The Busy Beaver Game

Presented at the book club at work. We read THE NEW TURING OMNIBUS, and you can find a description of The Busy Beaver Game there in chapter 39.

Busy Beavers describe a class of n-state, two-symbol Turing Machines whose aim is to write the greatest number of ones on an initial blank tape before halting. If we say that Σ(n) equals the greatest achievable number of ones for an n-state machine, then Σ can be shown to be non-computable. Moreover, it grows asymptotically faster than any other computable function.

Unfortunately, Speaker Deck doesn't seem to allow me to post the slide comments.

At the end of the deck I present plots on the number of ones and number of steps for 2 and 3-state Busy Beaver champions.

This time around I used Google Docs instead of Apple Keynote, and I think it's become a viable alternative. You can view the slides in your browser at https://t.co/WlQAhCuBZH

Transcript

1. The Busy Beaver Game Christian Stigen @ Roxar AS 2016-04-27
2. Tibor Radó Bell Labs, 1962
5. Recap Turing Machines • Tape • Current State • Current Symbol • Next Symbol • Next State • Next Movement
7. State Symbol 0 Symbol 1 A B1R D0L B C1R F0R C C1L A1L D E0L Z1L E A1L B0R F C0R E0R
8. State Symbol 0 Symbol 1 A B1R D0L B C1R F0R C C1L A1L D E0L Z1L E A1L B0R F C0R E0R Write 1, Move Left, Go to Z (halt)
9. n Σ(n)
10. n Σ(n) 0 0
11. n Σ(n) 0 0 1 1
12. n Σ(n) 0 0 1 1 2
13. (4(n+1))2n
14. (4(n+1))2n Current state Current symbol New symbol x New direction New state + Halt
15. • Input: The current non-halting state (n possibilities) • Input: The symbol in the current tape cell (2 possibilities) • Output: The new symbol to write (2 possibilities) • Output: The direction to move (2 possibilities) • Output: The new state to transition to (n+1 possibilities, Z=halting state) • Output = 2*2*(n+1), Output x Input = (4(n+1))2n (4(n+1))2n Current state Current symbol New symbol x New direction New state + Halt
16. Symbol 0 Symbol 1 State A State B
17. Symbol 0 Symbol 1 State A B1L State B Tape 0 1 0 0
18. Symbol 0 Symbol 1 State A B1L State B A1R Tape 0 1 0 0 1 1 0 0
19. Symbol 0 Symbol 1 State A B1L B1R State B A1R Tape 0 1 0 0 1 1 0 0 1 1 0 0
20. Symbol 0 Symbol 1 State A B1L B1R State B A1R Tape 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0
21. Symbol 0 Symbol 1 State A B1L B1R State B A1R Tape 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1
22. Symbol 0 Symbol 1 State A B1L B1R State B A1R Z1L Tape 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1
23. Tape 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 Tape 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1
24. Symbol 0 Symbol 1 State A B1R A1L State B A1R Z0L
25. n Σ(n) 0 0 1 1 2 4
26. n Σ(n) S(n) 0 0 0 1 1 1 2 4 6
27. A B1R D0L B C1R F0R C C1L A1L D E0L Z1L E A1L B0R F C0R E0R
29. n Σ(n) S(n) 0 0 0 1 1 1 2 4 6 3 6 21 4 13 107
30. n Σ(n) S(n) 0 0 0 1 1 1 2 4 6 3 6 21 4 13 107 5 >= 4098 >= 47 176 870 6 > 3.5x1018267 > 7.4 x 1036534
40. [...] the busy beaver functions offer an entirely new approach to solving pure mathematics problems. Many open problems in mathematics could in theory, but not in practice, be solved in a systematic way given the value of S(n) for a sufficiently large n.
41. • Σ(n) grows faster asymptotically than any other computable function (hence, it’s non-computable). • Thus, it’s undecidable whether an arbitrary Turing machine is a Busy Beaver. • Busy Beavers has implications for computability theory, the halting problem and complexity theory. It’s often used, among other tools, to prove and disprove stuff.