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? >> > >> > 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. >> > >> > _______________________________________________ >> > Glasgow-haskell-users mailing list >> > glasgow-haskell-us...@haskell.org >> > http://www.haskell.org/mailman/listinfo/glasgow-haskell-users >> >> _______________________________________________ >> Haskell-Cafe mailing list >> Haskell-Cafe@haskell.org >> http://www.haskell.org/mailman/listinfo/haskell-cafe > > > > > -- > Gregory Collins <g...@gregorycollins.net> _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
