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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures The Busy Beaver, the Placid Platypus and Other Crazy Creatures James Harland jah@cs.rmit.edu.au School of CS & IT RMIT University

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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 1’s that can be printed by a terminating n-state machine?

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures Known Values nbb(n)ff(n) 111 246 3621 413107 5≥ 4098≥ 47,176,870 6≥ 1.29×10 865 ≥ 3×10 1730 7!!!!!!!!????

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures Known Beaver Machines

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures Known Beaver Machines This can be represented in around 60 bits … 10 865 takes about 2,800 bits …

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures Monsters are rare … prod5678910111213 machines73,61713,02919814757913652 Of 117,440,512 4-state machines: 89,207 irredundant and terminate with prod ≥ 5 only 2,561 machines with prod > bb(3) loops abound!

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 5-state monsters prodmaxtransitions 409812,28847,176,870 40986,14411,798,826 40976,14323,554,764 40976,14311,798,796 40966,14311,804,910 40966,14311,804,896 14711,4742,358,064

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures Equivalence

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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 << bb(n) Restrictions on tape (1-sided, bounded, …) Relationship to 3n+1 problem ….

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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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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CATS’06 18/1/2006James Harland The Busy Beaver, the Placid Platypus and Other Crazy Creatures 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)

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