busy beaver, universal machines and the wolfram prize

CS&IT Seminar 7/9/2007

James Harland Busy Beaver, Universal machines and the Wolfram Prize

Busy Beaver, Universal Machines and the Wolfram

Prize

James [email protected]

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•‘Constructive’ Computability•Machine Learning possibilities



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



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?



Busy Beaver FunctionGrows faster than any computable function (!!)Proof: f computable ⇒ so is F(x) = Σ 0 ≤i≤x f(i) + i² ⇒ k-state machine MF: x 1's → F(x) 1's and

x-state machine X: blank → x 1’sM: 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



Busy Beaver FunctionM 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) ◊



Known Values (n states, m symbols)

n m name bb(n,m) ff(n,m)

2 2 blue bilby 4 6

3 2 blue bilby 6 21

2 3 blue bilby 9 38

4 2 ebony elephant

13 107

2 4 ebony elephant

≥ 2,050 ≥ 3,932,964

3 3 white whale ≥ 95,524,079

≥ 4.3×1015

5 2 white whale ≥ 4098 ≥ 47,176,870

2 5 white whale ≥ 1.7×1011 ≥ 7.1×1021

6 2 demon duck of doom

≥ 1.29×10865

≥ 3×101730



Search Method1.Generate next machine with n

states, m symbols2.Reject obvious non-terminators3.Store reasonable candidates4.Test for termination5.Attempt ‘sophisticated’ non-

termination analysis6.Give up on this machine7.Go to 1 unless finished



Search Resultsn m Machin

esTermites

Iguanas

Ducks Wild* Unicorns

2 2 13 8 3 2 0 0

3 2 2,435 817 774 594 25 0

2 3 1,921 605 139 1062 110 8

4 2 350,440

134,048

79,328

126,735

5,154

21

2 4 (26,911)

(10,172)

(402)

(13,651)

(2,273)

(413)

* Wombats, Snakes, Monkeys, Kangaroos, …



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



“Sophisticated” non-termination

• Use execution history for non-termination conjectures• Evaluate conjectures on a “hypothetical” engine• Automate the search as much as possible



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 …)



Killer Kangaroos

16{D}0 → 118{D}0 → 142{D}0 (!!!) → 190{D}0 130{D}0 does not occur …

1N{D}0 → 12N+6{D}0 or alternatively

1N{D}0 → (11)N111111{D}0 Then execute on engine as before



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 monkeyslithery snake → resilient reptile when |I| < |O|



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



Addictive Adders1111{C}11

1011{C}111110{C}1111

111111{C}11101111{C}11111011{C}11111110{C}11111

11111111{C}11



Addictive AddersConjecture is 1N1111{C}11 → 1N111111{C}11

“Secondary” induction of the form1N 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



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)



Wolfram 2,3 machine



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!



Universal MachinesStrong 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 haltsM halts on w ⇒ W on “M+w” prints TM doesn’t halt on w ⇒ W on “M+w” doesn’t print T



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



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



Finite Decision problemsClaim: Any finite decision problem is decidable (!!)

2N cases, and there is a TM for each case … We call this the bureaucratic TM

Case Decision

1 Yes/no

2 Yes/no

… …

N Yes/no



‘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?



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?



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



Any takers?

… so who wants to play?

busy beaver, universal machines and the wolfram prize

Documents