1. Proving Termination and Non-Termination of Busy Beaver Machines
Using an
to catch a
to chase a
otter
beaver
platypus
James Harland
Otter -> Beaver -> Platypus
20th November, 2009

2. What? Turing Machines of a particular type: Deterministic
Symbols are only 0 (blank) and 1 (initially)
Only consider blank input
n states plus a halt state means size is n a b c h
11R
01R
00L
10R
01L
Otter -> Beaver -> Platypus
20th November, 2009

3. What?
What is the largest number of 1’s that can be printed by a terminating n-state machine? n
#1’s (productivity)
#steps 1 2 4 6 3 21 13
107 5
≥ 4098
≥ 47,176,870 (??)
≥ 4.64×101439
≥ 2.58× (!!)
Otter -> Beaver -> Platypus
20th November, 2009

4. What? Otter -> Beaver -> Platypus 20th November, 2009 n m #1’s
#steps 2 4 6 3 21 9 38 13
107
2,050
3,932,964
374,676,383
119,112,334,170,342,540 5
≥ 4098
≥ 47,176,870
≥ 1.7 × 10352
≥ 1.9 × 10704
≥ 4.64 ×
≥ 2.58 ×
≥ ×
≥ ×
≥ ×
≥ ×
≥ ×
≥ ×
Otter -> Beaver -> Platypus
20th November, 2009

5. What?
Busy Beaver function is non-computable; it grows faster than any computable function (!!)
Various mathematical bounds known
All surpassed in practice
Seems hopeless for n ≥ 7
Values for n ≤ 5 seem settled (but as yet unproven)
Otter -> Beaver -> Platypus
20th November, 2009

6. Why? Some enormous numbers here!
`… the poetry of logical ideas’ (Einstein)
Evidence for results is somewhat lacking
(“The remaining 250+ cases were checked by hand”)
Should have test data and evidence available
Should be able to automate entire test
`Nothing capable of proof should be accepted as true without it’ (Frege)
`Nothing capable of automation should be accepted as finished without it’
Otter -> Beaver -> Platypus
20th November, 2009

7. Why? An n-state (2-symbol) machine of productivity k shows that
bb(n,2) ≥ k
at most n states are needed to print k 1’s
What is the minimum number of states (for an n-state 2- symbol machine) needed to print k 1’s?
We call this the placid platypus problem [ pp(k) ]
Otter -> Beaver -> Platypus
20th November, 2009

8. Why? bb(n,2) = k means that pp(k) ≤ n and pp(k+1) ≥ n+1
pp(2) = pp(3) = pp(4) = 2
pp(5) = pp(6) = 3
pp(k) = 4 for k є {7,8,9,10,11,12,13}
pp(k) ≥ 5 for k ≥ 14
…???
There seem to be no 5-state 2-symbol machines with productivity 74, 85, 92-7, , 113, 114, 115, …
Need complete classification to be sure
Otter -> Beaver -> Platypus
20th November, 2009

9. How? Easy! :-) To find bb(n,m): Generate all n-state m-symbol machines
Classifying them as terminating or non-terminating
Find the terminating one which produces the largest number of 1’s (or non-zero characters)
Only problem is that there are lots of traps …
Otter -> Beaver -> Platypus
20th November, 2009

10. How? There is a finite but very large set of machines for each size …
Number of machines of size n is O(nn) [ > O(n!) > O(2n) !!]
`Free’ generation of machines quickly becomes impractical
`Free’ generation creates too many trivial machines
01R
10R
11R a b
01R c h
00L, 10R
Otter -> Beaver -> Platypus
20th November, 2009

11. How? First transition is fixed Run partial machine
Add a new transition when a `missing’ case happens
Add `halt’ transition last a b c h
11L
01R
10R
00L
{a}0 -> 1{b}0 -> {a}1 -> {a}01 -> 1{b}1 -> 10{c}0
Generates machines in tree normal form (tnf)
Otter -> Beaver -> Platypus
20th November, 2009

12. How Many? n m
tnf
free 2 32
128 3
3,288
82,944
2,280 4
527,952
100,663,296
447,664
24,711,336 ?? 5
110,537,360
176,293,040
Otter -> Beaver -> Platypus
20th November, 2009

13. How? Some non-termination cases are easy
perennial pigeon: repeats a particular configuration
phlegmatic phoenix: tape returns to blank
road runner: “straight to the end of the tape”
meandering meerkat: terminal transition cannot be reached
Hence we check for these first
Otter -> Beaver -> Platypus
20th November, 2009

14. How? Otherwise .. (ie all the easy cases have been checked):
Run the machine for a given number of steps
If it halts, done.
Examine the execution trace for non-termination conjectures
Attempt to show that the hypothesis pattern repeats infinitely
Execute machine for a large number of steps
Give up!
Otter -> Beaver -> Platypus
20th November, 2009

15. How?
11{C}1 → 11{C}111 → 11{C}11111 …
Conjecture 11{C} 1 (11)N →  11{C} 1(11)N+1
Start engine in 11{C} 1 (11)N
Terminate with success if we reach 11{C} 1 (11)N+1
(or 11{C} 1 (11)N+2 or …)
Halt with failure after a certain number of steps
Main trick is how to deal with (11)N case where N is a variable
Otter -> Beaver -> Platypus
20th November, 2009

16. Observant Otter
Some surprises can be in store {D}0 → 118{D}0 → 142{D}0 (!!!) → 190{D}0 130{D}0 does not occur … (this is why two previous instances are needed) Conjecture in this case is 1N{D}0 → 12N+6{D}0 or alternatively 1N{D}0 → (11)N111111{D}0 These are known as killer kangaroos ...
Otter -> Beaver -> Platypus
20th November, 2009

