthermore, note that the use of parametricity here lets us make an
> >>>> "end run" around the usual problem of giving laws to Foldable --
> rather than
> >>>> providing an interaction with another class, we provide a claim about
> _all_
> >>>> functions of a particular type.
> >>>>
> >>>> Also note that the functions `g` we intend to quantify over are
> >>>> functions that _can be written_ -- so we can respect the property of
> data
> >>>> structures to abstract over information. Consider
> >>>>
> >>>> data Funny a = Funny {hidden :: a, public :: [a]}
> >>>>
> >>>> instance Foldable Funny where
> >>>>     foldMap f x = foldMap f (public x)
> >>>>
> >>>> Now, if it is truly impossible to ever "see" hidden (i.e. it is not
> >>>> exported, or only exported through a semantics-breaking "Internal"
> module),
> >>>> then the Foldable instance is legitimate. Otherwise, the Foldable
> instance
> >>>> is illegitimate by the law given above.
> >>>>
> >>>> I would suggest the law given is "morally" the right thing for
> Foldable
> >>>> -- a Foldable instance for `f` should suggest that it gives us "all
> the as
> >>>> in any `f a`", and so it is, in some particular restricted sense,
> initial
> >>>> among functions that extract as.
> >>>>
> >>>> I do not suggest we add this law right away. However, I would like to
> >>>> suggest considering it, and I believe it (or a cleaned-up variant)
> would
> >>>> help us to see Foldable as a more legitimately lawful class that not
> only
> >>>> provides conveniences but can be used to aid reasoning.
> >>>>
> >>>> Relating this to adjointness, as I do in the IFL preprint, remains
> >>>> future work.
> >>>>
> >>>> Cheers,
> >>>> Gershom
> >>>>
> >>>> _______________________________________________
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> >>>> Haskell-Cafe at haskell.org
> >>>> http://www.haskell.org/mailman/listinfo/haskell-cafe
> >>>>
> >>>
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> >>
> >
> >
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