Ciro Santilli @cirosantilli 36

Bibliography: discuss.bbchallenge.org/t/decider-cyclers/33

Example: bbchallenge.org/279081.

These are very simple, they just check for exact state repetitions, which obviously imply that they will run forever.

Unfortunately, cyclers may need to run through an initial setup phase before reaching the initial cycle point, which is not very elegant.

Also, we have no way of knowing the initial setup length of the actual cycle length, so we just need an arbitrary cutoff value.

And unfortunately, this can lead to misses, e.g. Skelet machine #1, a 5 state machine, has a (translated) cycle that starts at around 50-200M steps, and takes 8 trillion steps to repeat.

Non formal proof with a program March 2023: www.sligocki.com/2023/03/13/skelet-1-infinite.html Awesome article that describes the proof procedure.

Formal proof August 2023: discuss.bbchallenge.org/t/skelet-1-is-a-translated-cycler-coq-agrees/166

The proof uses Turing machine acceleration to show that Skelet machine #1 is a Translated cycler Turing machine with humongous cycle paramters:

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.