1. The Busy Beaver, the Placid Platypus and the Universal Unicorn
James Harland
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

2. Busy beaver problem and similar problems Machine execution techniques
Overview
Busy beaver problem and similar problems
Machine execution techniques
Searching large numbers of machines
Universal Turing machines
Connections with other areas
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

3. Busy beaver problem and similar problems Machine execution techniques
Overview
Busy beaver problem and similar problems
Machine execution techniques
Searching large numbers of machines
Universal Turing machines
Connections with other areas
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

4. ‘Organic’ theoretical computer science
Overview
‘Organic’ theoretical computer science
Exploration of theoretical concepts by software tools
Mix of empirical and theoretical methods
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

5. a b c h 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
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

6. 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 (??) 6
≥ 3.51×1018,276
≥ 7.41×1036, (!!)
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

7. HERE BE DRAGONS! What? 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
3.51×
≥ 7.41×
≥ ×
≥ ×
≥ ×
≥ ×
≥ ×
≥ ×
HERE BE DRAGONS!
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

8. Various mathematical bounds known All surpassed in practice
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)
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

9. Some enormous numbers here! `… the poetry of logical ideas’ (Einstein)
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’
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

10. An n-state (2-symbol) machine of productivity k shows that bb(n,2) ≥ k
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) ]
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

11. bb(n,2) = k means that pp(k) ≤ n and pp(k+1) ≥ n+1
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
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

12. Marxen and Buntrock (1990): macro machines
Executing Machines
Marxen and Buntrock (1990): macro machines
Treat a sequence of k characters as a single ‘symbol’
Avoids repetition of previous sequences
Significant speed up
Need to choose k in advance
Holkner (2004): loop prediction
{b}(01) > (11)100 {b}1
State
Input
InDir
Output
OutDir
NewState b 01 l 11 r
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

13. 20,000+ steps in 1 Executing Machines What do the monsters do?
5x2 state champion (terminates in 47,176,870 steps)
Step
Configuration
12393
001111{b}(001)66100
12480
001(111)6{b}(001)63100
12657
001(111)11{b}(001)60100 …
33120
001(111)106{b}(001)3100
101{h}(001)
20,000+ steps in 1
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

14. 5x2 state champion takes 11 otter steps
Executing Machines
5x2 state champion takes 11 otter steps
Otter
Steps Predicted 1
267 2
2,235 3
6,840 4
20,463 5
62,895 6
176,040 7
500,271 8
1,387,739 9
3,878,739 10
10,830,672 11
30,144,672
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

15. Executing Machines
Macro machines + loop prediction + observant otter can cope with all known monsters (!!)
Size
Ones
Hops
Otters
Otter Steps
2x4 90
7195 2
64%
2050 8
99%
3x3
17338 17
> 99%
(16 digits)
(18 digits) 60
5x2
4098
47,176,870 11
11,798,796 10
4097
23,554,764 74
2x5
(31 digits)
(62 digits)
168
(106 digits)
(212 digits)
593
(353 digits)
(705 digits)
1993
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

16. Executing Machines Size Ones Hops Otters Otter Steps 6x2
(18268 digits)
(36535 digits)
30345
> 99%
(10567 digits)
(21133 digits)
99697
(1440 digits)
(2880 digits)
8167
(882 digits)
(1763 digits)
2211
4x3
(7037 digits)
(14073 digits)
25255
(6035 digits)
(12069 digits)
34262
(4932 digits)
(9864 digits)
57368
3x4
(6519 digits)
(13037 digits)
13667
(2373 digits)
(4745 digits)
13465
(2356 digits)
(4711 digits)
107045
2x6
(4934 digits)
(9867 digits)
16394
49142
(822 digits)
(1644 digits)
2733
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

17. HERE BE DRAGONS! Executing Machines Size Ones Hops Otters Otter Steps
(18268 digits)
(36535 digits)
30345
> 99%
(10567 digits)
(21133 digits)
99697
(1440 digits)
(2880 digits)
8167
(882 digits)
(1763 digits)
2211
4x3
(7037 digits)
(14073 digits)
25255
(6035 digits)
(12069 digits)
34262
(4932 digits)
(9864 digits)
57368
3x4
(6519 digits)
(13037 digits)
13667
(2373 digits)
(4745 digits)
13465
(2356 digits)
(4711 digits)
107045
2x6
(4934 digits)
(9867 digits)
16394
49142
(822 digits)
(1644 digits)
2733
HERE BE DRAGONS!
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

