The Busy Beaver, the Placid Platypus and Other Crazy Creatures

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The Busy Beaver, the Placid Platypus and Other
Crazy Creatures
James Harland jah_at_cs.rmit.edu.au School of CS
IT RMIT University
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Introduction

• New twist on an old problem
• More questions than answers!
• Innocuous class of machines generate huge numbers
• Involves termination analysis and constraint
  programming
• Frustrating to the point of obsession

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Busy Beaver Turing Machines

• Two-way infinite tape
• Only tape symbols are B and 1
• Deterministic
• Blank on input
• Question What is the largest number of 1s that
  can be printed by a terminating n-state machine?

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Known Values
5
Known Beaver Machines
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Known Beaver Machines
This can be represented in around 60 bits
10865 takes about 2,800 bits
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Busy Beaver function

• Non-computable
• Grows faster than any computable function
• Various mathematical bounds known
• Seems hopeless for n 7
• Values for n 5 seem settled
• 3, 4, 5, 6 symbol versions are popular

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Monsters are rare

• Of 117,440,512 4-state machines
• 89,207 irredundant and terminate with prod 5
• only 2,561 machines with prod gt bb(3)
• loops abound!

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5-state monsters
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Platypus machines

• An n-state machine of productivity m shows
• bb(n)   m
• at most n states are needed to print m 1's
• Question what is the minimum number of states
  needed to print m 1's? We call this the placid
  platypus or pp(m)

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Known Platypus values

• 1-83 except 46, 48, 50, 74, 75, 77, 80, 82
• 87,88,89,91,99,112,
• ,1471, (..?...), 4096, 4097, 4098

Question Is it true that there is a 5-state
machine which prints m 1's for each bb(4) m
bb(5)? This is certainly false for bb(5) to
bb(6).
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Platypus questions

• Distribution of platypus machines for n 5
• Largest interval m1,m2 of existence?
• Largest interval m1,m2 of non-existence?
• Smallest m s.t. pp(m) 6?
• Distribution of platypus machines for n 6
• Smallest m s.t. pp(m) 7? (!!!)
• .

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Equivalence
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More questions

• Productivity for machines which are
• contiguous (always of the form B1 B)
• eager (output is only 1, never B)
• monotonic (no 1-to-B)
• Maximum productivity with 10,000 steps
• Restrictions for productivity ltlt bb(n)
• Restrictions on tape (1-sided, bounded, )
• Relationship to 3n1 problem
• .

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Other Crazy Creatures

• loops with maximum productivity per cycle
• frenetic phoenix (blank to blank)
• pseudo-random generator machines?
• maximum number of regions on tape
• does this function occur naturally?

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Conclusions Further Work

• Plenty of interesting questions
• Complete analysis of n 5 case
• Publish database for n 3,4,5
• Better evaluator needed
• Inductive prover for non-termination
• mine cases for 3,4,5 for attempt on n 6 (aka
  quest for the demon duck of doom)