This problem has been solved – see our TYPES 2018 abstract (attached).
Basically the idea is to define a relation between “pre-terms” and the semantics and then show that this is contractible for “well-typed objects”, this way you avoid the mutual dependency. This was an idea by Andras Kovacs. In this year’s TYPES there are two abstracts that show how this can be used to give a universal reduction from Inductive-Inductive types to indexed inductive types and hence W-types.
I have discussed this with Jesper when he was in Nottingham and I think our tentative conclusion was that this could be combined with his approach to provide a reduction for IITs. However, this needs to be checked.
Thorsten
From: <homotopyt...@
googlegroups.com > on behalf of Matt Oliveri <atm...@gmail.com>
Date: Tuesday, 21 May 2019 at 01:28
To: Homotopy Type Theory <homotopyt...@googlegroups.com >
Subject: Re: [HoTT] Semantics of QIITs ?
I'm not completely satisfied with Hugunin's technique, because it justifies only the "simple" elimination principle, rather than the general, "recursive-recursive" elimination principle implemented in Agda. As far as I can tell, realistic use cases for inductive-inductive families also need the recursive-recursive elimination principle, where the types of the maps out of later families depend on the maps out of earlier families. (I'm not sure how much of the other research on IIFs stops short of handling recursion-recursion, but I think it should be taken seriously.)
On Monday, May 20, 2019 at 7:26:12 PM UTC-4, Jon Sterling wrote:Echoing Andras, I recall that a new encoding due to Jasper Hugunin enable us to interpret IITs without using UIP, and it is an open question to determine whether Jasper's ideas can be extended to QIITs. I hope they can, and someone should find out :)
Best,
Jon