[Agda] [Coq-Club] Why dependent type theory?

[Dominique Unruh]

Hi,

following up with Thorsten's command about the word "dependent type 
theory", I would like to add a few observations I had in this discussion:

  * I think the word "type theory" itself is unclear in this context. At
    least several of the emails seem to use different but related ideas
    of what that means:
      o It could mean "something where everything has a type" (i.e., not
        the usual ZF). Then HOL would be type theory. (Thorsten's email
        quoted below makes sense in that setting because HOL avoids the
        problem described there.)
      o It could mean the above with dependent types (but not necessary
        the Curry-Howard thing from the next item)
      o It could mean "a system where we use the same language for
        propositions and proofs via Curry-Howard isomorphism" (I believe
        this will then also need dependent types since otherwise the
        proof terms are not powerful enough)
      o It could mean a system with strong normalization (so that
        everything terminates), at least some of the answers seem to see
        this as part of the meaning of "type theory".

Of course, there are many interaction between the different concepts, 
but if we talk about the costs or benefits of adopting type theory, it 
becomes quite important which aspect of it we are adopting. (E.g., if we 
have, say, a good reason why we need the second point, and a big cost 
for getting the four point, then it is important not to just say "type 
theory has the following good reason and the following cost".)

Maybe when discussing *why* type theory, we should prefix our answers by 
what we mean by type theory (e.g., one of the above). Otherwise it will 
be very confusing (at least to me).

Another question is also the context in which we want to use it:

  * Programming (with all the associated things like verification, type
    checking, etc.)
  * Math

These have different requirements, so making explicit which domain we 
are thinking of in our answer might make things clearer as well.

Just my personal thoughts, but I hope they may help to add some clarity 
to the discussion.

Best wishes,
Dominique.



On 3/4/20 11:42 AM, Thorsten Altenkirch wrote:
>
> First of all I don’t like the word “dependent type theory”. Dependent 
> types are one important feature of modern Type Theory but hardly the 
> only one.
>
> To me the most important feature of Type Theory is the support of 
> abstraction in Mathematics and computer science. Using  types instead 
> of sets means that you can hide implementation choices which is 
> essential if you want to build towers of abstraction. Set theory fails 
> here badly. Just as a very simple example: in set theory you have the 
> notion of union, so for example
>
> {0,1}  \cup {0,1,2,3} = {0,1,2,3}
>
> However, if we change the representation of the first set and use lets 
> say {true,false} we get a different result:
>
> {true , false}  \cup {0,1,2,3} = {true,false,0,1,2,3}
>
> This means that \cup exposes implementation details because the 
> results are not equivalent upto renaming. In Type Theory we have the 
> notion of sum, sometimes called disjoint union, which is well behaved
>
> {0,1}  + {0,1,2,3} = {in1 0,in1 1,in2 0,in2 1,in2 2,in2 3}
>
> {true , false}  + {0,1,2,3} = {in1 true,in1 false ,in2 0,in2 1,in2 
> 2,in2 3}
>
> Unlike \cup, + doesn’t reveal any implementation details it is a 
> purely structural operation. Having only structural operations means 
> that everything you do is stable under equivalence, that is you can 
> replace one object with another one that behaves the same. This is the 
> essence of Voevodsky’s univalence principle.
>
> There are other nice aspects of Type Theory. From a constructive point 
> of view (which should come naturally to a computer scientists) the 
> proporsitions as types explanation provides a very natural way to 
> obtain “logic for free” and paedagogically helpful since it reduces 
> logical reasoning to programming.
>
> There are performance issues with implementations of Type Theory, 
> however, in my experience (mainly agda) the execution of functions at 
> compile time isn’t one of them. In my experience the main problem is 
> to deal with a loss of sharing when handling equational constraints 
> which can blow up the time needed for type checking. I think this is 
> an engineering problem and there are some suggestions how to fix this.
>
> Thorsten
>
> *From: *"coq-club-request at inria.fr" <coq-club-request at inria.fr> on 
> behalf of Jason Gross <jasongross9 at gmail.com>
> *Reply to: *"coq-club at inria.fr" <coq-club at inria.fr>
> *Date: *Tuesday, 3 March 2020 at 19:44
> *To: *coq-club <coq-club at inria.fr>, agda-list 
> <agda at lists.chalmers.se>, "coq+miscellaneous at discoursemail.com" 
> <coq+miscellaneous at discoursemail.com>, lean-user 
> <lean-user at googlegroups.com>
> *Subject: *[Coq-Club] Why dependent type theory?
>
> I'm in the process of writing my thesis on proof assistant performance 
> bottlenecks (with a focus on Coq), and there's a large class of 
> performance bottlenecks that come from (mis)using the power of 
> dependent types.  So in writing the introduction, I want to provide 
> some justification for the design decision of using dependent types, 
> rather than, say, set theory or classical logic (as in, e.g., 
> Isabelle/HOL).  And the only reasons I can come up with are "it's fun" 
> and "lots of people do it"
>
> So I'm asking these mailing lists: why do we base proof assistants on 
> dependent type theory?  What are the trade-offs involved?
>
> I'm interested both in explanations and arguments given on list, as 
> well as in references to papers that discuss these sorts of choices.
>
> Thanks,
>
> Jason
>
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