Ciro Santilli @cirosantilli 35

Bibliography: discuss.bbchallenge.org/t/decider-translated-cyclers/34

Like a cycler, but the cycle starts at an offset.

To see infinity, we check that if the machine only goes left N squares until reaching the repetition, then repetition must only be N squares long.

Described at: www.sligocki.com/2022/06/10/ctl.html

The following things come to mind when you look into research in this area, especially the search for BB(5) which was hard but doable:

Bibliography:

If you can reduce a mathematical problem to the Halting problem of a specific turing machine, as in the case of a few machines of the Busy beaver scale, then using Turing machine deciders could serve as a method of automated theorem proving.

That feels like it could be an elegant proof method, as you reduce your problem to one of the most well studied representations that exists: a Turing machine.

However it also appears that certain problems cannot be reduced to a halting problem... OMG life sucks (or is awesome?): Section "Turing machine that halts if and only if Collatz conjecture is false".

ECDH has smaller keys. youtu.be/gAtBM06xwaw?t=634 mentions some interesting downsides:

Stopped 2019 apparently. Shame. We need something to be upstreamed!

bbchallenge.org/story#what-is-known-about-bb lists some (all?) cool examples,

wiki.bbchallenge.org/wiki/Cryptids contains a larger list. In June 2024 it was discovered that BB(6) is hard.

Turing machine acceleration refers to using high level understanding of specific properties of specific Turing machines to be able to simulate them much fatser than naively running the simulation as usual.

Acceleration allows one to use simulation to find infinite loops that might be very long, and would not be otherwise spotted without acceleration.

This is for example the case of www.sligocki.com/2023/03/13/skelet-1-infinite.html proof of Skelet machine #1.

Best busy beaver machine known since 1989 as of 2023, before a full proof of all 5 state machines had been carried out.

Entry on The Busy Beaver Challenge: bbchallenge.org/1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA

Paper extracted to HTML by Heiner Marxen: turbotm.de/~heiner/BB/mabu90.html

List on The Busy Beaver Challenge: bbchallenge.org/skelet

Bibliography:

mathoverflow.net/questions/309044/is-there-a-known-turing-machine-which-halts-if-and-only-if-the-collatz-conjectur suggests one does not exist. Amazing.

Intuitively we see that the situation is fundamentally different from the Turing machine that halts if and only if the Goldbach conjecture is false because for Collatz the counter example must go off into infinity, while in Goldbach conjecture we can finitely check any failures.

Amazing.

This could refer to several more specific courses, see the tagged articles for a list.