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Busy Beaver, Universal Machines and the Wolfram Prize

Busy Beaver, Universal Machines and the Wolfram Prize. James Harland jah@cs.rmit.edu.au www.cs.rmit.edu.au/~jah School of CS & IT RMIT University. Introduction. Busy Beavers and the Zany Zoo Small universal Turing machines Wolfram machines The Wolfram prize US$25,000

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Busy Beaver, Universal Machines and the Wolfram Prize

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  1. Busy Beaver, Universal Machines and the Wolfram Prize James Harland jah@cs.rmit.edu.au www.cs.rmit.edu.au/~jah School of CS & IT RMIT University James Harland Busy Beaver, Universal machines and the Wolfram Prize

  2. Introduction • Busy Beavers and the Zany Zoo • Small universal Turing machines • Wolfram machines • The Wolfram prize US$25,000 • ‘Constructive’ Computability • Machine Learning possibilities James Harland Busy Beaver, Universal machines and the Wolfram Prize

  3. 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 James Harland Busy Beaver, Universal machines and the Wolfram Prize

  4. Busy Beaver Problem (Rado, 1962) • Turing machine • Two-way infinite tape • Only tape symbols are 0 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? James Harland Busy Beaver, Universal machines and the Wolfram Prize

  5. Busy Beaver Function • Grows faster than any computable function (!!) • Proof: • f computable ⇒ so is F(x)  = Σ0 ≤i≤x f(i) + i² • ⇒ k-state machine MF: x 1's → F(x)  1's and • x-state machine X: blank →x 1’s • M:X then MF then MF • M first writes x 1's • M then writes F(x) 1's • M then writes F(F(x)) 1's James Harland Busy Beaver, Universal machines and the Wolfram Prize

  6. Busy Beaver Function M has x + 2k states ⇒bb(n+2k) ≥ 1's output by M = x + F(x) + F(F(x))Now F(x) ≥ x² > x + 2k, and F(x) > F(y)  when x > y, and so F(F(x)) > F(x+2k) > f(x+2k)So bb(x+2k) ≥ x + F(x) + F(F(x)) > F(F(x)) > F(x+2k) > f(x+2k) ◊ James Harland Busy Beaver, Universal machines and the Wolfram Prize

  7. Known Values (n states, m symbols) James Harland Busy Beaver, Universal machines and the Wolfram Prize

  8. Search Method • Generate next machine with n states, m symbols • Reject obvious non-terminators • Store reasonable candidates • Test for termination • Attempt ‘sophisticated’ non-termination analysis • Give up on this machine • Go to 1 unless finished James Harland Busy Beaver, Universal machines and the Wolfram Prize

  9. Search Results * Wombats, Snakes, Monkeys, Kangaroos, … James Harland Busy Beaver, Universal machines and the Wolfram Prize

  10. Dual machines (S1, In, Out, Dir, S2) {1..N} x {0,1} x {0,1} x {l,r} x {1..N} (a, 0, 1, r, c) (In, S1, S2, Dir, Out) {0,1} x {1..N} x {1..N} x {l,r} x {0,1} (0, a, c, r, 1) • (naïve) search spaces are the same size • Unclear what other relationship exists James Harland Busy Beaver, Universal machines and the Wolfram Prize

  11. “Sophisticated” non-termination • Use execution history for non-termination • conjectures • Evaluate conjectures on a “hypothetical” • engine • Automate the search as much as possible James Harland Busy Beaver, Universal machines and the Wolfram Prize

  12. Example • 11{C}1 →  11{C}111 →  11{C}11111 … • Conjecture is 11{C} 1 (11)N →  11{C} 111(11)N • Start engine in 11{C} 1 (11)N • Terminate with success if we reach • 11{C} 111 (11)N(or 11{C} 11 (11)N1 or …) James Harland Busy Beaver, Universal machines and the Wolfram Prize

  13. Killer Kangaroos 16{D}0 → 118{D}0 → 142{D}0 (!!!) → 190{D}0 130{D}0does not occur … 1N{D}0 → 12N+6{D}0 or alternatively 1N{D}0 → (11)N111111{D}0 Then execute on engine as before James Harland Busy Beaver, Universal machines and the Wolfram Prize