18. HERE BE HOBBITS! Executing Machines Size Ones Hops Otters Otter Steps
(18268 digits)
(36535 digits)
30345
> 99%
(10567 digits)
(21133 digits)
99697
(1440 digits)
(2880 digits)
8167
(882 digits)
(1763 digits)
2211
4x3
(7037 digits)
(14073 digits)
25255
(6035 digits)
(12069 digits)
34262
(4932 digits)
(9864 digits)
57368
3x4
(6519 digits)
(13037 digits)
13667
(2373 digits)
(4745 digits)
13465
(2356 digits)
(4711 digits)
107045
2x6
(4934 digits)
(9867 digits)
16394
49142
(822 digits)
(1644 digits)
2733
HERE BE HOBBITS!
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

19. Aberrent Albatrosses The observant otter is not a panacea …
Some machines do not produce “neat, compressible” patterns …
3x3 case terminates in 2,315,619 steps with 31 non-zeroes
2x5 case terminates in 26,375,397,569,930 steps with 143 non-zeroes
1004: {b}134334
5000: (011)2101(111)311{a}
10001: {a}
???
Beaver, Platypus, Unicorn, …
Monash June 2nd, 2014

20. Some non-termination cases are easy
Analysing Machines
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
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

21. How? Otherwise .. (ie all the easy cases have been checked):
Run the machine for a given number of steps
If it halts, done.
Check for non-terminating pattern
(Verify non-terminating pattern)
Execute machine for a larger number of steps
Eventually give up!
Beaver, Platypus, Unicorn, …
Monash June 2nd, 2014

22. Observant Otter Some surprises can be in store ....
16{D}0 → 118{D}0 → 142{D}0 (!!!) → 190{D}0
130{D}0 does not occur …
So always look for three instances of a pattern
Conjecture in this case is 1N{D}0 → 12N+6{D}0
or alternatively
1N{D}0 → (11)N111111{D}0
Clearly there is exponential growth here …
Beaver, Platypus, Unicorn, …
Monash June 2nd, 2014

23. Aberrent Albatrosses
These occur in non-termination cases as well … 36: {a} : {a} : {a} : {a} : {a} : {a} : {a} : {a} : {a} ??
Beaver, Platypus, Unicorn, …
Monash June 2nd, 2014

24. Aberrent Albatrosses
These occur in non-termination cases as well … 36: {a} : {a} : {a} : {a} : {a} : {a} : {a} : {a} : {a}
Beaver, Platypus, Unicorn, …
Monash June 2nd, 2014

25. Searching for Machines
To find bb(n,m):
Generate all n-state m-symbol machines
Classify 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 …
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

26. HERE BE DRAGONS! Executing Machines Size Ones Hops Otters Otter Steps
(18268 digits)
(36535 digits)
30345
> 99%
(10567 digits)
(21133 digits)
99697
(1440 digits)
(2880 digits)
8167
(882 digits)
(1763 digits)
2211
4x3
(7037 digits)
(14073 digits)
25255
(6035 digits)
(12069 digits)
34262
(4932 digits)
(9864 digits)
57368
3x4
(6519 digits)
(13037 digits)
13667
(2373 digits)
(4745 digits)
13465
(2356 digits)
(4711 digits)
107045
2x6
(4934 digits)
(9867 digits)
16394
49142
(822 digits)
(1644 digits)
2733
HERE BE DRAGONS!
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

27. Searching for Machines
There is a finite but very large set of machines for each size …
#machines of size n is O(nn) [ > O(n!) > O(2n) !!]
‘Free’ generation of machines quickly becomes cumbersome ..
‘Free’ generation creates too many trivial machines a b c h
11R
01R
00L, 10R
10R
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

28. Searching through Machines
First transition is fixed
Run partial machine
Add a new transition when a ‘missing’ case happens
Add ‘halt’ transition when appropriate
Only one transition left
At least n different states for an n-state machine 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)
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

