Using an otter to catch a beaver to chase a
James Harland james.harland@rmit.edu.au
platypus
Otter -> Beaver -> Platypus 20 th November, 2009

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
11R a
01R b
10R c
00L 01L
11R h
Otter -> Beaver -> Platypus 20 th November, 2009

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 1 1
2 4
3 6
6
21
4 13
5 ≥ 4098
6 ≥ 4.64
× 10 1439
107
≥ 47,176,870 (??)
≥ 2.58
× 10 2879 (!!)
Otter -> Beaver -> Platypus 20 th November, 2009

What? n
5
2
2
3
2
4
2
3
3
2
6
4 m
2
5
4
3
3
2
2
2
4
6
2
3
Otter -> Beaver -> Platypus
#1’s
4
6
9
13
2,050
374,676,383
≥ 4098
≥ 1.7 × 10 352
≥ 4.64 × 10 1439
≥ 1.383 × 10 7036
≥ 3.743 × 10 6518
≥ 1.903 × 10 4933
#steps
6
21
38
107
3,932,964
119,112,334,170,342,540
≥ 47,176,870
≥ 1.9 × 10 704
≥ 2.58 × 10 2879
≥ 1.025 × 10 14072
≥ 5.254 × 10 13036
≥ 2.415 × 10 9866
20 th November, 2009

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 20 th November, 2009

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 20 th November, 2009

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 2symbol machine) needed to print k 1’s?
We call this the placid platypus problem [ pp(k) ]
Otter -> Beaver -> Platypus 20 th November, 2009

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, 100-110, 113, 114, 115, …
Need complete classification to be sure
Otter -> Beaver -> Platypus 20 th November, 2009

How?
Easy! :-) To find bb(n,m):
1.
Generate all n -state m -symbol machines
2.
Classifying them as terminating or non-terminating
3.
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 20 th November, 2009

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

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

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

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 20 th November, 2009

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 20 th November, 2009

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 20 th November, 2009

Observant Otter
Some surprises can be in store ....
1 6 {D}0 → 1 18 {D}0 → 1 42 {D}0 (!!!) → 1 90 {D}0
1 30 {D}0 does not occur …
(this is why two previous instances are needed)
Conjecture in this case is 1 N {D}0 → 1 2N+6 {D}0 or alternatively
1 N {D}0 → (11) N 111111{D}0
These are known as killer kangaroos ...
Otter -> Beaver -> Platypus 20 th November, 2009

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

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

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

Observant Otter
11 {C} (11) N 01 → … →
111 {C} 1(11) N-1 01 → … →
1111 {C} 1(11) N-2 01 → … →
11 (1) N {C} 01 → …
Otter -> Beaver -> Platypus 20 th November, 2009

Observant Otter
X {S} I N 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 20 th November, 2009

Observant Otter
1111{C}11
1011{C}111
110{C}1111
111111{C}11
101111{C}111
11011{C}1111
1110{C}11111
11111111{C}11
Sometimes this is not as simple as it may seem …
Otter -> Beaver -> Platypus 20 th November, 2009

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 20 th November, 2009

Blue Bilby n -state m -symbol machines, n x m ≤ 6 tnf n
2
3
2 m
2
2
3
Total
32
3,288
2,280
Terminated Going
8 24
831
616
2,457
1,664 free n
2
3
2 m
2
2
3
Total
128
Terminated Going
35
82,944 22,034
93
60,910
82,944 18,613 64,331
Otter -> Beaver -> Platypus 20 th November, 2009
Unclassified
0
0
0
Unclassified
0
0
0

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

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

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


 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 20 th November, 2009

2-D machines
Otter -> Beaver -> Platypus
0
1
1
2
0
20 th November, 2009

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 20 th November, 2009







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 20 th November, 2009

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’ 
Otter -> Beaver -> Platypus 20 th November, 2009

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 20 th November, 2009