  14. Engine Design • L {S}IN R → ???? • Run L {S}I R and look for “repeatable” parts • L {S}I R→ L O {S} R wild wombat • L {S}I R→ L’ O {S} R slithery snake • L {S}I R→ L’ O {S} R’ maniacal monkey • slithery snake → resilient reptile when |I| < |O| James Harland Busy Beaver, Universal machines and the Wolfram Prize

  15. Engine State • Around 4,000 lines of Ciao Prolog • Available on my web page • Includes all three heuristics • Some killer kangaroos still escape … • Analysis does not terminate for all machines (yet!) • At least one further heuristic needed James Harland Busy Beaver, Universal machines and the Wolfram Prize

  16. Addictive Adders 1111{C}11 1011{C}111 110{C}1111 111111{C}11 101111{C}111 11011{C}1111 1110{C}11111 11111111{C}11 James Harland Busy Beaver, Universal machines and the Wolfram Prize

  17. Addictive Adders Conjecture is 1N1111{C}11 → 1N111111{C}11 “Secondary” induction of the form 1N 0(11)K {C} 1M → 1N+10(11)K-1 {C} 1M+1 The forthcoming observant otter heuristic will evaluate this as 1N+K 0 {C} 1M+K James Harland Busy Beaver, Universal machines and the Wolfram Prize

  18. Small Universal Turing machines (Shannon 1956, Watanabe 1961) Minsky 7-state 4-symbol machine (1962) Machines known for the cases: (18,2), (9,3), (6,4), (4,6), (3,9), (2,18) Weakly universal machines known for: (6,2), (3,3), (2,4), (Neary & Woods, Cook) (2,5) (Wolfram) James Harland Busy Beaver, Universal machines and the Wolfram Prize

  19. Wolfram 2,3 machine James Harland Busy Beaver, Universal machines and the Wolfram Prize

  20. Wolfram 2,3 machine James Harland Busy Beaver, Universal machines and the Wolfram Prize

  21. Wolfram 2,3 Machine • Doesn’t terminate • Simulates termination by generation a particular set of tape symbols • Prove universal by encoding a known universal machine • Prove non-universal with more care! James Harland Busy Beaver, Universal machines and the Wolfram Prize

  22. Universal Machines Strong case: M on w ⇒ U on “M+w” M halts on w iff U halts on “M+w” Weak case: M on w ⇒ W on “M+w”, which never halts M halts on w ⇒ W on “M+w” prints T M doesn’t halt on w ⇒ W on “M+w” doesn’t print T James Harland Busy Beaver, Universal machines and the Wolfram Prize

  23. Blank vs. Arbitrary input • Equivalent for termination in general case • Not equivalent on size-restricted machines • M on w ⇒ M’M on blank where M’ prints w. • As w is arbitrary, M’ can be arbitrarily large James Harland Busy Beaver, Universal machines and the Wolfram Prize

  24. Universal? • For: Seems complex (??) • Against: • Search results suggest low complexity • 2,3 class is decidable (paper in Russian), so there is no strongly universal machine • Simple reduction would have been found by now  James Harland Busy Beaver, Universal machines and the Wolfram Prize

  25. Finite Decision problems Claim: Any finite decision problem is decidable (!!) 2N cases, and there is a TM for each case … We call this the bureaucratic TM James Harland Busy Beaver, Universal machines and the Wolfram Prize

  26. ‘Short’ programs Chaitin: An elegant program is the shortest one producing the required output. An algorithmic program is one which is shorter than the bureaucratic program for the same problem. So how do we generate algorithmic programs? James Harland Busy Beaver, Universal machines and the Wolfram Prize

  27. Machine Learning Possibilities Only about 2% of machine searched required sophisticated techniques (so 98% of cases were trivial) Can we use data mining or learning techniques to find a heuristic to reduce the search space? James Harland Busy Beaver, Universal machines and the Wolfram Prize

  28. Conclusions & Further Work • Plenty of interesting questions … • Algorithmic solution (?) for n x m <= 8 • Wolfram Prize question • Monster termination on other inputs • Placid platypus? • “Constructive” computability • “mine” cases for 3,4,5 for attempt on n = 6 James Harland Busy Beaver, Universal machines and the Wolfram Prize

  29. Any takers? … so who wants to play? James Harland Busy Beaver, Universal machines and the Wolfram Prize

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