CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Busy Beaver, Universal Machines and the Wolfram Prize James Harland School of CS & IT RMIT University

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Introduction Busy Beavers and the Zany Zoo Small universal Turing machines Wolfram machines The Wolfram prize US$25,000 ‘Constructive’ Computability Machine Learning possibilities

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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?

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Busy Beaver Function Grows faster than any computable function (!!) Proof: f computable ⇒ so is F(x) = Σ 0 ≤i≤x f(i) + i ² ⇒ 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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) ≥ x ² > 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) ◊

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Known Values (n states, m symbols) nmnamebb(n,m)ff(n,m) 22blue bilby blue bilby938 42ebony elephant ebony elephant≥ 2,050≥ 3,932,964 33white whale≥ 95,524,079≥ 4.3× white whale≥ 4098≥ 47,176,870 25white whale≥ 1.7×10 11 ≥ 7.1× demon duck of doom≥ 1.29× ≥ 3×

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Search Method 1.Generate next machine with n states, m symbols 2.Reject obvious non-terminators 3.Store reasonable candidates 4.Test for termination 5.Attempt ‘sophisticated’ non-termination analysis 6.Give up on this machine 7.Go to 1 unless finished

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Search Results nmMachinesTermitesIguanasDucksWild*Unicorns , , ,440134,04879,328126,7355, (26,911)(10,172)(402)(13,651)(2,273)(413) * Wombats, Snakes, Monkeys, Kangaroos, …

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize “Sophisticated” non-termination Use execution history for non-termination conjectures Evaluate conjectures on a “hypothetical” engine Automate the search as much as possible

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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) N 1 or …)

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Killer Kangaroos 1 6 {D}0 → 1 18 {D}0 → 1 42 {D}0 (!!!) → 1 90 {D} {D}0 does not occur … 1 N {D}0 → 1 2N+6 {D}0 or alternatively 1 N {D}0 → (11) N {D}0 Then execute on engine as before

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Engine Design L {S}I N 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|

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Addictive Adders 1111{C} {C} {C} {C} {C} {C} {C} {C}11

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Addictive Adders Conjecture is 1 N 1111{C}11 → 1 N {C}11 “Secondary” induction of the form 1 N 0(11) K {C} 1 M → 1 N+1 0(11) K-1 {C} 1 M+1 The forthcoming observant otter heuristic will evaluate this as 1 N+K 0 {C} 1 M+K

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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)

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Wolfram 2,3 machine

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Wolfram 2,3 machine

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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!

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Finite Decision problems Claim: Any finite decision problem is decidable (!!) 2 N cases, and there is a TM for each case … We call this the bureaucratic TM CaseDecision 1Yes/no 2 …… N

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize ‘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?

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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?

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize 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

CS&IT Seminar 7/9/2007James Harland Busy Beaver, Universal machines and the Wolfram Prize Any takers? … so who wants to play?