A Modest Records Proposal
Gershom B
gershomb at gmail.com
Sun Apr 1 08:14:58 CEST 2012
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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