Hi Dan,
Of course. I'm thinking primarily of composition for Glue given in the CCHM
paper you linked, reproduced below.
As you know, the single potential issue is that we need pres of a
degenerate filling problem and function to be reflexivity. I claim that
this holds by regularity of composition in T and A, partly as a consequence
of the fact that regularity of composition implies regularity of filling
(that fill of a degenerate system is refl), which certainly holds for fill
defined by connections (and I believe also holds for fill as defined in
ABCFHL).
(a)
Given i |- B = Glue [ phi |-> (T, f) ] A, with psi, i |- b : B and b0 :
B(i0)[ psi |-> b(i0) ], we want to compute b1 = comp^i B [ psi |-> b ] b0 :
B(i1)[ psi |-> b(i1) ].
We set a := unglue b and a0 := unglue b0.
Set delta := forall i. phi.
Then we take:
a1' := comp^i A [ psi |-> a ] a0
delta |- t1' := comp^i T [ psi |-> b ] b0
delta |- omega := pres^i f [ psi |-> b ] b0
phi(i1) |- (t, alpha) := equiv f(i1) [ delta |-> (t1', omega), psi |->
(b(i1), refl a1') ] a1'
a1 := hcomp^j A(i1) [ phi(i1) |-> alpha j, psi |-> a(i1) ] a1' (note that
in the regular setting the psi face is redundant)
b1 := glue [ phi(i1) |-> t1 ] a1
With given i |- f : T -> A, with psi, i |- b : T and b0 : T(i0)[ psi |->
b(i0) ], we define
pres^i f [ psi |-> b ] b0 = comp^i A [ psi |-> f b, j = 1 |-> f (fill^i
T [ psi |-> b ] b0) ] (f(i0) b0).
(b)
Now consider the regular case, where phi, T, f, and A are independent of i.
We want that b1 = b0.
We have that a is independent of i, and delta = phi.
First consider delta (= phi) |- pres^i f [ psi |-> b ] b0. (This is the
explanation of your first dash)
Note that if comp^i A is regular, then fill^i A [ psi |-> b ] b0 = b
This is comp^i A [ psi |-> f b, j = 1 |-> f (fill^i T [ psi |-> t ] t0)
] (f t0) = comp^i A [ psi |-> f b, j = 1 |-> f t0 ] (f t0) = f t0.
Thus pres of a degenerate filling problem and function is reflexivity.
Going back to composition of Glue,
a1' = a0
phi |- t1' = b0
phi |- omega = refl (f b0)
phi |- (t1, alpha) = (t1', omega) (since delta = phi, so we end up in the
delta face of equiv)
a1 = a1' (the only dependence on j is via (alpha j), but alpha = omega =
refl, so this filling problem is degenerate)
b1 = glue [ phi |-> t1 ] a1 = glue [ phi |-> b0 ] a0 = glue [ phi |-> b0 ]
(unglue b0) = b0 (by eta, see Figure 4 in CCHM)
Thus this algorithm for composition of Glue is regular.
Other algorithms, such as the one in ABCFHL, may not be, but I am prone to
believe that there exist regular algorithms in other settings including
Orton-Pitts and Cartesian cubes.
Best regards,
- Jasper Hugunin
On Mon, Sep 16, 2019 at 12:18 PM Licata, Dan wrote:
> Hi Jasper,
>
> It would help me follow the discussion if you could say a little more
> about (a) which version of composition for Glue exactly you mean (because
> there is at least the one in the CCHM paper and the “aligned” one from
> Orton-Pitts, which are intensionally different, as well as other possible
> variations), and (b) include some of your reasoning for why you think
> things are regular, to make it easier for me and others to reconstruct.
>
> My current understanding is that
>
> - For CCHM proper https://arxiv.org/pdf/1611.02108.pdf the potential
> issue is with the ‘pres’ path omega, which via the equiv operation ends up
> in alpha, so the system in a1 may not be degenerate even if the input is.
> Do you think this does work out to be degenerate?
>
> - For the current version of ABCFHL
> https://github.com/dlicata335/cart-cube/blob/master/cart-cube.pdf which
> uses aligning “all the way at the outside”, an issue is with the adjust_com
> operation on page 20, which is later used for aligning (in that case beta
> is (forall i phi)). The potential issue is that adjust_com uses a
> *filler*, not just a composition from 0 to 1, so even if t doesn’t depend
> on z, the filling does, and the outer com won’t cancel. In CCHM, filling
> is defined using connections, so it’s a little different, but I think there
> still has to be a dependence on z for it to even type check — it should
> come up because of the connection term that is substituted into the type of
> the composition problem. So I’d guess there is still a problem in the
> aligned algorithm for CCHM.
>
> However, it would be great if this is wrong and something does work!
>
> -Dan
>
> > On Sep 15, 2019, at 10:18 PM, Jasper Hugunin
> wrote:
> >
> > 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
> > 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
> > >
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