```
From: Jasper Hugunin <jasperh@cs.washington.edu>
To: HomotopyTypeTheory@googlegroups.com
Subject: Fwd: [HoTT] Re: A question about the problem with regularity in CCHM cubical type theory
Date: Sun, 15 Sep 2019 22:18:37 -0400
Message-ID: <CAGTS-a8SZx8PiaD-9rq5QWffU75Wz8myrXD1g5P3DCjSO=fvOQ@mail.gmail.com> (raw)
In-Reply-To: <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com>
[-- Attachment #1: Type: text/plain, Size: 5022 bytes --]
This doesn't seem right; as far as I can tell, composition for Glue types
in CCHM preserves regularity and reduces to composition in A on phi.
- Jasper Hugunin
On Sun, Sep 15, 2019 at 3:28 AM Anders Mortberg <anders.mortberg@math.su.se>
wrote:
> Hi Jasper,
>
> Indeed, the problem is to construct an algorithm for comp (or even
> coe/transp) for Glue that reduces to the one of A when phi is true
> while still preserving regularity. It was pointed out independently by
> Sattler and Orton around 2016 that one can factor out this step in our
> algorithm in a separate lemma that is now called "alignment". This is
> Thm 6.13 in Orton-Pitts and discussed in a paragraph in the end of
> section 2.11 of ABCFHL. Unless I'm misremembering this is exactly
> where regularity for comp for Glue types break down. In this step we
> do an additional comp/hcomp that inserts an additional forall i. phi
> face making the comp/coe irregular.
>
> One could imagine there being a way to modify the algorithm to avoid
> this, maybe by inlining the alignment step... But despite considerable
> efforts no one has been able to figure this out and I think Swan's
> recent paper (https://arxiv.org/abs/1808.00920v3) shows that this is
> not even possible!
>
> Another approach would be to have weak Glue types that don't strictly
> reduce to A when phi is true, but this causes problems for the
> composition in the universe and would be weird for cubical type
> theory...
>
> In light of Swan's negative results I think we need a completely new
> approach if we ever hope to solve this problem. Luckily for you Andrew
> Swan is starting as a postdoc over in Baker Hall in October, so he can
> explain his counterexamples to you in person.
>
> Best,
> Anders
>
> On Sun, Sep 15, 2019 at 7:57 AM Jasper Hugunin
> <jasperh@cs.washington.edu> wrote:
> >
> > Offline, Carlo Angiuli showed me that the difficulty was in part 1,
> because of a subtlety I had been forgetting.
> >
> > Since types are *Kan* cubical sets, we need that the Kan operations
> agree as well as the sets.
> > So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A,
> compGlue) = (A, compA), and getting that the Kan operations to agree was/is
> difficult.
> > (Now that I know what the answer is, it is clear that this was already
> explained in the initial discussion.)
> >
> > Humbly,
> > - Jasper Hugunin
> >
> > On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <
> jasperh@cs.washington.edu> wrote:
> >>
> >> Hello all,
> >>
> >> I've been trying to understand better why composition for the universe
> does not satisfy regularity.
> >> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |->
> equiv^i E ] A, I would expect regularity to follow from two parts:
> >> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of
> regularity condition for the Glue type constructor itself)
> >> 2. That equiv^i (refl A) reduces to equivRefl A
> >> I'm curious as to which (or both) of these parts was the issue, or if
> regularity for the universe was supposed to follow from a different
> argument.
> >>
> >> Context:
> >> I've been studying and using CCHM cubical type theory recently, and
> often finding myself wishing that J computed strictly.
> >> If I understand correctly, early implementations of ctt did have strict
> J for Path types, and this was justified by a "regularity" condition on the
> composition operation, but as discussed in this thread on the HoTT mailing
> list, the definition of composition for the universe was found to not
> satisfy regularity.
> >> I don't remember seeing the regularity condition defined anywhere, but
> my understanding is that it requires that composition in a degenerate line
> of types, with the system of constraints giving the sides of the box also
> degenerate in that direction, reduces to just the bottom of the box. This
> seems to be closed under the usual type formers, plus Glue, but not the
> universe with computation defined as in the CCHM paper (for trivial reasons
> and non-trivial reasons; it gets stuck at the start with Glue [ phi |->
> equiv^i refl ] A not reducing to anything).
> >>
> >> Best regards,
> >> - Jasper Hugunin
> >
> > --
> > You received this message because you are subscribed to the Google
> Groups "Homotopy Type Theory" group.
> > To unsubscribe from this group and stop receiving emails from it, send
> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
> > To view this discussion on the web visit
> https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a9oS0CQDGKgj7ghCh8%2BZwAcAefiTg4JJVHemV3HUPcPEg%40mail.gmail.com
> .
