The Busy Beaver Game - Speaker Deck

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The Busy Beaver Game Christian Stigen @ Roxar AS 2016-04-27

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Tibor Radó Bell Labs, 1962

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Recap Turing Machines ● Tape ● Current State ● Current Symbol ● Next Symbol ● Next State ● Next Movement

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State Symbol 0 Symbol 1 A B1R D0L B C1R F0R C C1L A1L D E0L Z1L E A1L B0R F C0R E0R

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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)

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n Σ(n)

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n Σ(n) 0 0

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n Σ(n) 0 0 1 1

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n Σ(n) 0 0 1 1 2

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(4(n+1))2n

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(4(n+1))2n Current state Current symbol New symbol x New direction New state + Halt

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● 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

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Symbol 0 Symbol 1 State A State B

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Symbol 0 Symbol 1 State A B1L State B Tape 0 1 0 0

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Symbol 0 Symbol 1 State A B1L State B A1R Tape 0 1 0 0 1 1 0 0

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Symbol 0 Symbol 1 State A B1L B1R State B A1R Tape 0 1 0 0 1 1 0 0 1 1 0 0

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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

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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

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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

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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

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Symbol 0 Symbol 1 State A B1R A1L State B A1R Z0L

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n Σ(n) 0 0 1 1 2 4

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n Σ(n) S(n) 0 0 0 1 1 1 2 4 6

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A B1R D0L B C1R F0R C C1L A1L D E0L Z1L E A1L B0R F C0R E0R

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n Σ(n) S(n) 0 0 0 1 1 1 2 4 6 3 6 21 4 13 107

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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

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[...] 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.

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● 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.

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