Download Searching for Machines

The Busy Beaver, the Placid
Platypus and the Universal
Unicorn
James Harland
[email protected]
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
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
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
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
00L
Beaver, Platypus, Unicorn …
10R
01L
c
Monash
11R
h
June 2nd, 2014
What?
What is the largest number of 1’s that can
be printed by a terminating n-state
machine?
n
1
2
#1’s (productivity)
1
4
#steps
1
6
3
4
6
13
21
107
5
6
≥ 4098
≥ 3.51×1018,276
≥ 47,176,870 (??)
≥ 7.41×1036,534
Beaver, Platypus, Unicorn …
Monash
(!!)
June 2nd, 2014
What?
n
m
#1’s
#steps
2
2
4
6
3
2
6
21
2
3
9
38
4
2
13
107
2
4
2,050
3,932,964
3
3
374,676,383
119,112,334,170,342,540
5
2
≥ 4098
≥ 47,176,870
2
5
≥ 1.7 × 10352
≥ 1.9 × 10704
6
2
3.51×1018276
≥ 7.41×1036534
3
≥ 1.383 × 107036
≥ 1.025 × 1014072
4
≥ 3.743 × 106518
≥ 5.254 × 1013036
6
≥ 1.903 × 104933
≥ 2.415 × 109866
4
3
2
HERE BE DRAGONS!
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
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
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 nstate 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
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
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
State
Input
InDir
Output
OutDir
NewState
b
01
l
11
r
b
{b}(01)1001 -> (11)100 {b}1
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
Executing Machines
What do the monsters do?
5x2 state champion (terminates in 47,176,870 steps)
Step
12393
Configuration
001111{b}(001)66100
12480
12657
…
001(111)6{b}(001)63100
001(111)11{b}(001)60100
…
33120
…
001(111)106{b}(001)3100
…
47176870
101{h}(001)4095100
Beaver, Platypus, Unicorn …
Monash
20,000+ steps
in 1
June 2nd, 2014
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
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%
2x4
2050
3932964
8
99%
3x3
17338
127529109
17
> 99%
3x3
95524079
(16 digits)
17
> 99%
3x3
374676383
(18 digits)
60
> 99%
5x2
4098
47,176,870
11
> 99%
5x2
4098
11,798,796
10
> 99%
5x2
4097
23,554,764
74
> 99%
2x5
(31 digits)
(62 digits)
168
> 99%
2x5
(106 digits)
(212 digits)
593
> 99%
2x5
(353 digits)
Beaver, Platypus, Unicorn …
(705 digits)
1993
Monash
> 99%
June 2nd, 2014
Executing Machines
Size
Ones
Hops
Otters
Otter Steps
6x2
(18268 digits)
(36535 digits)
30345
> 99%
6x2
(10567 digits)
(21133 digits)
99697
> 99%
6x2
(1440 digits)
(2880 digits)
8167
> 99%
6x2
(882 digits)
(1763 digits)
2211
> 99%
4x3
(7037 digits)
(14073 digits)
25255
> 99%
4x3
(6035 digits)
(12069 digits)
34262
> 99%
4x3
(4932 digits)
(9864 digits)
57368
> 99%
3x4
(6519 digits)
(13037 digits)
13667
> 99%
3x4
(2373 digits)
(4745 digits)
13465
> 99%
3x4
(2356 digits)
(4711 digits)
107045
> 99%
2x6
(4934 digits)
(9867 digits)
16394
> 99%
2x6
(4932 digits)
(9864 digits)
49142
> 99%
2x6
(822 digits)
(1644 digits)
Beaver, Platypus, Unicorn …
2733
Monash
> 99%
June 2nd, 2014
Executing Machines
Size
Ones
Hops
Otters
Otter Steps
6x2
(18268 digits)
(36535 digits)
30345
> 99%
6x2
(10567 digits)
(21133 digits)
99697
> 99%
6x2
(1440 digits)
(2880 digits)
8167
> 99%
6x2
(882 digits)
(1763 digits)
2211
> 99%
4x3
(7037 digits)
(14073 digits)
25255
> 99%
4x3
(6035 digits)
(12069 digits)
34262
> 99%
4x3
(4932 digits)
(9864 digits)
57368
> 99%
3x4
(6519 digits)
(13037 digits)
13667
> 99%
3x4
(2373 digits)
(4745 digits)
13465
> 99%
3x4
(2356 digits)
(4711 digits)
107045
> 99%
2x6
(4934 digits)
(9867 digits)
16394
> 99%
2x6
(4932 digits)
(9864 digits)
49142
> 99%
HERE BE DRAGONS!
