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From: "Anders Mörtberg" <andersmortberg@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] The Brunerie number is -2
Date: Tue, 24 May 2022 02:46:35 -0700 (PDT)	[thread overview]
Message-ID: <36c1e828-af3a-49c2-840b-1fae9897eeban@googlegroups.com> (raw)
In-Reply-To: <CAMWCppmqhhiHkpgHPqoSMvdyQUsT=hFevG-LJ8FPhtrLpNtdbQ@mail.gmail.com>

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I did some digging and found the first commit with an attempt to compute 
the number written by Guillaume Brunerie, Thierry Coquand and Simon Huber 
already in December 2014:


This code is written for the predecessor of cubicaltt called "cubical" and 
Thierry reminded me that it was based on the buggy regularity evaluator 
which we eventually fixed in cubicaltt. I also think this cubical code is 
what became Appendix B in Guillaume's thesis (or maybe Guillaume already 
had a draft at the time, I don't remember). Anyway, Guillaume gave a nice 
talk in 2017 with an overview of the attempts up to then and the problems 
we had encountered:


This was before Cubical Agda was invented and there are some tricks in 
Cubical Agda that might make a difference compared to cubicaltt (in 
particular the "ghcomp" trick which I learned about from 
Angiuli-Favonia-Harper and which eliminates empty systems in hcomps). Now 
that we have a computation that actually terminates I'm looking forward to 
seeing if any of these tricks actually were necessary or if it was "just" a 
matter of simplifying the definition of the number. 


On Monday, May 23, 2022 at 11:00:17 PM UTC+2 Anders Mörtberg wrote:

> Thanks Nicolai! And yes, our β' is a different definition of the order of 
> pi_4(S^3). In fact, the number β in the Summary file is not exactly the 
> same number as in Guillaume's Appendix B either for various reasons. For 
> instance, Guillaume only uses 1-HITs while we are quite liberal in using 
> higher HITs as they are not much harder to work with in Cubical Agda than 
> 1-HITs. Also Guillaume of course defines everything with path-induction 
> while we use cubical primitives and the maps in appendix B are quite 
> unnecessarily complex from a cubical point of view (for instance, the 
> equivalence S^3 = S^1 * S^1 can be written quite directly in Cubical Agda 
> while in Book HoTT it's a bit more involved and Guillaume uses a chain of 
> equivalences to define it).
> One could of course define the number exactly like Guillaume does and try 
> to compute it, but I don't find that very interesting now that we have a 
> much simpler definition which is fast to compute. However, we have come up 
> with various other interesting numbers that we can't get Cubical Agda to 
> compute, so there's definitely room to make cubical evaluation faster. 
> Surprisingly enough though, one doesn't need to do this in order to get 
> Cubical Agda to compute the order of pi_4(S^3)   :-)
> --
> Anders
> On Mon, May 23, 2022 at 10:23 PM Nicolai Kraus <nicola...@gmail.com> 
> wrote:
>> Congratulations! It's great that this number finally computes in practice 
>> and not just in theory, after all these years. :-)
>> And it's impressive how short the new proof is! But this still doesn't 
>> mean that Cubical Agda passes the test that Guillaume formulates in 
>> Appendix B of his thesis, right? Because this test refers to the Brunerie 
>> number β (in the Summary.agda file you linked), and not to β'.
>> In any case, that's a fantastic result!
>> Best,
>> Nicolai
>> On Mon, May 23, 2022 at 8:30 PM Anders Mortberg <andersm...@gmail.com> 
>> wrote:
>>> We're very happy to announce that we have finally managed to compute the 
>>> Brunerie number using Cubical Agda... and the result is -2! 
>>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>>> The computation was made possible by a new direct synthetic proof that 
>>> pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of 
>>> new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we 
>>> got the one called β' in the file above to reduce to -2 in just a few 
>>> seconds. With some work we then managed to prove that pi_4(S^3) = Z / β' 
>>> Z, leading to a proof by normalization of the number as conjectured in 
>>> Brunerie's thesis.
>>> Axel's new proof is very direct and completely avoids chapters 4-6 in 
>>> Brunerie's thesis (so no cohomology theory!), but it relies on chapters 1-3 
>>> to define the number. It also does not rely on any special features of 
>>> cubical type theory and should be possible to formalize also in systems 
>>> based on Book HoTT. For a proof sketch as well as the formalization of the 
>>> new proof in just ~700 lines (not counting what is needed from chapters 
>>> 1-3) see: 
>>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>>> So to summarize we now have both a new direct HoTT proof, not relying on 
>>> cubical computations, as well as a cubical proof by computation.
>>> Univalent regards,
>>> Anders and Axel
>>> PS: the minus sign is actually not very significant and we can get +2 by 
>>> slightly modifying β', but it's quite funny that we ended up getting -2 
>>> when we finally got a definition which terminates!
>>> -- 
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>>> .

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  reply	other threads:[~2022-05-24  9:46 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2022-05-23 19:30 Anders Mortberg
2022-05-23 19:38 ` Steve Awodey
2022-05-23 20:22 ` Nicolai Kraus
2022-05-23 20:59   ` Anders Mortberg
2022-05-24  9:46     ` Anders Mörtberg [this message]
2022-05-24  9:49       ` Anders Mörtberg

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