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2023/02/05/shift-overflow

Shift Overflow Counters | sligocki

In my last post, I described the peculiar behavior of Skelet #34. After sharing that post on the bbchallenge.org Discord, Justin Blanchard and @IIjIl shared some interesting machines with similar looking behavior. I will call these Shift Overflow Counters. They seem to characterized by having a completely orderly “Counter Phase” in which they implement basic double counter until one of the sides overflows (expands) at which point they shift the block offset leading to the other side counter needing to be “reparsed” (in Skelet #34 this shifted 1000 -> 0100).

https://www.sligocki.com//2023/02/05/shift-overflow.html

2023/10/19/greens-machines

Green’s Machines | sligocki

In 1964 (only 2 years after Tibor Radó first described the Busy Beaver game), Milton W. Green of the Stanford Research Institute hand-crafted a family of fast-growing Turing Machines with (n) states, 2 symbols for all (n \ge 4).1 At the time of publication, Green’s Machines were the Busy Beaver champions for (BB(n)) for all (n \ge 6). This family grows roughly as fast as the Ackermann function Milton W. Green. “A Lower Bound on Rado’s Sigma Function for Binary Turing Machines”, Preceedings of the IEEE Fifth Annual Symposium on Switching Circuits Theory and Logical Design, 1964, pages 91–94, doi: 10.1109/SWCT.1964.3. ↩

https://www.sligocki.com//2023/10/19/greens-machines.html

2022/07/11/bb-16-graham

BB(16) > Graham’s Number | sligocki

Almost exactly 12 years ago, I wrote an article1 describing an idea on how to hand-build a a Turing Machine which runs for more steps than Graham’s number (g_{64}) (A famously huge number). Doing so would give a constructive upper bound to the (N) such that (BB(N) > g_{64}) and perhaps provide a little more intuition on how the Busy Beaver function grows. The original article was actually posted on my Wikipedia User page of all places! And, honestly, that location has served me well. That article has been preserved and is still available 12 years later, at no cost, ad free and even with a detailed change history! That said, I did recently copy it to this blog. ↩

https://www.sligocki.com//2022/07/11/bb-16-graham.html

2023/10/16/bb-3-3-is-hard

BB(3, 3) is Hard | sligocki

I’m excited to share a 3 state, 3 symbol Turing Machine that cannot be proven to halt or not (when starting on a blank tape) without solving a Collatz-like problem. Therefore, solving the (BB(3, 3)) problem is at least as hard as solving this Collatz-like problem, a class of problem for which Paul Erdős famously said: “Mathematics may not be ready for such problems.”

https://www.sligocki.com//2023/10/16/bb-3-3-is-hard.html

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