2x6
(822 digits)
(1644 digits)
Beaver, Platypus, Unicorn …
2733
Monash
> 99%
June 2nd, 2014
Executing Machines
Size
Ones
Hops
Otters
Otter Steps
6x2
(18268 digits)
(36535 digits)
30345
> 99%
6x2
(10567 digits)
(21133 digits)
99697
> 99%
6x2
(1440 digits)
(2880 digits)
8167
> 99%
6x2
(882 digits)
(1763 digits)
2211
> 99%
4x3
(7037 digits)
(14073 digits)
25255
> 99%
4x3
(6035 digits)
(12069 digits)
34262
> 99%
4x3
(4932 digits)
(9864 digits)
57368
> 99%
3x4
(6519 digits)
(13037 digits)
13667
> 99%
3x4
(2373 digits)
(4745 digits)
13465
> 99%
3x4
(2356 digits)
(4711 digits)
107045
> 99%
2x6
(4934 digits)
(9867 digits)
16394
> 99%
2x6
(4932 digits)
(9864 digits)
49142
> 99%
HERE BE HOBBITS!
2x6
(822 digits)
(1644 digits)
Beaver, Platypus, Unicorn …
2733
Monash
> 99%
June 2nd, 2014
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:
012010150130170{b}134334
5000: (011)2101(111)311{a}0433135441100
10001: 01401013010160{a}331374300
???
Beaver, Platypus, Unicorn, …
Monash
June 2nd, 2014
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
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
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
Aberrent Albatrosses
These occur in non-termination cases as well …
36:
{a}0333
54:
{a}030323
74:
{a}032333
88:
{a}033333
110:
{a}03030323
??
136: {a}03232333
150: {a}03332333
168: {a}03303333
204: {a}0332330323
Beaver, Platypus, Unicorn, …
Monash
June 2nd, 2014
Aberrent Albatrosses
These occur in non-termination cases as well …
36:
{a}0333
54:
{a}030323
74:
{a}032333
88:
{a}033333
110:
{a}03030323
136: {a}03232333
150: {a}03332333
168: {a}03303333
204: {a}0332330323
Beaver, Platypus, Unicorn, …
Monash
June 2nd, 2014
Searching for Machines
To find bb(n,m):
1. Generate all n-state m-symbol machines
2. Classify 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 …
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
Executing Machines
Size
Ones
Hops
Otters
Otter Steps
6x2
(18268 digits)
(36535 digits)
30345
> 99%
6x2
(10567 digits)
(21133 digits)
99697
> 99%
6x2
(1440 digits)
(2880 digits)
8167
> 99%
6x2
(882 digits)
(1763 digits)
2211
> 99%
4x3
(7037 digits)
(14073 digits)
25255
> 99%
4x3
(6035 digits)
(12069 digits)
34262
> 99%
4x3
(4932 digits)
(9864 digits)
57368
> 99%
3x4
(6519 digits)
(13037 digits)
13667
> 99%
3x4
(2373 digits)
(4745 digits)
13465
> 99%
3x4
(2356 digits)
(4711 digits)
107045
> 99%
2x6
(4934 digits)
(9867 digits)
16394
> 99%
2x6
(4932 digits)
(9864 digits)
49142
> 99%
HERE BE DRAGONS!
2x6
(822 digits)
(1644 digits)
Beaver, Platypus, Unicorn …
2733
Monash
> 99%
June 2nd, 2014
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
11R
a
01R
b
01R c
00L, 10R
Beaver, Platypus, Unicorn …
Monash
10R
h
June 2nd, 2014
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
11L
a
01R
00L
b
10R
c
h
{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
Searching through Machines
n
m
tnf
free
2
2
36
64
3
2
3,522
55,296
2
3
2,765
41,472
4
2
507,107
??
2
4
501,232
??
3
3
25,955,301
??
5
2
99,174,375
??
2
5
≥ 115,000,000 ??
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
Blue Bilby
n-state m-symbol machines, n x m ≤ 6
n
m
Total
Terminated
Going
Unclassified
2 2
36
14
22
0
3 2
3,522
1,188
2,334
0
2 3
2,765
839
1,926
0
The free and arbitrary classes have also
been classified for these cases
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
0
2
4
501,232
168,674
331,960
598
1's
1
2
3
#
3836
23161
37995 31023
4
5
6
7
8
15131
5263
1487
357 74 11
Beaver, Platypus, Unicorn, Zoo …
9
Monash 29th
June,2011
10
11 12
13
5
2
3
White Whale
n-state m-symbol machines, n x m = 9 or 10
n
m Total
3
3
25,955,301
5
2
99,174,375
2
5
≥ 115,000,000
Need to improve the technology
before taking this on ...
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
Demon Duck of Doom
n-state m-symbol machines, n x m = 12 (!??!!)
n m Total
6 2 ??
4 3 ??
3 4 ??
2 6 ??
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
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
2-D machines
0
2
1
0
1
Beaver, Platypus, Unicorn …
Monash
June 2nd, 2014
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
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
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
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
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
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
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