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 g...@gregorycollins.net wrote:
Whoosh? :-)
On Sun, Apr 1, 2012 at 3:54 PM, Greg Weber g...@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 gersh...@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?
a a aa = a{a = a, aaa = aa}
Using standard Haskell typeclasses this is a difficult question to
answer. The types of 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