29. Searching through Machines
tnf
free 2 36 64 3
3,522
55,296
2,765
41,472 4
507,107 ??
501,232
25,955,301 5
99,174,375
≥ 115,000,000
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

30. Around 5,000 lines of SWI Prolog Versions available at my website
Implementation
Around 5,000 lines of SWI Prolog
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!)
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

31. Blue Bilby n-state m-symbol machines, n x m ≤ 6
Total
Terminated
Going
Unclassified 2 36 14 22 3
3,522
1,188
2,334
2,765
839
1,926
The free and arbitrary classes have also been classified for these cases
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

32. Ebony Elephant n-state m-symbol machines, n x m = 8 n m Total
Terminated
Going
Unclassified 4 2
507,107
150,897
356,210
501,232
168,674
331,960
598
1's 1 2 3 4 5 6 7 8 9 10 11 12 13 #
3836
23161
37995
31023
15131
5263
1487
357 74
Beaver, Platypus, Unicorn, Zoo …
Monash 29th June,2011

33. White Whale n-state m-symbol machines, n x m = 9 or 10 n m Total 3
25,955,301 5 2
99,174,375
≥ 115,000,000
Need to improve the technology
before taking this on ...
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

34. Demon Duck of Doom n-state m-symbol machines, n x m = 12 (!??!!) n m
Total 6 2 ?? 4 3
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

35. Universal Turing machines
Quest for the smallest universal TM goes on …
Involves searching similar (but larger) 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
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

36. Universal Turing machines
What exactly is the definition of a universal Turing machine?
How can such definitions be used to identify universal machines “in the wild”?
What constraints are there on the coding function?
Does a UTM have to terminate?
Must a UTM terminate on code(M)code(w) exactly when M terminates on w?
What is an appropriate “architecture” for a UTM? (code(M)code(w) vs code(w)code(M) )
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

37. 2-D machines 2 1 1
Monash June 2nd, 2014
Beaver, Platypus, Unicorn …

38. 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
Strict generalisation of one-dimensional case
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

39. 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 ...
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

40. Finite Decidability Anomaly
Termination for TMs on blank input is undecidable
Calculating bb(n) involves “only” a finite # machines …
So there is an algorithm for computing bb(n) for each n (!!)
Claim: Any decision problem over any finite set is decidable
Proof Sketch: For a set of size n, there are only 2n possible decisions. Each such decision can be implemented by a Turing machine, ie. a “table” machine that outputs the decision value for each element i є {1 … n}. This is only a “table”, not an “algorithm”, but it is a Turing machine … QED
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

41. Finite Decidability Anomaly
Termination for TMs on blank input is undecidable
Calculating bb(n) involves “only” a finite # machines …
So there is an algorithm for computing bb(n) for each n (!!)
Claim: Any decision problem over any finite set is decidable
Proof Sketch: For a set of size n, there are only 2n possible decisions. Each such decision can be implemented by a Turing machine, ie. a “table” machine that outputs the decision value for each element i є {1 … n}. This is only a “table”, not an “algorithm”, but it is a Turing machine … QED
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

42. Finite Decidability Anomaly
Note: A decision procedure for a decidable problem over an infinite set is much shorter than the set itself
So for a finite set, we should require that the Turing machine that decides it should be quantifiably shorter than the “table” which specifies the decision …
An “algorithm” should be predictive, ie should work for more values than those specified in the table …
(more thought needed!)
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

43. Conclusions & Further Work
Ebony elephant is close to capture
White whale should be alert and alarmed (“Call me Ishmael’’ )
Demon Duck of Doom … [watch this space]
Machine learning techniques may identify criteria which can reduce search space
Duality between n-state 2-symbol and 2-state n-symbol cases
Otter needs to be “remembered”
Compress machines rather than fix macro size in advance
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014

44. Conclusions & Further Work
Weakening of conjectures for aberrent albatrosses
Proofs desired for
3 instances for otter is sufficient
Calculation of #steps (2nd order difference equations)
Champion machines are always exhaustive
Quadruple vs Quintuple machines
Better “pruning” of search space
Beaver, Platypus, Unicorn …
Monash June 2nd, 2014