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State of the art in dependent typing?

To: <types@cis.upenn.edu>

Subject: State of the art in dependent typing?

From: "Tim Sweeney" <tim@epicgames.com>

Date: Wed, 21 Jun 2000 07:32:58 0400

Importance: Normal

Inreplyto: <15292.961506527@saul.cis.upenn.edu>
I'm a game programmer, but lately I found out about Haskell, Cayenne,
and dependent type systems. As a beginner, I have a few questions
about dependent types and proofsastypes systems. This is for fun
and enlightenment rather than important research, so please don't feel
obliged to respond unless you enjoy doing so.
1. In propositional logic, one can use {x:Natx<10} to express the
subset of natural numbers less than 10. HindleyMilner extensions
have been proposed to support subset types, such as Sokolowski's:
http://www.ipipan.gda.pl/~stefan/Papers/research.html
This seems very useful; is there a good reason why this hasn't been
adopted by languages like Haskell and ML?
2. In languages like Augustsson's Cayenne:
http://www.cs.chalmers.se/~augustss/cayenne/
One can model both programs and proofs in a uniform language. Why
isn't this an area of huge interest, i.e. with wouldbe Sun
Microsystems trying to popularize new mainstream languages based on
this technology?
While Java is receiving tremendous hype, it's mostly an incremental
improvement over C++, whereas this stuff is *revolutionary*, having
the potential to make commercial software far more reliable. It seems
surprising there isn't widespread interest.
Is it that people are turned off by the theoretical undecidability of
powerful dependent type systems? This doesn't seem too likely to
cause realworld problems  reasonable programs tend to be easy for a
typechecker to analyze.
3. Is there anything preventing the programswithproofs concept from
being taken to the extreme? For example, building abstract
definitions of:
 sets
 monads
 groups
 lambda calculi
 categories
 sorting functions
and concrete instances containing proofs that they obey the
appropriate laws. I've seen a few assorted systems along these lines,
but nothing universal. Has anyone attempted to build a comprehensive
set of types and proofs for mathematical objects or categories this
way?
4. I noticed the following analogy between ordinary
types and dependent types:
A=>B <> All(x:A).B
A*B <> Exists(x:A).B
A+B <> ___________
Can we fill in the blank meaningfully, i.e. is there a useful notion
of "dependent sum"?
Would it be valid to say that "A+B" is not actually a primitive type
itself, but instead corresponds to the existential subset type:
Exists( {tt==A or t==B)} ).t
Where the {xP(x)} represents the notion of "subset types" as
described above? One can read this as: there exists a type t, equal
to either A or B, such that we encode a single value of type t. This
encoding of sums is (I think) a more flexible version of Cardelli's
"Any/TypeOf/ValueOf" definitions:
http://research.microsoft.com/Users/luca/Papers/TypeType.ps
5. Cayenne has a construct for defining records:
s = struct {a=1; s="hello";}
Has anyone tried defining such records as functions from strings
(corresponding to tags) to dependent types? i.e.:
s :: (x::String).
if x="A" then Int;
else if x="s" then String;  else __
s = \(x:String)>
if x="A" then 1;
else if x="s" then "hello";
Such functions seem to obey the ordinary rules of records.
A variation on this theme is to represent certain records as
"functions from types to dependent types". These seem to be
signaturefree records, which may be useful for composing
Cayennestyle proofs without committing to specific proof names.
6. Looking at the bigger picture, are "programs with proofs" of
practical value in future commercial software? I have to wonder if
we're simply trading "bugs in programs" for "bugs in proof types".
For example, I can easily write a "qsort" function, test it with
sample data, and be pretty confident it's ok.
But, what is the "proof type which all sorting functions must
inhabit"? I have no idea, but suspect it's very complex and dependent
on other proofs such as those governing ordering relationships. Given
human fallability, it seems just as likely to write the wrong "proof
type" (hence prove the wrong thing) as write an buggy program. Is
this a problem in practice?
Thanks,
Tim