[HN Gopher] Which Collatz numbers do Busy Beavers simulate (if a...
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       Which Collatz numbers do Busy Beavers simulate (if any)?
        
       Author : Fibra
       Score  : 38 points
       Date   : 2025-10-18 09:54 UTC (5 days ago)
        
 (HTM) web link (gbragafibra.github.io)
 (TXT) w3m dump (gbragafibra.github.io)
        
       | LegionMammal978 wrote:
       | To the extent that some champion machines have Collatz-like
       | behavior, it's usually not by iterating a function with
       | randomized scale factors until the value hits zero (like in the
       | original Collatz iteration), but instead by iterating a function
       | with a fixed scale factor until it hits a certain modular value.
       | 
       | E.g., the BB(5) machine [0] repeatedly multiplies a unary counter
       | by 5/3 and adds a small offset, until the value becomes 2 mod 3
       | and the machine halts. In further domains, these kinds of "halt
       | once a certain modular value is reached" conditions are used to
       | terminate iterated exponentiation and higher operations.
       | 
       | The main similarity with the original Collatz iteration is that
       | the machines repeatedly divide out a certain factor to produce a
       | pseudorandom stream of remainders. In some cases, a machine
       | measures a statistical property of the stream that is expected to
       | hold forever, but cannot easily be proven to do so, as in the
       | infamous Hydra/Antihydra machines [1].
       | 
       | Somewhat curiously, a candidate for the BB(3,3) champion machine
       | [2] does not use any sort of Collatz-like modular-value
       | conditions. Instead, it repeatedly samples points and compares
       | them against the subdivisions of a sort of Cantor set, until one
       | of the points perfectly hits the edge of a subdivision. It's
       | expected to halt after roughly 10^135 iterations.
       | 
       | [0] https://wiki.bbchallenge.org/wiki/5-state_busy_beaver_winner
       | 
       | [1] https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html
       | 
       | [2] https://wiki.bbchallenge.org/wiki/1RB2LC1RC_2LC---
       | 2RB_2LA0LB...
        
       | pclalv wrote:
       | It's interesting that the Collatz tapes look like wavy sheets,
       | like a photo of a curtain with a shadow cast over one half. I
       | wonder if there is such a higher-dimensional shape that's
       | underlying these 2D plots, and what that shape might look like
       | from different points of view.
        
         | rcxdude wrote:
         | I think a large part of that is the read head wandering left
         | and right. For any machine that halts in a nontrivial number of
         | steps it would be expected that the head doesn't move
         | monotonically.
        
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       (page generated 2025-10-23 23:01 UTC)