[Haskell-cafe] A Modest Records Proposal
Greg Weber
greg at gregweber.info
Sun Apr 1 19:03:29 CEST 2012
Obviously Gregory is not familiar with Homotopy. In fact, its
isomorphism predicts that if someone named Greg is involved in a
discussion, someone named Gregory will also become involved.
Or that is what I get for responding to an e-mail without reading it
on April 1st :)
On Sun, Apr 1, 2012 at 7:40 AM, Gregory Collins <greg at gregorycollins.net> wrote:
> Whoosh? :-)
>
> On Sun, Apr 1, 2012 at 3:54 PM, Greg Weber <greg at gregweber.info> wrote:
>>
>> Hi Gershom,
>>
>> This sounds very interesting even if I have no idea what you are
>> talking about :)
>> Please create a proposal linked from this page:
>> http://hackage.haskell.org/trac/ghc/wiki/Records
>> The first thing you should probably do is explain the programmer's
>> point of view. That ensures that we are all going through the
>> requirements phase correctly.
>> I can assure you that haskell prime would not accept a records change
>> until it is first implemented in GHC or another Haskell compiler.
>>
>> Thanks,
>> Greg Weber
>>
>> On Sat, Mar 31, 2012 at 11:14 PM, Gershom B <gershomb at gmail.com> wrote:
>> > The records discussion has been really complicated and confusing. But
>> > I have a suggestion that should provide a great deal of power to
>> > records, while being mostly[1] backwards-compatible with Haskell 2010.
>> > Consider this example:
>> >
>> > data A a = A{a:a, aa::a, aaa :: a -> A (a -> a)}
>> > data B a = B{aaa :: a -> A (a -> a), a :: A}
>> >
>> > Now what is the type of this?
>> >
>> > aaaa aaaaa a aa = aaaaa{a = a, aaa = aa}
>> >
>> > Using standard Haskell typeclasses this is a difficult question to
>> > answer. The types of aaaa for A and B do not unify in an obvious way.
>> > However, while they are intensionally quite distinct, they unify
>> > trivially extensionally. The obvious thing to do is then to extend the
>> > type system with extensional equality on record functions.
>> >
>> > Back when Haskell was invented, extensional equality was thought to be
>> > hard. But purity was thought to be hard too, and so were Monads. Now,
>> > we know that function existentionality is easy. In fact, if we add the
>> > Univalence Axiom to GHC[2], then this is enough to get function
>> > existensionality. This is a well-known result of Homotopy Type
>> > Theory[3], which is a well-explored approach that has existed for at
>> > least a few years and produced more than one paper[4]. Homotopy Type
>> > Theory is so sound and well understood that it has even been
>> > formalized in Coq.
>> >
>> > Once we extend GHC with homotopies, it turns out that records reduce
>> > to mere syntactic sugar, and there is a natural proof of their
>> > soundness (Appendix A). Furthermore, there is a canonical projection
>> > for any group of fields (Appendix B). Even better, we can make "."
>> > into the identity path operator, unifying its uses in composition and
>> > projection. In fact, with extended (parenthesis-free) section rules,
>> > "." can also be used to terminate expressions, making Haskell friendly
>> > not only to programmers coming from Java, but also to those coming
>> > from Prolog!
>> >
>> > After some initial feedback, I'm going to create a page for the
>> > Homotopy Extensional Records Proposal (HERP) on trac. There are really
>> > only a few remaining questions. 1) Having introduced homotopies, why
>> > not go all the way and introduce dependent records? In fact, are HERP
>> > and Dependent Extensional Records Proposal (DERP) already isomorphic?
>> > My suspicion is that HERP is isomorphic, but DERP is not. However, I
>> > can only get away with my proof using Scott-free semantics. 2) Which
>> > trac should I post this too? Given how well understood homotopy type
>> > theory is, I'm tempted to bypass GHC entirely and propose this for
>> > haskell-prime. 3) What syntax should we use to represent homotopies?
>> > See extend discussion in Appendix C.
>> >
>> > HTH HAND,
>> > Gershom
>> >
>> > [1] To be precise, 100% of Haskell 2010 programs should, usually, be
>> > able to be rewritten to work with this proposal with a minimal set of
>> > changes[1a].
>> >
>> > [1a] A minimal set of changes is defined as the smallest set of
>> > changes necessary to make to a Haskell 2010 program such that it works
>> > with this proposal. We can arrive at these changes by the following
>> > procedure: 1) Pick a change[1b]. 2) Is it minimal? If so keep it. 3)
>> > are we done? If not, make another change.
>> >
>> > [1b] To do this constructively, we need an order. I suggest the lo
>> > mein, since noodles give rise to a free soda.
>> >
>> > [2] I haven't looked at the source, but I would suggest putting it in
>> > the file Axioms.hs.
>> >
>> > [3] http://homotopytypetheory.org/
>> >
>> > [4] http://arxiv.org/
>> >
>> >
>> > *Appendix A: A Natural Proof of the Soundness of HERP
>> >
>> > Take the category of all types in HERP, with functions as morphisms.
>> > Call it C. Take the category of all sound expressions in HERP, with
>> > functions as morphisms. Call it D. Define a full functor from C to D.
>> > Call it F. Define a faithful functor on C and D. Call it G. Draw the
>> > following diagram.
>> >
>> > F(X)----F(Y)
>> > | |
>> > | |
>> > | |
>> > G(X)----G(Y)
>> >
>> > Define the arrows such that everything commutes.
>> >
>> >
>> > *Appendix B: Construction of a Canonical Projection for Any Group of
>> > Fields.
>> >
>> > 1) Take the fields along the homotopy to an n-ball.
>> > 2) Pack them loosely with newspaper and gunpowder.
>> > 3) Project them from a cannon.
>> >
>> > In an intuitionistic logic, the following simplification is possible:
>> >
>> > 1) Use your intuition.
>> >
>> >
>> > *Appendix C: Homotopy Syntax
>> >
>> > Given that we already are using the full unicode set, what syntax
>> > should we use to distinguish paths and homotopies? At first, I thought
>> > we could avoid providing any syntax for homotopies at all. Haskell is
>> > a language with type inference, so we should just be able to infer
>> > paths and homotopies behind the scenes by adding homotopies to the
>> > type system. That's a very nice answer in theory. But in the real
>> > world, when we're writing code that solves actual problems,
>> > theoretical niceties break down. What if a user wants to use a
>> > nonstandard homotopy?
>> >
>> > Why should we stop them just because we're too lazy to come up with a
>> > good syntax? I then realized that we keep running out of syntax
>> > because we've limited ourselves to unicode. Instead, I propose we add
>> > a potentially infinite universe of identifiers: youtube videos. For
>> > example, the higher inductive type for a circle can be written as:
>> >
>> > homotopyType http://www.youtube.com/watch?v=dQw4w9WgXcQ where
>> > Base ::: http://www.youtube.com/watch?v=dQw4w9WgXcQ
>> > Loop ::: http://www.youtube.com/watch?v=J---aiyznGQ Base Base
>> >
>> > Note that the urls do not use SSL. For portability reasons, I propose
>> > that SSL only be enabled as an optional extension.
>> >
>> > _______________________________________________
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>> > Glasgow-haskell-users at haskell.org
>> > http://www.haskell.org/mailman/listinfo/glasgow-haskell-users
>>
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>
>
>
> --
> Gregory Collins <greg at gregorycollins.net>
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