>
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<div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div dir="ltr">This doesn't seem right; as far as I can tell, composition for Glue types in CCHM preserves regularity and reduces to composition in A on phi.<div><br></div><div>- Jasper Hugunin</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Sep 15, 2019 at 3:28 AM Anders Mortberg <<a href="mailto:anders.mortberg@math.su.se" target="_blank">anders.mortberg@math.su.se</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Hi Jasper,<br>
<br>
Indeed, the problem is to construct an algorithm for comp (or even<br>
coe/transp) for Glue that reduces to the one of A when phi is true<br>
while still preserving regularity. It was pointed out independently by<br>
Sattler and Orton around 2016 that one can factor out this step in our<br>
algorithm in a separate lemma that is now called "alignment". This is<br>
Thm 6.13 in Orton-Pitts and discussed in a paragraph in the end of<br>
section 2.11 of ABCFHL. Unless I'm misremembering this is exactly<br>
where regularity for comp for Glue types break down. In this step we<br>
do an additional comp/hcomp that inserts an additional forall i. phi<br>
face making the comp/coe irregular.<br>
<br>
One could imagine there being a way to modify the algorithm to avoid<br>
this, maybe by inlining the alignment step... But despite considerable<br>
efforts no one has been able to figure this out and I think Swan's<br>
recent paper (<a href="https://arxiv.org/abs/1808.00920v3" rel="noreferrer" target="_blank">https://arxiv.org/abs/1808.00920v3</a>) shows that this is<br>
not even possible!<br>
<br>
Another approach would be to have weak Glue types that don't strictly<br>
reduce to A when phi is true, but this causes problems for the<br>
composition in the universe and would be weird for cubical type<br>
theory...<br>
<br>
In light of Swan's negative results I think we need a completely new<br>
approach if we ever hope to solve this problem. Luckily for you Andrew<br>
Swan is starting as a postdoc over in Baker Hall in October, so he can<br>
explain his counterexamples to you in person.<br>
<br>
Best,<br>
Anders<br>
<br>
On Sun, Sep 15, 2019 at 7:57 AM Jasper Hugunin<br>
<<a href="mailto:jasperh@cs.washington.edu" target="_blank">jasperh@cs.washington.edu</a>> wrote:<br>
><br>
> Offline, Carlo Angiuli showed me that the difficulty was in part 1, because of a subtlety I had been forgetting.<br>
><br>
> Since types are *Kan* cubical sets, we need that the Kan operations agree as well as the sets.<br>
> So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A, compGlue) = (A, compA), and getting that the Kan operations to agree was/is difficult.<br>
> (Now that I know what the answer is, it is clear that this was already explained in the initial discussion.)<br>
><br>
> Humbly,<br>
> - Jasper Hugunin<br>
><br>
> On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <<a href="mailto:jasperh@cs.washington.edu" target="_blank">jasperh@cs.washington.edu</a>> wrote:<br>
>><br>
>> Hello all,<br>
>><br>
>> I've been trying to understand better why composition for the universe does not satisfy regularity.<br>
>> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts:<br>
>> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)<br>
>> 2. That equiv^i (refl A) reduces to equivRefl A<br>
>> I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.<br>
>><br>
>> Context:<br>
>> I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.<br>
>> If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in this thread on the HoTT mailing list, the definition of composition for the universe was found to not satisfy regularity.<br>
>> I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in the CCHM paper (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything).<br>
>><br>
>> Best regards,<br>
>> - Jasper Hugunin<br>
><br>
> --<br>
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br>
> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br>
> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a9oS0CQDGKgj7ghCh8%2BZwAcAefiTg4JJVHemV3HUPcPEg%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a9oS0CQDGKgj7ghCh8%2BZwAcAefiTg4JJVHemV3HUPcPEg%40mail.gmail.com</a>.<br>
</blockquote></div></div>
</div></div>
<p></p>
-- <br />
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```

next prev parent reply indexThread overview:13+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-09-13 6:10 [HoTT] " Jasper Hugunin 2019-09-15 5:57 ` [HoTT] " Jasper Hugunin [not found] ` <CAMWCppk4PWzfZ1HKNLMdAZ=RBC-ARxtJXR8okvwO3raea5gC8Q@mail.gmail.com> [not found] ` <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com>2019-09-16 2:18 ` Jasper Hugunin [this message]2019-09-16 16:18 ` [HoTT] " Licata, Dan 2019-09-16 17:09 ` Jasper Hugunin 2019-09-16 19:01 ` Licata, Dan 2019-09-16 20:17 ` Jasper Hugunin 2019-09-18 12:16 ` Anders Mortberg 2019-09-18 12:52 ` Thierry Coquand 2019-09-15 11:55 ` [HoTT] " Andrew Swan 2019-09-15 22:38 ` Jasper Hugunin 2019-09-16 1:04 ` Jon Sterling [not found] ` <A605E6EE-0101-4390-B50D-A6AEB36FDCC2@icloud.com> 2019-09-16 1:44 ` Jon Sterling

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