In recent years there has been a movement to collaborate on math proofs via blueprints (dependency graphs) in the Lean language, which seems related.For example:<a href="https:&#x2F;&#x2F;teorth.github.io&#x2F;equational_theories&#x2F;" rel="nofollow">https:&#x2F;&#x2F;teorth.github.io&#x2F;equational_theories&#x2F;</a><a href="https:&#x2F;&#x2F;teorth.github.io&#x2F;pfr&#x2F;" rel="nofollow">https:&#x2F;&#x2F;teorth.github.io&#x2F;pfr&#x2F;</a>

The BB Challenge site is really well structured:<a href="https:&#x2F;&#x2F;bbchallenge.org&#x2F;13650583" rel="nofollow">https:&#x2F;&#x2F;bbchallenge.org&#x2F;13650583</a>

what&#x27;s most interesting to me about this research is that it is an online collaborative one. I wonder how many more project such as this there are, and if it could be more widespread, maybe as a platform.

Last but not least, those deciders were implemented and verified in the Rocq proof assistant, so we know they are correct.

Not quite, I think this is the relevant part of the paper:&gt; Structure of the proof. The proof of our main result, Theorem 1.1, is given in Section 6. The structure of the proof is as follows: machines are enumerated arborescently in Tree Normal Form (TNF) [9] – which drastically reduces the search space’s size: from 16,679,880,978,201 5-state machines to “only” 181,385,789; see Section 3. Each enumerated machine is fed through a pipeline of proof techniques, mostly consisting of deciders, which are algorithms trying to decide whether the machine halts or not. Because of the uncomputability of the halting problem, there is no universal decider and all the craft resides in creating deciders able to decide large families of machines in reasonable time. Almost all of our deciders are instances of an abstract interpretation framework that we call Closed Tape Language (CTL), which consists in approximating the set of configurations visited by a Turing machine with a more convenient superset, one that contains no halting configurations and is closed under Turing machine transitions (see Section 4.2). The S(5) pipeline is given in Table 3 – see Table 4 for S(2,4). All the deciders in this work were crafted by The bbchallenge Collaboration; see Section 4. In the case of 5-state machines, 13 Sporadic Machines were not solved by deciders and required individual proofs of nonhalting, see Section 5.So, they figured out how to massively reduce the search space, wrote some generic deciders that were able to prove whether large amounts of the remaining search spaces would halt or not, and then had to manually solve the remaining 13 machines that the generic deciders couldn&#x27;t reason about.

I think you have to exhaustively check each 5-state TM, but then for each one brute force will only help a bit - brute force can&#x27;t tell you that a TM will run forever without stopping?

Isn&#x27;t it rather that the Busy Beaver function is uncomputable, particular values can be calculated - although anything great than BB(5) is quite a challenge!<a href="https:&#x2F;&#x2F;scottaaronson.blog&#x2F;?p=8972" rel="nofollow">https:&#x2F;&#x2F;scottaaronson.blog&#x2F;?p=8972</a>

The busy beaver numbers form an uncomputable sequence.For BB(5) the proof of its value is an indirect computation. The verification process involved both computation (running many machines) and proofs (showing others run forever or halt earlier). The exhaustiveness of crowdsourced proofs was a tour de force.

They are at the very boundary of what is computable!

You can not rely on brute force alone to compute these numbers. They are uncomputable.

I can&#x27;t quite understand - did they use brute force?

Determination of the fifth Busy Beaver value