17. How? Three main cases: Wild Wombat Stretching Stork Observant Otter
(soon also the Ossified Ocelet)
slithering snake
resilient reptile
maniacal monkey
Otter -> Beaver -> Platypus
20th November, 2009

18. Wild Wombat X {S} IN Y → ?? {S} I → J {S} Pointer remains within I
First exits I to the right
State on exit is S
``Context-free''
X {S} IN Y → X JN {S} Y
X IN {S} Y → X {S} JN Y
11 {C} (11)N 011 → 11 (00)N {C} if
{C}11 → 0{D}1 →{B}01 → 0{B}1 → 00 {C}
Otter -> Beaver -> Platypus
20th November, 2009

19. Stretching Stork X {S} IN Y → ??
If the wombat doesn't apply and N > 0
Condition N > 0 required to stop this being infinitely applicable
X {S} IN Y → X {S} I IN-1 Y
11 {C} (11)N 011 → 11 {C} 11 (11)N-1 011
Otter -> Beaver -> Platypus
20th November, 2009

20. Oi! There is a pattern here!
Observant Otter
11 {C} (11)N 01 → … →
111 {C} 1(11)N → … →
1111 {C} 1(11)N-2 01 → … →
Oi! There is a pattern here!
11 (1)N {C} 01 → …
Otter -> Beaver -> Platypus
20th November, 2009

21. Observant Otter X {S} IN Y → ??
Looks through execution history to find a similar pattern
Requires two previous instances before a match is made
Currently only allows decrement of 1 
Calculates new state from patterns
Ossified Ocelet will store these patterns for potential future use
This technique used by Wood in 2008 to evaluate monsters ...
Otter -> Beaver -> Platypus
20th November, 2009

22. Observant Otter Sometimes this is not as simple as it may seem …
1111{C} {C} {C} {C} {C} {C} {C} {C}11
Otter -> Beaver -> Platypus
20th November, 2009

23. Implementation Around 8,000 lines of Ciao Prolog
Needs a thorough clean up; necessary code is around 40% of this size
Versions available at my website
Data available as well
Documentation is a little lacking …
Intention is that other researchers can use data and verify results (this is empirical science!)
Otter -> Beaver -> Platypus
20th November, 2009

24. Blue Bilby n-state m-symbol machines, n x m ≤ 6 tnf free n m Total
Terminated
Going
Unclassified 2 32 8 24 3
3,288
831
2,457
2,280
616
1,664
tnf n m
Total
Terminated
Going
Unclassified 2
128 35 93 3
82,944
22,034
60,910
18,613
64,331
free
Otter -> Beaver -> Platypus
20th November, 2009

25. Ebony Elephant n-state m-symbol machines, n x m = 8 n m Total
Terminated
Going
Unclassified 4 2
527,952
118,348
408,876
728
447,664 ??
1's 1 2 3 4 5 6 7 8 9 10 11 12 13 #
3836
23161
37995
31023
15131
5263
1487
357 74
Otter -> Beaver -> Platypus
20th November, 2009

26. White Whale n-state m-symbol machines, n x m = 9 or 10 n m Total 3
24,711,336 5 2
110,537,360
176,293,040 (??)
Need to improve the technology
before taking this on ...
Otter -> Beaver -> Platypus
20th November, 2009

27. Demon Duck of Doom n m Total 6 2 ?? 4 3
Otter -> Beaver -> Platypus
20th November, 2009

28. Universal Turing machines
Quest for the smallest universal TM goes on …
Involves searching similar spaces
Alain Colmerauer (KR’08 talk)
U on code(M)code(w) simulates M on w
Let M = U
U on code(U)code(w) simulates U on w
Let w = blank (and assume code(blank) = blank)
U on code(U) simulates U on blank
Hence pseudo-universality test: M is pseudo-universal if M on code(M) simulates M on blank
Otter -> Beaver -> Platypus
20th November, 2009

29. 2-D machines 2 1 1 Otter -> Beaver -> Platypus2 1 1
Otter -> Beaver -> Platypus
20th November, 2009

30. 2-D machines Can interpret numbers as images …
Have generated some of the smaller cases here
Very basic implementation of emulation
Potentially very interesting loop behaviour here
Strict generalisation of one-dimensional case
No existing work that I have been able to find …
Otter -> Beaver -> Platypus
20th November, 2009

31. Learning?
Ebony Elephant case suggests that `most' machines are boring …
Only 2,667 are either high productivity or unclassified
At most 100,000 of the non-terminators are difficult
The remaining 80% are unexceptional …
Can we find a criterion to identify (at least a substantial fraction) of the unexceptional machines?
Need to identify possible features of machines ...
Otter -> Beaver -> Platypus
20th November, 2009

32. Conclusions & Further Work
Otter still needs some refinements
needs to `dovetail’ with the rest of the engine
some small number of machines resistant to otter (so far)
Some parameters (eg maximum search steps) need to be tweaked
Connections to universal Turing machines and 2-D machines need investigation
New (journal) paper will be out `soon’ 
Ebony elephant is close to capture
White whale should be alert and alarmed
Otter -> Beaver -> Platypus
20th November, 2009

33. Conclusions & Further Work
Ebony elephant is close to capture
White whale should be alert and alarmed (``Call me Ishmael’’ )
Demon Duck of Doom should be alert but not alarmed yet
Machine learning techniques may identify criteria which can reduce search space
Duality between n-state 2-symbol and 2-state n-symbol cases
Lock away the beaver and play with the platypus 
Otter -> Beaver -> Platypus
20th November, 2009