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 wrote:

Whoosh? :-)

On Sun, Apr 1, 2012 at 3:54 PM, Greg Weber 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 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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-- Gregory Collins