The main idea for recognising ``Translated Cyclers’’ is to find two configurations that break a record (i.e. visit a memory cell that was never visited before) in the same state (here state D, color green) such that the content of the memory tape at distance L from the record positions is the same in both record configurations. Distance L is defined as being the maximum distance to record position 1 (green cell in record 1) that was visited between the configuration of record 1 and record 2.
01/04/22 results were run on the 3,575,540 machines remaining after [Decider] Cyclers and the first passes of translated cyclers and after discarding the machines decided by [Decider] Backward reasoning because of a bug needing to be resolved in it.
05/03/22 results were run on all machines that remained undecided after [Decider] Cyclers, the first results of this decider and the backward reasoning decider. This corresponds to a total of 2,323,786 machines tested.
29/01/22 results were run on the machines starting at index 14,322,029 until final index 88,664,063 of the seed database which are the machines that exceeded the 12k memory limit. Indeed, it is unlikely that a translated cycler would exceed 47M time limit before reaching 12k visited memory cells. This corresponds to a total of 74,342,035 machines tested.
I have written a Translated Cyclers Decider in C++, which outputs a Decider Verification File (dvf) as well as an Undecided Machines File (umf). I also wrote a Verifier, which checks the dvf for correctness. You can find them at bbchallenge/TranslatedCyclers at main · TonyGuil/bbchallenge · GitHub.
The Decider was run on the output file Cyclers.umf from the Cyclers Decider, with time limit 10,000 and space limit 1,000. It found 73,860,604 Translated Cyclers, as expected. Afterwards I ran it with time limit 4,000,000, and it found 566 more Translated Cyclers (in about 12 hours). The longest of these Translated Cyclers was found after 3,092,791 steps, so there might well be more out there. TC_4000000.dvf is the Decider Verification File for this run; you can check the new Translated Cyclers for yourselves using the data in this file. The format of the dvf is described in the README file at the link. My TranslatedCyclersVerifier program takes about 0.4 seconds to verify this file.
See the README file at the link for the format of the dvf.
@cosmo: I am having trouble verifying the machines at that link. It seems to me that the repeated states are not occurring at record values of the tape head, which means they can’t be Translated Cyclers. Have I misunderstood something? If only they provided proper Verification Data!
I’ll let @sligocki jump in as he found those machines but I believe that his output contains verification data:
This notation means that the TM has period 7,129,704, offset 512 (to the right) and has started cycling by step 536,870,912 (This is not a minimum start time, but by then it is cycling for sure).
So the main issue in finding this one is running your simulator for half a billion steps! This, specifically, is the most extreme result I found before giving up my search for long periods.
Thanks for that! I can verify your machine now. The tape head after 536,870,912 is indeed not a record, but that doesn’t matter because we can verify that the tape contents to the right of the tape head are the same after 536,870,912 and (536,870,912 + 7,129,704) steps.
Note that the method based on records that I have presented in this post and implemented is not universal and @sligocki’s implementation is probably based on a different idea.
